ABSTRACT
Title of dissertation: APPLICATION OF DISPERSIVE PDE
TECHNIQUES TO THE STUDIES OF
THE TIME-DEPENDENT HARTREE-FOCK
-BOGOLIUBOV SYSTEM FOR BOSONS
Jacky Jia Wei Chong
Doctor of Philosophy, 2019
Dissertation directed by: Professor Manoussos Grillakis
Professor Matei Machedon
Department of Mathematics
The thesis provides rigorous quantitative analyses for studying quantum fluc-
tuations of a non-relativistic Bose gas about a Bose-Einstein condensate. In recent
years, the dynamics of the condensate and the excitations of a Bose gas was shown to
be well approximated by the quasifree dynamics with governing equations given by
a system of coupled nonlinear dispersive PDEs called the time-dependent Hartree-
Fock-Bogoliubov (HFB) system (c.f. [GM13a, GM17, BBC+18, BSS18]). However,
both the quantitative and qualitative analysis of the time-dependent HFB system
are still in their early stages of development. Thus, the primary purposes of the
thesis are to further the development of some analytic tools necessary for studying
the time-dependent HFB system and use these effective equations to provide quanti-
tative estimates for the true dynamics of the Bose gas at absolute zero temperature
in Fock space norm.
The thesis comprises the entirety of the author?s current and past projects
on the time-dependent HFB system. Each project falls into one or more of two
categories: studying the local or global well-posedness of the time-dependent HFB
system, or obtaining global-in-time Fock space estimates for the error terms of the
quasifree approximation to the dynamics of a system of interacting bosons. In the
former category, the author employs techniques of dispersive PDE theory developed
in the past three decades along with classical methods of harmonic analysis to study
lower regularity solutions to the time-dependent HFB system. The lower regular-
ity well-posedness of solutions for the time-dependent HFB system is necessary for
studying norm approximation of the dynamics of a dilute Bose gas with strong in-
teractions. In the latter category, we prove global a-priori estimates for the solutions
to the HFB system and use them to obtain estimates for the error terms of the Fock
space approximation.
APPLICATION OF DISPERSIVE PDE TECHNIQUES
TO THE STUDIES OF THE TIME-DEPENDENT
HARTREE-FOCK-BOGOLIUBOV SYSTEM FOR BOSONS
by
Jacky Jia Wei Chong
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2019
Advisory Committee:
Professor Manoussos Grillakis, Co-Chair/Co-Advisor
Professor Matei Machedon, Co-Chair/Co-Advisor
Professor Charles David Levermore
Professor Dionisios Margetis
Professor John D. Weeks, Dean?s Representative
?c Copyright by
Jacky Jia Wei Chong
2019

Acknowledgments
Let me take this opportunity to express my deepest gratitude towards my two
advisors, Manoussos Grillakis and Matei Machedon, for their guidance and support.
I believe one could not wish for better advisors than these two fine gentlemen. They
have taught me a great deal of mathematics. It was also a great learning experience
for me to see how these two master craftsmen of mathematics work relentlessly
together to chip away on a big problem one small piece at a time. In many ways,
they have shaped my taste for mathematics and help me understand how to ask
meaningful questions. I attribute much of my success in graduate school to their
guidance and support if not all.
Several other professors have also played significant roles during my graduate
school career. First, I would like to thank Professor David C. Levermore. In many
ways, Professor Levermore has helped me improve both as a researcher and as an
educator. In terms of research, Professor Levermore plays a big role in helping me
hone my presentation skills. In the many talks that I have given at his Applied PDE
RIT (Research Interaction Team) seminar, Professor Levermore was always very
enthusiastic and encouraging, which gave me confidence when giving talks outside
of UMD. As an educator, I have benefited a great deal from the many discussions I
had with him about teaching Math 246 (elementary differential equations).
Next, I would like to thank Professor Dionisios Margetis and Professor Kon-
stantina Trivisa. I thank Professor Margetis for the many insightful discussions we
have had. I would also like to thank him for making time to serve as a committee
ii
member on both my preliminary oral exam and final oral defense. Moreover, I am
truly grateful for his careful review of my thesis manuscript. I thank Professor Triv-
isa for her constant encouragement and warm welcoming personality which always
brightens my day. Also, I am indebted to her for writing my letter of recommenda-
tion under short notice. She definitely helped reduce a lot of my stress during my
postdoc application process.
I want to thank the Dean?s representative Professor John Weeks for agree-
ing to serve on my final defense committee. I had the pleasure to meet Professor
Weeks through auditing his graduate thermodynamics course. He is a fantastic ed-
ucator with very deep insight on the subject. I attended all his lectures with great
anticipation and he never fails to deliver.
Many thanks to my friends and colleagues who were with me since the very
beginning of my graduate school career: Patrick Daniels, Danul Gunatilleka, Siming
He, Ian Johnson, Minsung Kim, Weilin Li, Mark Magsino, Xuesen Na, Yousheng Shi,
Weikun Wang, Zhenfu Wang, Dong Xin, Tao Zhang, Jing Zhou and many others
which I have not named. I also had the good fortune to meet many wonderful
junior graduate students along the way. I wish them the best of luck as they head
toward the finish line. I would also like to thank Elif Kuz and Zhenfu Wang for
their hospitalities during my visits to the University of Iowa and the University of
Pennsylvania.
Finally, I am grateful to the Graduate School for awarding me the 2018-2019
Anne G. Wylie Fellowship to fund for my living expenses during the preparation of
the dissertation.
iii
Table of Contents
Dedication ii
Acknowledgements ii
Table of Contents iv
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Mean-Field Model . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Short-Range Scaling . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Initial Condition: the Ground State . . . . . . . . . . . . . . . 8
1.1.4 Recent Advancements . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Mathematical Framework . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 Fock Space Formalism . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 Uncoupled Time-Dependent HFB System . . . . . . . . . . . . 20
1.2.3 (Coupled) Time-Dependent HFB System . . . . . . . . . . . . 23
1.3 Outline and Main Results of the Thesis . . . . . . . . . . . . . . . . . 28
2 Uncoupled Time-Dependent HFB systems 30
2.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 Estimates for the Solution to the Hartree-Type Equations . . . . . . 33
2.2.1 Uniform in N Global Well-posedness of the Hartree Equations 33
2.2.2 Decay Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Estimates for the Pair Excitations . . . . . . . . . . . . . . . . . . . . 43
2.4 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4.1 List of Error Terms . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4.2 Estimates for the Error Terms . . . . . . . . . . . . . . . . . . 52
2.5 Application: Derivation of The Focusing NLS in R3 . . . . . . . . . . 54
2.5.1 Pair Excitation Method . . . . . . . . . . . . . . . . . . . . . 54
2.5.2 Pickl?s Method . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Uniform in N Global Well-posed of the Time-Dependent HFB system in R1+1 62
3.1 Notations and Main Statements . . . . . . . . . . . . . . . . . . . . . 64
3.2 Estimates for the Homogeneous ? Equation . . . . . . . . . . . . . . 69
3.3 Estimates for the Inhomogeneous ? Equation . . . . . . . . . . . . . 75
3.4 Application of the Inhomogeneous ? Estimates . . . . . . . . . . . . 78
iv
3.5 Homogeneous ? Equation . . . . . . . . . . . . . . . . . . . . . . . . 82
3.6 Inhomogeneous ? Equation . . . . . . . . . . . . . . . . . . . . . . . 89
3.7 The time-dependent HFB System in 1D . . . . . . . . . . . . . . . . 93
3.7.1 Proofs of Estimates (3.46b) and (3.46c) . . . . . . . . . . . . . 96
3.7.2 Proof of Estimate (3.46a) . . . . . . . . . . . . . . . . . . . . 99
3.7.3 Global Well-Posedness of the Time-Dependent HFB Equations 100
4 Global Well-posed of the Time-Dependent HFB system in R1+3 and Fock
Space Estimate 103
4.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 Global Estimates for the Time-Dependent HFB Equations . . . . . . 105
4.3 Global Well-posedness of the Time-Dependent HFB System . . . . . 115
4.4 Estimates for sh(2k) . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5 Morawetz Estimates of the time-dependent HFB System 136
5.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.2 Proof of Theorem 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.3 Morawetz Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.4 Proof of the Interaction Morawetz Estimate for ? . . . . . . . . . . . 156
6 Collapsing Estimates on Closed Manifolds 167
6.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.2 Collapsing Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.2.1 Estimates for (6.1) . . . . . . . . . . . . . . . . . . . . . . . . 170
6.2.2 Estimates for the ? Equation . . . . . . . . . . . . . . . . . . 183
6.2.3 Bourgain Refinement of the Collapsing Estimates . . . . . . . 187
6.2.4 Proof of Theorem 6.1 for (6.1) . . . . . . . . . . . . . . . . . . 193
6.2.5 Proof of Theorem 6.1 for (6.2) . . . . . . . . . . . . . . . . . . 198
7 Conclusion and Discussion 201
Bibliography 205
v
Chapter 1: Introduction
1.1 Background
The studies of dynamical behaviors of systems with many interacting bod-
ies from first principles of quantum mechanics are of paramount interest in many
branches of physics and chemistry. A prominent example is the studies of systems of
interacting bosons. More precisely, the studies of Bose-Einstein Condensate (BEC),
a state of matter of a dilute gas of bosons when cooled to near absolute zero temper-
ature, has gained eminence in the world of experimental physics after its initial real-
ization in atomic gases a little over two decades ago [AEM+95,BSTH95,DMA+95].
Notably, for the groundbreaking achievement of exhibiting condensation limits in
dilute gases of alkali atoms, Eric Cornell and Carl Wieman of JILA/NIST and
Wolfgang Ketterle of MIT were awarded the 2001 Nobel Prize in Physics1. The
experimental success has and continues to garner substantial attention of scientists
and mathematicians.
Due to the subsequent voluminous influx of research activities in the field of
many-body boson systems, the demand for a firm mathematical foundation also
grew. Moreover, a rigorous understanding of the dynamics of such systems is one
1https://www.nobelprize.org/prizes/physics/2001/press-release
1
of the main challenges of modern mathematical physics and provides fundamental
insights into quantum mechanical systems, as well as offering potential applications
to the sciences. However, the immediate pressing difficulty one encounters when
studying large particle systems is often the size of the system. In many applications
of chemistry or physics, the size of the system of interest typically ranges between
thousands to an Avogadro?s number (? 1023) of particles. Hence, even if one man-
ages to exhibit a many-body wave function which solves the many-body Schr?dinger
equation analytically, the sheer number of particles of the system will render the us-
age of the wave function to analyze the dynamical behaviors of the system obsolete,
or at the very least, not very effective with the current available tools.
It is no news that the size of the system presents a formidable obstacle for
studying dynamical properties of the system. Indeed, it is rudimentary knowledge
among researchers that the wave function contains more information than one could
process since it encapsulates all the microscopic details of the many-body system.
Even in practice, experiments are conducted and measured at a macroscopic scale
where a lot of the quantum information are overlooked. Thus, to compare the
experimental data against the full-theoretical description of the system from the
wave function is impractical. As a matter of fact, based on heuristics and scaling
arguments, many areas of chemistry and physics employ macroscopic equations to
approximate the behaviors of the system. Therefore, it seems fair to study effec-
tive descriptions of the many-body system, which allows one to approximate the
macroscopic dynamics of the large particle system but with much lesser variables,
rather than the full quantum mechanical description. However, to understand the
2
validity and qualities of any of these approximations, a rigorous derivation of effec-
tive macroscopic equations from quantum mechanical laws is essential, but this, in
general, is a challenging task.
In this thesis, we study a coupled nonlinear system of effective macroscopic
equations, parametrized by the particle-numberN , called the time-dependent Hartree-
Fock-Bogoliubov (HFB) system (equations), for describing the quantum fluctuations
about a BEC and use them to obtain global quantitative estimates for the true
dynamics of the many-body system in Fock space. The main contribution of our
work is the nonlinear analysis of the coupled system through the lens of dispersive
PDE theory. We show that by employing dispersive PDE techniques to our analysis
of the coupled system we could improve upon results which only uses ?standard?
mathematical physics techniques. More specifically, by applying dispersive PDE
techniques, we were able to obtain nonlinear approximations to the dynamics of
a N -body quantum system which are valid for a longer period of time and more
singular interaction potentials without imposing further assumptions or restrictions
on the approximation. In fact, we show the reader glimpses of the harmonious
marriage between dispersive PDE theory and the studies of many-body interacting
boson systems.
3
1.1.1 Mean-Field Model
We begin by considering a system of N interacting non-relativistic spinless
bosons2 in three dimensional space whose evolution is governed by the N -body
linear Schr?dinger equation
( )
N
1 ? ?? 1 ??x + vN(xi ? xj) ?N(t, x1, . . . , xN) = 0 (1.1)
i ?t j N
j=1 i>j
where x ? R3i for 1 ? i ? N and vN(x) := N3?v(N?x) for 0 ? ? ? 1 where
v ? C?0 (R). The reader should note that as N ? ? we have that vN(x) ? c?(x),
in the sense of distribution, for some constant c. However, it should also be noted
that in our work N is typically large but fixed. In the literature, it is common to
refer
?N 1 ?
HN,mf := ? ?x + v (x ? x ) (1.2)j N i jN
j=1 i>j
as the mean-field Hamiltonian and (1.1) the mean-field model.
There are many interpretations for for the coupling constant N?1 in front of
the interaction potential. The simplest argument for having N?1 is based on the
heuristic that the coupling constant allows for the balance between the kinetic energy
2In relativistic quantum mechanics, bosons are classified, by the Spin-Statistic theorem, to be
particles with integer intrinsic spin. However, in this thesis, we work in the realm of non-relativistic
quantum physics where the bosonic property of a system of particles is captured by the symmetric
structure of the wave function.
4
and the interaction potential energy. More precisely, since a general Hamiltonian
?N ?
HN := ?x ? ? v(xi i ? xj) (1.3)
i=1 1?i<j?N
scales like O(N) + ?O(N2), then the energy of each particle is O(1) provided the
coupling constant ? is O(N?1), which we called the mean-field scaling of (1.3).
With this scaling, we define the mean-field limit to be the singular limit of (1.3) as
the particle-number N tends to infinity. The coupling constant N?1 could also be
understood with the law of large numbers. If the particles are independent and
identically distributed according to an underlying density distribution ? then we see
that
?N ?1
vN(xi ? xj) ' vN(xi ? y)?(y) dy = (vN ? ?)(xi) ' c?(xi),
N
j=1 R3
j=6 i
that is, the interaction potential has a non-zero limit with respect to the particle
limit.
To physically motivate the mean-field model, let us consider N particles inside
a fixed box3 with volume V = `d subjected to either Robin or Neumann boundary
conditions. Furthermore, assume the particles interact through a two-body repulsive
potential v (with coupling constant ? set to 1). Then the particles will uniformly
3The box model is used to simplify the exposition. Alternatively, we could have considered N
particles in Rd subjected to some harmonic trapping potential, i.e.
?N ?
HN = {?x ? Vext(xi)} ? v(xi ? x )i j
i=1 1?i<j?N
where Vext is small inside the box [?L,L] and large otherwise.
5
spread themselves inside the box with an average interparticle separation distance of
N?1/d` since the average volume occupied by a particle is N?1`d. In particular, we
are interested in the dilute gas model, that is the case when N?1/d` a, where a is
the scattering length. Following a scaling argument, one can show that the dynamics
generated by the Hamiltonian (1.3) is equivalent to the dynamics generated by the
rescaled Hamiltonian4
1 ?N 1 ?
? ?y ? vN(yi ? yj) (1.4)N3 d i N
i=1 1?i<j?N
provided we set the length scale of y 5i to order 1 . In the case d = 3, we see that
(1.4) gives us a mean-field model for the particles in a unit box with interactions
vN . Finally, if we take the dilute limit, N?1/d` ? ?, in the box `3, we essentially
recover the mean-field limit of N weakly interacting particles in the unit box. In
particular, the 3D mean-field model in the unit box is equivalent to the strongly
interacting dilute gas model in a box. The studies of weakly interacting bose gases
can be trace back to the pioneering work of Bogoliubov in [Bog47], and later by
Lee, Wang, and Yang in [LHY57] and as well as Dyson in [Dys57]. We refer the
interested reader to [Lew15,LSSY05,Gol16] for more in-depth discussions.
1.1.2 Short-Range Scaling
The reader should take note of the two scaling processes that are involved
in the interactions of this mean-field model. Aside from the obvious mean-field
4To preserve the dynamics, we will need to rescale the time by a factor of N?2.
5Here we are assuming xi is on the length scale ` ? N .
6
scaling, we also have the short-range scaling of the interaction v given by vN with
the tuning parameter ? > 0. Let us consider the dynamics generated by the mean-
field Hamiltonian and let ?N be the solution to
1 ?
?N = HN,mf?N (1.5)
i ?t
then by rescaling the solution, i.e. defining ?(?, y) = ?N(N?2??,N??y), we see the
dynamics of the rescaled system is governed by the equation
N
1 ? ? ?
? = ? ??N (d?2)??1y v(yi ? yj)? (1.6)
i ?? i
i=1 1?i<j?N
In the instance of d = 3, we see, at least heuristically, the appearance of a crit-
ical scaling when ? = 1, which we called the Gross-Pitaveskii scaling. With this
consideration, we restrict ourselves to the case 0 ? ? ? 1, at least for the initial
investigation. In some sense, the parameter ? is a mathematical apparatus which
was introduced to aid with the study of (1.1); in fact, the analysis of the dynamics of
the system becomes more difficult as ? approaches 1. Nevertheless, the are physical
interpretations for ?. Physically, for 0 < ? < 1 , we are in the regime of weakly
3
interacting dense gas; since the scale size is N? which means the size of the volume
is effectively V ' N3?, then we see that the interparticle separation distance is given
by `/N?1/3 = N?/N?1/3, which tends to 0 as N ?? provided 0 < ? < 1 . But once
3
1 ? ? ? 1, we enter the self-interacting regime or sometimes called the strongly
3
interacting diluted gas regime, depending on the modeling situation. In this regime,
7
each particle is said to only feel the potential generated by itself; in some sense, this
is a reminiscence of the Gross-Pitaevskii (GP) theory which proposed to model the
many-body effects by a nonlinear strong on-site self interaction of a complex order
parameter (the ?condensation wave function?) [Gro61, Gro63, Pit61]. But, strictly
speaking, only ? = 1 captures the 2 scale structure postulated by the GP theory.
See Chapter 6 of [LSSY05] for a survey of the GP theory for trapped bosons.
1.1.3 Initial Condition: the Ground State
The main interests of the thesis are to study the effective dynamics describing
the evolution of the above many-body system, parametrized by N and ?, and pro-
vide a quantitative method for tracking the evolution of the many-body quantum
system in state space. Unfortunately, the problem of tracking the exact dynamics
of bosonic systems in state space with arbitrary initial condition, at least to the
author?s knowledge, is still not tractable with the current available tools. Neverthe-
less, if we restrict ourselves to a special class of initial datum, then we are able to
obtain some positive results in the direction of understanding the exact evolution in
state space via studying some effective dynamics of the system.
In this thesis, we consider the evolution problem (1.1) with initial condition
given by the ground state of the mean-field hamiltonian, i.e., the lowest energy state
of (1.2). This choice of initial condition is natural for modeling Bose gases at lower
temperature. In fact, by cooling a system of bosons to the absolute-zero tempera-
ture, we are essentially forcing the system to its ground state since the temperature
8
of the system is directly proportional to its kinetic energy. However, solving the
static (eigenvalue) problem for a many-body hamiltonian to determine the ground
state remains a highly non-trivial open problem if not completely intractable. Nev-
ertheless, based on the experimental observation of formation of BEC inside a trap
potential at lower temperature, that is, the individual particles of the Bose gas co-
alesce into a single quantum entity, one could speculate that the ground state of a
boson system has the form
?N, ground(x1, . . . , x ) = ?
?N
N 0 + important corrections (1.7)
?
where ??N := N0 j=1 ?0(xj) for some ?0 which we called the condensate wave function.
Furthermore, it is also speculated that in the particle limit the correction terms will
tend to zero, i.e., ? ?NN, ground ' ?0 .
In fact, shortly after the discovery of atomic BEC in laboratory, Lieb, Seiringer,
and Yngvason were able to rigorously verified the GP theory for the ground state of
a dilute trapped Bose gas in [LSY00], i.e., they were able to show that the energy
per particle of the ground state in the particle limit satisfies a variational principle,
min{EGP(?) | ?? ? = 1}, where
? { }
EGP(?) = dx |??(x)|2 + Vext(x)|?(x)|2 + 4?a|?(x)|4 (1.8)
R3
is the Gross-Pitaevskii (GP) energy functional. Here, a is the scattering length
associated to the potential v. Subsequently, Lieb and Seiringer prove the existence
9
of BEC in a dilute trapped gas at absolute zero temperature [LS02] by demonstrating
that the 1-particle marginal density (operator), denoted by (1)?N , with kernel
?
(1)
? ?N (x, x ) := dxN?
? ?
1 ?N(x,xN?1)?N(x ,xN?1)? ?GP(x)?GP(x )
R3(N?1)
in trace norm where ?GP is the minimizer of (1.8). This formulation of the definition
of BEC in terms of the marginal density was first stated by Penrose and Onsager
in [PO56].
Hence it strongly suggest that the ground state is well-approximated by tensor
products of the form ??N0 where ?0 satisfies a variational principle.
1.1.4 Recent Advancements
In recent years, many have contributed to the studies of effective dynamics for
many particle systems. In the case of ? = 0 with repulsive Coulomb interactions,
Erd?s and Yau in [EY01] prove the qualitative result, via the method of BBGKY
hierarchy, that the one-particle marginal density (1)?N,t associated to the wave function
?N,t with asymptotically factorized initial state, i.e. ?N,0 ? ??N as N ? ?,
converges to |?t???t| in trace norm in the mean-field limit of N ? ? where ?t
satisfies the Hartree equation
( )
1 ? 1
?(t, x)??x?(t, x) + ? |? 2t| ?(t, x) = 0. (1.9)
i ?t | ? |
10
In fact, the approach mentioned above is based on the earlier work of Spohn [Spo80]
proving the statement for bounded potentials in the case ? = 0. Using the Fock
space method introduced by Hepp in [Hep74] and subsequently extended by Ginibre
and Velo in [GV79a, GV79b], Rodnianski and Schlein in [RS09] provide a rate of
convergence of the one-particle marginal associated to the many-body quantum
system towards the Hartree dynamics in trace norm, that is6
??? ? Kt(1) ? eTr ?N,t ? |?t???t|? . ?
N
for some constant K > 0 independent of N . The estimate was later improved
to eKtN?1 in [ES09, CLS11]. Using a second-order correction Fock space method
introduced by Grillakis, Machedon, and Margetis in [GMM10, GMM11], Kuz in
[Kuz15b] provides a rate of convergence of the many-body quantum system to the
Hartree dynamics in the sense of Fock space marginal density7. Consequently, Kuz
shows that
??? ? ?(1) ? 1 + tTr ?N,t ? |?t???t|? . 1
N 4
which in turn establishes the validity of the approximation for time t of the order
?
N . Similar results are derived in [FKS09,KP10] but the approaches are completely
6We adopt the standard notation A . B to mean there exists a constant, depending on some
parameters, such that A ? CB.
7One should note the main result in Rodnianski and Schlein?s paper is their result on the rate
of convergence of the one-particle Fock marginal towards the Hartree dynamics. Whereas, the
significance of Kuz?s paper is that she was able to show that the mean-field estimate is actually
valid for a much longer period of time then most proceeding results had indicated.
11
different from the above methods.
For the case 0 < ? ? 1, Erd?s, Schlein, and Yau in a series of papers
[ESY06, ESY07, ESY10, ESY09] show qualitatively that the many-body dynam-
ics with asymptotically factorized initial data converges to the cubic nonlinear
Schr?dinger dynamics when 0 < ? < 1 or the Gross-Pitaevskii dynamics when
? = 1. More precisely, they prove that (1)?N,t ? |?t???t| in trace norm where ?t
satisfies
????? (? )1 ? ? v |? 2t| ?t if 0 < ? < 1
?(t, x)??x?(t, x) =
i ?t ?????8?a|? |2t ?t if ? = 1
where a is the scattering length corresponding to the potential v. Results on the
rate of convergence of Fock space marginals can be can be found in [KP10,BdOS15,
Kuz15b].
Despite the success founded in the rigorous study of the mean-field behaviors
of BEC, recent experiments suggest that mean-field dynamics may not account for
the depletion of the condensate, the phenomenon where particles in the condensate
escape to higher energy states [XLM+06,LEN+17]. Hence, this warrants the rigorous
studies of quantum fluctuations about the mean-field dynamic of BEC. A natural
setting to account for the fluctuation is in the bosonic (Symmetric) Fock space
? ( )
Fs(h) = C? Sym h?
n?1
12
where h := L2(R3). Introducing Fs allows us to deal with states with varying number
of particles. Recent works on evolution of coherent states in Fock space with quan-
tum fluctuations can be found in [RS09,GMM10,GMM11,Che12,GM13a,GM13b,
Kuz15b, Kuz15a, BCS17, NN17, Cho16]. Hence, by accounting for some quantum
fluctuation, one is able to estimate the evolution of the coherent state in Fock space
norm, which in effect allows one to obtain L2-norm approximation of the evolution
of many-body quantum system with factorized initial data. This is the main setting
of the thesis which we will elaborate more on in the next section.
We refer the reader to [LSSY05,Gol16,GMM17] for a complete survey of the
subject.
1.2 Mathematical Framework
1.2.1 Fock Space Formalism
In this section, we provide the reader with a brief account of the main mathe-
matical framework for the thesis. For a more comprehensive treatment of the second
quantization formalism, we refer the reader to [Ber66].
Let us introduce the mathematical setting for our work. The one-particle base
space, denoted by h = L2(R3, dx), is a complex separable Hilbert space endowed
with the inner product ??, ??h which is linear in the second variable and conjugate
linear (or anti-linear) in the first variable 8.
8This is the physicists? inner product.
13
We define the bosonic Fock space over h to be the closure of
??
Fs(h) = Fs := C? Sym(h?n)
n=1
with respect to the norm induced by the Fock inner product
??
??, ??F = ??0?0 + ??n, ?n?h?n
n=1
where ? = (?0, ?1, . . .), ? = (?0, ?1, . . .) ? Fs(h). For convenience, we shall refer Fs
simply as the Fock space henceforth. The vacuum, denoted by ?, is define to be the
Fock vector (1, 0, 0, . . .) ? Fs.
For every field ? ? h we can define the associated creation and annihilation
operators on Fs, denoted respectively by a?(?) and a(??), as follow
n
1 ?
(a?(?)?)n(x1, . . . , xn) :=? ?(xj)?n?1(x1, . . . , x?j, . . . , xn) (1.10a)
n
j=1
? ?
(a(??)?)n(x1, . . . , xn) := n+ 1 dx ??(x)?n+1(x, x1, . . . , xn). (1.10b)
with the property that a(?)? = 0. We can also define the corresponding creation
and annihilation distribution-valued operators associated to (1.10a) and (1.10b),
denoted by a?x and ax, as follow
?n1
(a?x?)n := ? ?(x? xj)?n?1(x1, . . . , x?j, . . . , xn) (1.11a)n
j=1
?
(ax?)n := n+ 1?n+1(x, x1, . . . , xn). (1.11b)
14
In short, we have the relations
? ?
a?(?) = dx {?(x)a?x} and a(??) = dx {??(x)ax}.
Let us note that the creation and annihilation operators a(??) and a?(?) associated
to the field ? are unbounded, densely defined, closed operators. Moreover, one can
easily verify, formally, (a?x, ax) satisfy the canonical commutation relation (CCR):
[a ?x, ay] = ?(x? y), [ax, ay] = [a?x, a? 9y] = 0 , and the number operator defined by
?
N := dx a?xax (1.12)
is a diagonal operator on F that counts the number of particles in each sector.
As mentioned in the introduction we are interested in studying the time evo-
lution of the coherent state in Fock space. Before doing so, let us define the initial
datum, the coherent state and the Fock Hamiltonian. For each ? ? h, we associate
the corresponding unique closure of the operator
A(?) = a(??)? a?(?) (1.13)
9The reader should note for any f, g ? h the CCR for a?(f) and a(g) are not well defined since
there are domain issues that need to be resolved for the given unbounded operators. For an exotic
example of an ill-defined commutator of unbounded operators, we refer the reader to Chapter
VIII.5 of [RS80].
15
then the Weyl operator 10 is defined to be
?
e? NA(?). (1.14)
Let us note the operator A(?) is a skew-Hermitian unbounded operator which means
the corresponding Weyl operator is unitary. The coherent state associated to ? is
given by
?
?(?) := e? NA(?)?. (1.15)
Using the Baker-Campbell Hausdorff formula, one can show
?
? A ( ) ( )e N (?)? = . . . , c ??n 2n , . . . where cn = e?N?? ?hNn/n! .
For a fixed N ? N, we defined the Fock Hamiltonian associated to N , denoted by
HN , to be the diagonal operator on the Fock space given by
(? ? )n n
(H 1N?)n = ?x ? vN(xi ? xj) ?n = Hj N,n?nN
j=1 i<j
where v (x) = N3?N v(N?x). Rewrite HN using creation and annihilation operators
10To avoid the unfavorable technicality associated with the unbounded natural of our creation
and annihilation operators one often choose to work with the corresponding Weyl algebra, the
C?-algebra generated by the exponential of A(?) where ? ? h (cf. chapter 9 of [DG13] and chapter
5.2 of [BR03]).
16
we get
H 1N := H? 1 ? V (1.16a)N
H1 := ?dxdy {?x?(x? y)a
?
xay} and (1.16b)
V 1:= dxdy {vN(x? y)a? ?xayaxay}. (1.16c)2
In light of (1.16a), we are interested in the solution to the following Cauchy problem
in Fock space
1 ? ?
? = H ? with initial datum ? = e? NA(?0)N 0 ? (1.17)
i ?t
which we write
?
? = eitHN e? NA(?0)exact ?. (1.18)
An important fact to note about the Fock Hamiltonian is its action on the
Nth sector of the Fock space. There, the Fock Hamiltonian acts as a mean-field
Hamiltonian for the N -particle system, that is
(?N ? )N
(H 1N?)N = ?x ? vN(xi ? xj) ?N = HN,mf?N . (1.19)j N
j=1 i<j
1
Since theNth coefficient cN could be approximated, using Stirling?s formula, byN? 4
and the coherent state is a simple N -tensor of ? in the N sector, then heuristically
we see how by understanding the evolution of the coherent state we would also
17
understand the mean-field evolution of the N -particle factorized state.
Based on the earlier works of Hepp and Ginibre & Velo in [Hep74, GV79a,
GV79b], Rodnianski and Schlein in [RS09] studies the one-particle Fock marginal,
which is defined as follows: for every ? ? Fs the one-particle Fock marginal of ?,
denote by (1)?? , is a positive trace class integral operator on h with kernel given by
(1) ??, a?xay??F 1?? (x, y) = = ??, a
?a ?? . (1.20)
??,N?? x y FF N
They were about to show that the one-particle Fock marginal with an initial co-
herent state converges to the Hartree dynamics in trace norm for the case ? = 0.
Furthermore, they were also able to obtain a rate of convergence
??? ? Kt(1)Tr ?N,t ? |?t???t|?? e. (1.21a)N
where (1)?N,t denotes the one-particle Fock marginal for ?exact and ?t satisfies the
Hartree equation. Later, Kuz in [Kuz15b] improved the estimate substantially in
time and obtain the estimate
??? ?(1) ? tTr ?N,t ? |?t???t|? . . (1.21b)N
Unlike the approach of Rodnianski and Schlein which uses the mean-field approxi-
18
mation of the form
? ?
? = e? NA(?t)mf ? = e
? NA(t)?, (1.22)
Kuz uses the method of second-order correction introduced in the works of Grillakis,
Machedon, and Margetis in [GMM10,GMM11,GM13a] to establish (1.21b), which
relies on tracking the exact dynamics of the evolution of the coherent state in Fock
space.
To track the exact dynamics in Fock space, we need to introduce the pair
excitation function, k(x, y) = k(y, x), and its corresponding quadratic operator B(k)
with kernel
?
B(kt)(x, y) = B(t)(x, y) = dxdy {k?(t, x, y)a ? ?xay ? k(t, x, y)axay}. (1.23)
From the pair excitation, we concoct a new approximation scheme, which is a second
order correction to the mean field (1.22), given by
?
? iN?(t) ? NA(t) ?B(t)approx = e e e ? (1.24)
where ?(t) is some phase factor to be determined. Approximation (1.24) is inspired
by the earlier work of Wu in [Wu61] on weakly interacting bose gas in non-periodic
settings. In the literature, the unitary operator eB(kt) is called the Segal-Shale-Weil
metaplectic representation or Bogoliubov transformation by physicists (c.f. [Sha62],
chapter 4 of [Fol89], and chapter 11 of [DG13]). With some appropriate choice of
19
evolution equations for ? and k we will later see that (1.24) will indeed allow use to
track the exact dynamics of the evolution of coherent state (or quasifree state) in
Fock space.
1.2.2 Uncoupled Time-Dependent HFB System
Incidentally, one could show via a Lie algebra isomorphism argument estab-
lished in [GM13a,GM13b,GM17] that the evolution of k could be described by some
nonlinear evolution equations of
1 1
sh(k) := k + k ? k? ? k + k ? k? ? k ? k? ? k + . . . (1.25a)
3! 5!
1 1
ch(k) := ? + k? ? k + k? ? k ? k? ? k + . . . (1.25b)
2! 4!
where ? denotes the composition of operators. Moreover, in [GM13b], Grillakis and
Machedon show, by using a specific coordinate, that the nonlinear equation of the
pair excitation could be express as a system of coupled linear equations in sh(2k)
and ch(2k).
Let us introduce some notation to help us compactly write out the evolution
20
equations for ? and k
gN(t, x, y) :=??x?(x? y) + (vN ? |?|2)(t, x)?(x? y)
+ vN(x? y)??(t, x)?(t, y)
mN(t, x, y) :=? vN(x? y)?(t, x)?(t, y)
1
S Told(s) := ?ts+ gN ? s+ s ? gN (Schr?dinger-type operator)i
1
Wold(p) := ?tp+ [g
T
N , p] (Wigner-type operator)i
then the desired evolution equations of ? and k are given by
1
?t???x?+ (vN ? |?|2)? = 0 (Hartree-type equation) (1.26a)
i
Sold(sh(2k)) = mN ? ch(2k) + ch(2k) ?mN (1.26b)
Wold(ch(2k)) = mN ? sh(2k)? sh(2k) ?mN . (1.26c)
The system of equations (1.26) is referred to as the uncoupled time-dependent Hartree-
Fock-Bogoliubov system in contrast to the coupled system introduced in [GM13a,
GM17] where the equation for ? and the pair excitation equations are coupled.
Now, let us summarize the results in [GM13b,Kuz15b], which built on earlier
works by Grillakis, Machedon, and Margetis in [GMM10,GMM11].
Theorem 1.1 (Grillakis & Machedon ?13, Kuz ?16). Let v ? C1c (R3) and v ?
0. Assume ? and k satisfy (1.26) with initial conditions ?(0, ?) = ? ? L2(R20 ) ?
Wm,1(R3) for some sufficiently large m and k(0, ?) = 0. If ?exact and ?approx are
21
defined by (1.18) and (1.24) respectively, then we have the following estimate
? (1 + t) log
4(1 + t)
?exact(t) ? ?approx(t) ?F .
N (1?
(1.27)
3?)/2
provided 0 < ? < 1 . Moreover, if (?t sh(2k))(0, ?) is sufficiently regular, then for3
any  > 0 and j a positive integer, we have
??exact(t)? ?approx(?t) ?F????N?1/2+?(1+) 1? ? ? <
2j ,
j+3 ? 3 (1?2+4j). t 62 log (1 + t) ??? (1.28)?3+7?N +(j?1)(?1+2?) 2j2 ? ? ? < 1+2j .(1 2+4j) 3+4j
Remark 1.2. It should be noted that the assumption (?t sh(2k))(0, ?) must be suffi-
ciently regular imposes a restriction on the form of the initial condition; in particu-
lar, k(0, ?) cannot be zero. Due to the restriction, we could not choose the coherent
?
state as our initial condition since e? NA0e?B0? is a coherent state if and only if
k(0, ?) = 0.
Remark 1.3. In ?2 of [Kuz15a], Kuz provides a heuristic argument showing that the
system (1.26) has limitations. In fact, Kuz argued that (1.26) will not be able to
provide any Fock space estimate for ? ? 1 which indicates a revision to (1.26) is
2
necessary in order to study the case of large ?.
22
1.2.3 (Coupled) Time-Dependent HFB System
?
Let M = e? NAe?B. Following [GM13a,GM17], we work with the reduced
dynamic. More specifically, sinceM is unitary then it follows
??? ? ?? ? ? ?? iN t ?0(s) ds ? ?iN t ?0(s) ds ?exact(t)? e 0 ?approx(t)? = ? e 0 ?red(t)? ??
F F
where
? ?
? (t) = eB(t)e NA(t)eitHe? NA0e?B0red ?. (1.29)
Then by considering the evolution equation of ?red given by
1 ? 1
?red = Hred?red where Hred = (?M?t )M+M?HM (1.30)
i ?t i
we see that
( )
1 ? ( ? )t?H +X e?iN 0 ?0(s) dsred 0 ?red ? ? = Hred??X0?
i ?t
with
Hred? = (X0, X1, X2, X3, X4, 0, 0, . . .). (1.31)
Thus, to estimate the Fock space error, we need to be able to controlHred?. A direct
calculation reveals that X3 and X4 are heuristically small since they are proportional
23
to N?1/2 and N?1, respectively. On the other hand, X1 and X2 are proportional
to N1/2 and constant, respectively. Hence, X1 = X2 = 0 are natural conditions to
impose on ?t and kt.
Following [GM17], we define the monomial Pn,m := a?x ? ? ? a?x ay1 ? ? ? aym and1 n
consider the L-matrices whose kernels are defined by
L 1n,m(t, x1, . . . , xn; y1, . . . , ym) = ?M?, Pn,mM??. (1.32)
N (n+m)/2
In particular, let us focus on the matrices L0,1,L1,1 and L0,2, which we will denote
by ?,? and ? respectively. It is shown in [GM17] that the conditions X1 = X2 = 0
is equivalent to the fact that (?,?,?) forms a closed system of coupled nonlinear
equations
{ } ?
1 ?
? ??x1 ?(x1) = ? dy {vN(x1 ? y) diag ?(t, y)} ? ?(x1) (1.33a)i ?t
? ? dy {vN(x1 ? y)(?(y, x1)? ??(y)?(x1))?(y)}
{? dy {vN(x1 ? y})(?(x1, y)? ?(y)?(x1))??(y)}
1 ?
???x1 + ?x2 ?(x1, x2) (1.33b)i ?t
= ?? dy {(vN(x1 ? y)? vN(x2 ? y))?(x1, y)?(y, x2)}
? ? dy {(vN(x1 ? y)? vN(x2 ? y))?(x1, y)?(y, x2)}
? ?dy {(vN(x1 ? y)? vN(x2 ? y)) diag ?(t, y)?(x1, x2)}
+ 2 dy {(v 2N(x1 ? y)? vN(x2 ? y))|?(y)| ??(x1)?(x2)}
24
{ }
1 ? ? ? 1? ?x1 ?x2 + vN(x1 ? x2) ?(x1, x2) (1.33c)i ?t N
= ?? dy {(vN(x1 ? y) + vN(x2 ? y)) diag ?(t, y)?(x1, x2)}
? ? dy {(vN(x1 ? y) + vN(x2 ? y))?(x1, y)?(y, x2)}
? ?dy {(vN(x1 ? y) + vN(x2 ? y))??(x1, y)?(y, x2)}
+ 2 dy {(vN(x1 ? y) + vN(x2 ? y))|?(y)|2?(x1)?(x2)}
where diagF (t, x) = F (t, x, x)11. Note, we have suppressed the time dependence
to compactify the notation. We refer (1.33) as the time-dependent Hartree-Fock-
Bogoliubov (HFB) system. It is also instructive to consider v(x) = g?(x) which
yields the following system
{ }
1 ? ??x ?(t, x) = ?g diag ?(t, x)??(t, x) (1.34a)
i ?t
{ ? 2g diag ?(t, x})?(t, x) + 2g|?(t, x)|2?(t, x)
1 ? ??x + ?y ?(t, x, y) (1.34b)
i ?t
= ?g diag ?(t, x)?(x, y) + g diag ?(t, y)?(t, x, y)
? 2g{{diag ?(t, x)? diag ?(}t, y)}?(t, x, y)
+ 2g |?(t, x)|2 ? |?(t, y)|2 ??(t, x)?(t, y)
11In the literature, it is common to denote (diag ?)(t, x) by ?(t, x), which we will also use. In
general, we called the restricted kernels Schr?dinger-type densities.
25
{ }
1 ? ??x ?
g
?y ?(t, x, y) = ? diag ?(t, x) (1.34c)
i ?t N
? g diag ?(t, x)?(t, x, y)? g diag ?(t, y)?(t, x, y)
? 2g{diag ?(t, x) + diag ?(t, y)}?(t, x, y)
+ 2g{|?(t, x)|2 + |?(t, y)|2}?(t, x)?(t, y).
Remark 1.4. The physical interpretation of (?(t),?(t),?(t)) is as follows: The func-
tion ?(t) is the one-particle wave function called the condensate wave function which
describes the BEC. Following [BBC+18], ?(t, x, y) := N(?(t, x, y)??(t, x)??(t, y)) =
sh(k)?sh(k) and ?(t, x, y) := N(?(t, x, y)??(t, x)?(t, y)) = sh(k)?ch(k) describe the
dynamics of sound waves in the quasifree approximation; in particular, diag ?(t, x)
determines the density of the ?thermal cloud? of atoms, i.e. the excitation density
of the Bose gas. (In the physics literature, n = diag ? and m = diag ? are called the
non-condensate density and anomalous density, respectively.)
By direct calculation, it is shown in [GM17] that
1 ( )
?(t, x, y) = ??(t, x)?(t, y) + sh(k) ? sh(k) (t, x, y) (1.35a)
N
1
?(t, x, y) = ?(t, x)?(t, y) + sh(2k)(t, x, y). (1.35b)
2N
The local well-posedness of (1.33) were established in [GM17] using techniques from
dispersive PDEs. Consequently, the authors were able to obtain a Fock space esti-
mate for small time. The following theorem summarizes the main result of [GM17]
Theorem 1.5 (Grillakis & Machedon ?17). Let 1 ? ? < 2 and v ? S a nonnegative
3 3
26
interaction potential satisfying the condition that |v?| ? w? for some w ? S. Suppose
(?t,?t,?t) are solutions to the time-dependent HFB system with some smooth initial
conditions (?0,?0,?0) satisfying the following regularity condition uniformly in N :
for some ? > 0 and 0 ? i ? 1, 0 ? j ? 2
?? ? ??? ?1/2+? i? x ?t?
j?(t, ?)? ?x 2 . 1t=0 L (dx)
? ? ??? ?1/2+?? x ?? ?
1/2+??i?jy t x+y?(t, ?)? ??t=0 ?L2 . 1(dxdy)? ?? ?1/2+??? ?1/2+??i?j ? ?? x y? ?
t x+y?(t, ?) t=0 L2 . 1(dxdy)
?jx+y sh(2k)(0, x, y)? 2 . 1.L (dxdy)
Then there exists constants ? = ?(?), ? = ?(?), C = C(?, ?), a phase function ?(t),
depending on N , and T0 (T0 ? 1) independent of N such that we have the Fock space
estimate
??? ? ? ?eitHe? NA(?0)e?B(k0)?? ei?(t)e? N(?t)e?B(k Ct) ??? ?
F N1/6
for all 0 ? t ? T0.
These estimates were later extended by the author to a global-in-time result
in [Cho17], which is also the main focus of Chapter 4 of the thesis. More recently,
Grillakis and Machedon extended the local well-posedness of the time-dependent
Hartree-Fock-Bogoliubov system to the case 2 < ? < 1 in [GM18].
3
Independently and in a different frame work, Bach, Breteaux, Chen, Fr?hlich,
and Sigal derived equations closely related to the above equations in [BBC+18]. In
27
particular, the two sets of equations are equivalent in the case of pure states.
More recently, Benedikter, Sok, and Solovej use the reformulated Dirac-Frenkel
variational principle in the space of reduced density matrices to geometrically ap-
proximate12 the dynamics of both the bosonic and fermionic many-body systems
in [BSS18]. Using the variational principle, they provide a rigorous derivation of
both the time-dependent HFB equations and the Bogoliubov-de-Gennes equations,
also known as the fermionic time-dependent HFB equations, and show that the
equations are optimal approximations of the many-body dynamics when restricted
to the manifold of quasifree states13. We also refer the interested reader to [HLLS10]
for a study of the pseudo-relativistic version of the Bogoliubov-de-Gennes equations.
1.3 Outline and Main Results of the Thesis
In chapter 2, we study the uncoupled HFB equations in the case of attractive
interaction potentials. The main results of the chapter is Theorem 2.1 and Theorem
2.5. Theorem 2.1 generalizes the Fock space estimate in [GM13a,Kuz15a] to the case
of attractive boson systems. In fact, using Theorem 2.1, we provide two derivations
of the focusing NLS from a quantum many-body system with attractive interactions
for 0 < ? < 1/6, which is the result of Theorem 2.5. This chapter is based on the
author?s paper [Cho16].
In chapter 3, we study the uniform in N global well-posedness of the time-
dependent HFB system in 1D. The main result of this chapter is Theorem 3.3. More
12They were able to show that the Dirac-Frenkel variational principle implies the quasifree
reduction principle which was used in [BBC+18].
13See ?10 in [Sol14] for a definition of quasifree states.
28
precisely, we show for any ? > 0 the corresponding time-dependent HFB system is
uniform in N globally well-posed. It should also be noted that the main tools used
in the proving Theorem 3.3 are the linear estimates in ?3.2, 3.3, 3.5, and 3.6. This
chapter is based on the author?s paper [Cho18]
In chapter 4, we extend the local-in-time [GM17] Fock space estimate to a
global-in-time for a system of bosons in R3 for 0 < ? < 2 . The main result of the
3
chapter is Theorem 4.1. This chapter is based on the author?s paper [Cho17]
In chapter 5, we study some global estimates for the time-dependent HFB
system. The main results of the chapter are Theorem 5.3 and Theorem 5.5 which
are natural generalization of Morawetz identity and interaction Morawetz estimates
for the cubic NLS in R3.
In chapter 6, we study collapsing estimates for Schr?dinger-type densities on
closed Riemannian manifolds, which are crucial to proving local well-posedness of
the time-dependent HFB system closed manifolds. The main result of the chapter
is Theorem 6.1.
29
Chapter 2: Uncoupled Time-Dependent HFB systems
2.1 Main Results
One of the main purposes of this chapter is to extend the results in [GM13b,
Kuz15b,Kuz15a] to the case of arbitrary v ? C?0 with sufficiently small L1-norm
allowing non-positive v. Let us state the first main statement.
Theorem 2.1. Let v ? C?(R30 ). Assume ? and k satisfy (1.26) with initial condi-
tions ? ? L2(R2) ?Wm,10 (R3) for some sufficiently large m and sufficiently small
1/2
H?x -norm, depending on v, and k(0, ?) = 0. If ?exact and ?approx are defined by
(1.18) and (1.24) respectively, then we have the following estimate
? t?exact(t)? ?approx(t) ?F . ? (2.1)N (1 3?)/2
provided 0 < ? < 1 . Moreover, if (?t sh(2k))(0, ?) is sufficiently regular, then for3
30
any  > 0 and j a positive integer, we have
??exact(t)? ?approx(?t) ?F????N?1/2+?(1+) 0 < ? < 2j ,
j+3 (1?2+4j)
. t 2 log6(1 + t) ????? (2.2)?3+7?N +(j?1)(?1+2?) 2j ? ? < 1+2j2
(1? .2+4j) 3+4j
Remark 2.2. Let us note that there is a tradeoff between the size of the data ?0 and
the size of the interaction potential v (c.f. Remark 2.9). Due to the nature of our
proof, if we want to assume ?0 is large, i.e. ?? ? = 1 and ??1/20 L2 ?0 ?L2 large ,
then we need to restrict the L1-norm of the potential v, and vice versa.
Remark 2.3. A similar result was obtained in [NN17] for the case of repulsive in-
teraction. As stated in Remark 4 in [NN17], their method also extends to the case
of attractive interaction provided the uniform in N well-posedness and decay esti-
mates for the corresponding Hartree equations hold, which we will show in the next
section.
Remark 2.4. The second estimate in Theorem 2.1 could be improved. In particular,
we can get rid of the logarithmic terms. However, to keep the organization of the
chapter simple, we decided to keep the logarithmic terms. Nevertheless, we have
included a proof of how to remove the logarithmic terms in ?2.4.
The second purpose of the chapter is to derive the focusing cubic NLS in R3
from a many-body boson system as in [CH16a,CH17,CH16b]. For this purpose, we
assume v ? 0, i.e. the interaction is attractive. In this case, we have the following
31
statement.
Theorem 2.5. (Factorized Initial Condition) Assume v ? C? 3c (R ) and v ? 0.
Suppose ?N(t,x) solves the initial value problem
1
?t?N(t,x) = H
?N
N,mf?N(t,x), ?N(0, ?) = ?
i 0
where ?0 satisties the same conditions as in Theorem 2.1 and ??0 ?L2(dx) = 1. De-
note the one-particle density associated to (1)?N(t, x) by ?N,t. Then we have the esti-
mate
??? ?(1)Tr ?N (t, ?)? |?t?? ?? ?t|? . N
for some ? < 0 provided 0 < ? < 1 .
6
Remark 2.6. The reader should note that Theorem 2.5 only addresses the derivation
of the focusing NLS for a system of weakly-interacting dense bose gas since ? ?
(0, 1).
6
Remark 2.7. As pointed out by the referee, the case at hand deals with the situation
where (1.26a) does not exhibit soliton solutions. C.f. Remark 2.11 and Remark 2.21.
32
2.2 Estimates for the Solution to the Hartree-Type Equations
Let us consider the following family of Hartree-type PDE
1
?t???x?+ (vN ? |?|2)? = 0 (2.3)
i
?(0, ?) = ? s0 ?0 ? H (R3)
where vN(x) = N3?v(N?x) for 0 ? ? ? 1 and v ? C? 30 (R ) is not necessary
nonnegative. In this section we prove the uniform inN well-posedness of the Hartree-
type equation for small data and the corresponding decay estimates.
2.2.1 Uniform in N Global Well-posedness of the Hartree Equations
In this subsection we prove the uniform in N global well-posedness of (2.3)
assuming small data. Let us recall the Strichartz norm. We said a pair of numbers
(q, r) is admissible provided q, r ? 2 and
2 3 3
+ = .
q r 2
Then the Strichartz norm is defined by
?? ?S0 := sup ?? ?LqLr (R?R3).t x
(q,r) admissible
Proposition 2.8 (a-priori estimates). Let ? be a solution to (2.3), then we have
33
the estimate
1 1
? |?x| 2? ?S0 . ?? ? 30 1/2 + ? v ?L1(dx)? |?x| 2? ?S0 (2.4)H?x
which is independent of N . Moreover, if ? v ?L1 is sufficiently small then we obtain
the estimate
1
? |?x| 2? ?S0 . 1 (2.5)
which depends only on ??0 ? 1 and independent of N . Similar estimates holds for
H? 2x
time and higher spatial derivatives, that is
? ?mt |?x|s? ?S0 . 1 (2.6)
where the estimate only depends on m, s and the initial datum.
Remark 2.9. Observe (2.5) is a consequence of the following elementary observation:
if F is continuous on [0,?) with F (0) = A and F (x) ? A+xF (x)3 then there exists
? = ?(A) > 0 such that F (x) ? 2A whenever x ? ?.
Proof. Similar to the local estimate, we begin by differentiating (2.3)
1 1 1 1
?t|?x| ??? |? | ?+ |? | 22 x x 2 x 2 ((vN ? |?| ) ? ?) = 0
i
34
where
1 1 1
|?x| 2 ((vN ? |?|2) ? ?) = (vN ? |?|2) ? |?x| 2?+ (vN ? |? | 2x 2 |?| ) ? ?
+ ?lower order" terms.
Applying the L2L6/5- endpoint Strichartz estimate of [KT98] and the fractional
Leibniz rule, we obtain the following estimate
1 1
? |?x| 22? ?S0 . ??0 ? 1/2 + ? vN ? |?| ?L2H? (dt)L3(dx)? |?x| 2? ?L?(dt)L2(dx)x
1
+ ? v 2N ? |?x| 2 |?| ?L2(dtdx)?? ?L?L3t (dx).
For the first forcing term we have the estimate
2 1? vN ? |?| ?L2(dt)L3(dx)? |?x| 2? ?L?(dt)L2(dx)
1
. ? v ?L1(dx)?? ?2L4(dt)L6 2(dx)? |?x| ? ?L?(dt)L2(dx)
1 1
. ? v ? 2L1(dx)? |?x| 2? ? 4 3 ? |?x| 2L (dt)L (dx) ? ?L?(dt)L2(dx)
1
. ? v ? 3L1(dx)? |?x| 2? ?S0 .
The other term can be estimated in a similar fashion. Moreover, estimate (2.6)
follows from the observation
?
? ?m|? |s? ? . ? ?m
1
t x S0 t |? sx| ?? ? m s 22t=0 L (dx) + ? v ?L1(dx)? ?t |?x| ? ?S0? |?x| 2? ?S0 .
35
As an immediate corollary of Proposition 2.8, we have
Corollary 2.10 (Uniform in N global well-posedness). For any R > 0 there exists
? = ?(R) > 0, independent of N , such that when ? v ?L1 < ? then the family of
Hartree equations is uniform in N globally well-posed. More precisely, we have for
any ?0 ? { ? 1/2? H?x | ?? ? 1/2 < R} there exists a unique solution to (2.3) withH?x
initial data ?0 satisfying
1/2
?t ? C([0,?)? H?x ) ? S0.
Remark 2.11. In this paper, we always consider sufficiently smooth initial data. In
particular, we could take any ?0 ? H1x and obtain a uniform inN local well-posedness
of solutions to the family of Hartree-type equations. Of course, the tradeoff is that
we can only have uniform in N local well-posedness for short time. C.f. chapter 3.3
Proposition 3.19 in [Tao06].
2.2.2 Decay Estimates
In this subsection we prove the uniform in N decay estimates for ?t following
the approach in [GM13b], which is in the spirit of [LS78]. Before we begin let us
make a note on the notation used in this section. The notation ?? means ?? ? for
some fixed 0 < ? 1.
Proposition 2.12. Suppose ? k,10 ? Wx for some sufficiently large k. Let ? be a
solution to (2.3) with sufficiently small potential v, depending on the size of data.
36
Then we have the decay estimate
? 1?(t, ?) ?L?(dx) .
1 + t3/2
which only depends on ??0 ?Wk,1 and independent of N .x
Let us first prove the following lemmas.
Lemma 2.13. Assuming the same conditions as in Proposition 2.12. Then ??(t, ?) ?L?(dx) ?
0 as t??.
Proof of lemma 2.13. By Proposition 2.8 and Sobolev embedding, we have the es-
timates
?? ? 10/3 ? C and ?? ?L? (R?R3) . ? ?t?x? ?L2(dt)L6(dx) ? C.Lt,x t,x
Hence by interpolation we have
?? ? 1??Lp([n,n+1])Lp ? ?? ? 10/3 ?? ???x L ([n,n+1])L10/3(dx) Lt ([n,n+1])L?(dx)t
? ?? ?1?? ?10/3 ? ?t?x? ? 2 6 ? 0
Lt ([n,n+1])L
10/3(dx) Lt ([n,n+1])L (dx)
as n?? for all 10/3 < p <?. Letting p?? yields the desired result.
Remark 2.14. The slight analytic gymnastic is a consequence of the fact that we do
not have the endpoint Sobolev estimate.
Lemma 2.15. Assuming the same conditions as in Proposition 2.12. There exists
37
k ? L1([0,?)) and ? > 0 such that
? ei(t?s)?((v ? |?|2N ) ? ?(s)) ?L?(dx) ? k(t? s)??(s, ?) ?1+?L?(dx). (2.7)
Proof of Lemma 2.15. Using the L?L1 -decay and conservation of probability, we
have
? ei(t?s)?((vN ? |?|2) ?
1
?(s)) ? 2L?(dx) . ? (vN ? |?| ) ? ?(s) ? 1| ? |3/2 L (dx)t s
(2.8)
1
. ??(s, ?) ?L?| ? |3/2 (dx)t s
On the other hand, applying Sobolev embedding, L3+L3/2?- decay estimate and
interpolation yields
? ei(t?s)?((vN ? |?|2) ? ?(s) ? i(t?s)?L?(dx) . ??xe ((vN ? |?|2) ? ?(s)) ?L3+(dx)
1
. ?? ? ? 2 ?? ?2
| ? x L (dx)t s|1/2+ L12?(dx) (2.9)
1
. ? ?13/9??
| ? |1/2+ L?t s (dx)
.
In the case |t ? s| < 1, we could simply take k(t ? s) = |t ? s|1/2+. In the case
|t? s| ? 1 we interpolate estimates (2.8) and (2.9).
Proof of Proposition 2.12. Let ?0 be a test function and write (for t > 0)
[? t/2 ? ]t
?(t) = eit?? ? i + ei(t??)?0 (vN ? |?(?)|2)?(?) d?.
0 t/2
38
Taking the L? norm yields
[? ]?? ? t/2 ? t0 L1? (dx)?(t) ? . + + ? ei(t??)?(v ? |?|2L?(dx) N )?(?) ?L?3/2 (dx) d?t 0 t/2
where the first term is a consequence of the L?L1-decay estimate for the free evo-
lution. For the second term, we apply the L?L1-decay estimate and Young?s con-
volution estimate to get
? t/2 ? t/2 2
1
? ei(t??)? 2
? (vN ? |?| )?(?) ?L (dx)
(vN ? |?| )?(?) ?L?(dx) d? ? d?3/2
0 0 ? |t? ? |
1 t/2
. ??(?) ?L?3/2 (dx) d?.t 0
Lastly, by Lemma 2.15 there exists k ? L1([0,?]) and ? > 0 such that
? t ? t
? ei(t??)?(vN ? |?|2)?(?) ?L?(dx) d? . k(t? ?)??(?) ?1+?L?(dx) d?.
t/2 t/2
Combining all the estimates, we have
? ? t/2 ?? t
? 0
?L1(dx) ??(?) ?L?(dx)
?(t) ? 1+?L?(dx) . + d? + k(t? ?)??(?) ?
t3/2 t3/2 L
?(dx) d?
0 t/2
which holds for all t > 0.
Since we care about large time behavior we may assume t ? 1. In particular,
39
we get the equivalent estimate
? ?
?? ? t/20 ??(?) ? tL?? (dx)?(t) ?L?(dx) . + d? + k(t? ?)??(?) ?1+?
1 + t3/2 1 + t3/2 L
?(dx) d?.
0 t/2
(2.10)
Multiplying estimate (2.10) by 1 + t3/2 yields
? t/2
(1 + t3/2)??(t) ?L?(dx) . ??0 ?L1(dx) +? ??(?) ?L?(dx) d?0t
+ (1 + t3/2) ? k(t? ?)??(?) ?
1+?
L?(dx) d?
t/2
t/2
. ??0 ?L1(dx) + ??(?) ?L?(dx) d?
0
+ sup (1 + s3/2)??(s) ?1+?L?(dx)
t/2?s?t
since k ? L1([0,?)). Next, by Lemma 2.13, there exists T > 0 such that
? t/2
(1 + t3/2)??(t) ?L?(dx) ? c??0 ?L1(dx) + c ??(?) ?L?(dx) d?
0
1
+ sup (1 + s3/2)??(s) ?L?(dx)
2 t/2?s?t
whenever t ? 2T for some constant c > 0.
LetM(t) := supT?s?t(1+s3/2)??(s) ?L?(dx) and C := sup 3/20?s?2T (1+s )??(s) ?L?(dx)
then for all t ? T we have either
? t/2
(1 + t3/2)??(t) ?L?(dx) ?
M(?) 1
c??0 ?L1(dx) + c d? + M(t)
0 1 + ?
3/2 2
40
or M(t) ? C. Note, for all T < s < t we also have the following estimate
( ? )t/2
(1 + s3/2)? ? ? ? ? M(?) 1?(s) L?(dx) max c ?0 L1(dx) + c d? + M(t), C .
0 1 + ?
3/2 2
Hence it follows
( ? )t/2
? ? ? M(?) 1M(t) max c ?0 L1(dx) + c d? + M(t), C
0 1 + ?
3/2 2
for all t ? T . Then by Gronwall?s inequality, we have the estimate
( (? t ) )
M(t) . max ? d??0 ?L1(dx) exp , C . 1.
0 1 + ?
3/2
Thus, we have proved
sup (1 + s3/2)??(s) ?L?(dx) . max(M(t), C) . 1.
0?s?t
Corollary 2.16. Assume the same conditions as Proposition 2.12 then there exists
a constant C depending only on ??0 ?Wk,1 and ? ?t?0 ?Wk,1 such that
? 1?t?(t, ?) ?L?(dx) . . (2.11)
1 + t3/2
41
Proof. Begin by taking the time derivative of (2.3)
1 ?
?t???x?t?+ ?t(vN ? |?|2)? = 0.
i ?t
Then applying the L?L1 decay estimate yields (t ? 1)
?
? ?? ?(t)? ? t/21
? ? ?(t) ? t. t=0 L (dx) i(t??)?t L?(dx) ? + ? e ?? (vN ? |?|
2)?(?) ?L?(dx) d?
t3/2 0
t
+ ? ei(t??)??? (vN ? |?|2)?(?) ?L?(dx) d?.
t/2
For the first integral, we shall apply the L?L1-decay estimate and Proposition 2.12
to get
? t/2 ? t/2 2
? i(t??)?
? ?
? | |2 ? ?
(vN ? |?| )?(?) ?L1(dx)
e ?? (vN ?? )?(?) L?(dx) d? . d?0 0 |t? ? |3/2
1 t/2
. ? ??(?) ?L?(dx)??(?) ?L2(dx)? ???(?) ?L23/2 (dx) d?1 + t 0
1 t/2 d? 1
. . .
1 + t3/2 1 + ? 3/2 1 + t3/20
Note we have used the fact ? ?m?st x? ?S0 .m,s 1.
For the second integral, we use Sobolev embedding and L3+L3/2? decay esti-
42
mate to obtain the bound
? t
? ei(t??)?? ?? [(vN ? |?|
2)?(?)] ?L?(dx) d?
t/2
t 1
. ? ??
5/3?
x?t? ?L2(dx)?? ?L?(dx) d?
t/2 |t? ? |1/2+
t 1 1/3+
+ ? ? ? ? ? ? ? ?2/3? ?? ? ? 2 ?? ? ? d?.
t/2 |
t 2 t ? x L (dx) L (dx)
t? ? |1/2+ L (dx) L (dx)
Note the last inequality is a consequence of H?lder inequalities and space interpo-
lation. Since ?t? is bounded by Proposition 2.8, then by Proposition 2.12 it follows
? t ? t
? ei(t??)?? (v ? |?|2 1? N ? )?(?) ?L?(dx) d? . ??(?) ?L?(dx) d?1/2+t/2 t/2 |t? ? |
1 t 1 1
. d? . .
1 + t3/2 1/2+ 3/2t/2 |t? ? | 1 + t
2.3 Estimates for the Pair Excitations
There are two goals in this section. The first goal is to extend the estimates
for sh(2k) to the case of non-positive interacting potential, which we will see only
depends on the decay estimate of ?. The other goal is to provide a way to improve
the estimate in Theorem 2.1 which we have mentioned in Remark 2.4. However,
for the sake of simplicity, we will not propagate the improvement to the rest of the
paper and happily leave it as an exercise(s) for the interested reader.
Let us define the shorthand notation ch(k) := ? + p1, sh(k) := s1, and also
43
ch(2k) := ? + p2, sh(2k) := s2.
Proposition 2.17. Assume ? ? W k,10 for k sufficiently large. The following esti-
mates hold:
? s2(t, ?) ?L2(dxdy) + ? p2(t, ?) ?L2(dxdy) . 1
where the estimate only depends on ??0 ?Wk,1 for some k.
To prove the above proposition, we begin by proving a few preliminary lemmas.
Lemma 2.18. Let mN(t, x, y) := ?vN(x ? y)?(t, x)?(t, y). Then we have the fol-
lowing estimates
?
|m?N(t, ?, ?)|2
d?d? . ??(t, ?) ?4
| |2 | |2 2 L3(dx)
(2.12)
( ? + ? )
and
?
|? m? (t, ?, ?)|2t N
d?d? . ? ? ?(t, ?) ?2 ??(t, ?) ?2 . (2.13)
|???|>1 (|?|2 | |2 2
t L4(dx) L4+ ? ) (dx)
Proof. The proof of the first estimate can be found in [GM13b]. We shall focus on
the proof of the second estimate where the proof is a slight modification of the first.
First, observe
?
vN(x? y)?(x)?(y) = ?(x? y ? z)vN(z)?(x)?(y) dz
44
then the Fourier transform of ?(x? y ? z)?(x)?(y) is given by
?
e?i(x??+y??)? ?(x? y ? z)?(x)?(y) dxdy
= e?i((x??+(y?z)??)? ?(x? y)?(x)?(y ? z) dxdy
= eiz?? e?ix?(?+?)?(x)?(x? z) dx = eiz?????z(t, ? + ?)
which means
|? m? (t, ?, ?)|2 = ????? ??2eiz??t N vN?(z)??t(??z)(t, ? + ?) dz??
. ? v ?L1(dx) |vN(z)||??t(??z)(t, ? + ?)|2 dz.
Then it follows
? ? ?
|?tm?N(t, ?, ?)|2 |??t(??z)(t, ? + ?)|2
d?d? . |v (z)| d?d?dz
2 2 2 N 2 2 2
|???|>1 (|?| + |?| ) ? ?|???|>1 (|?| + |?| )
| | |??t(??z)(t, ?
?)|2
. v ? ?? N(z) ? ? d? d? dz2 2 2|??|>1 (|? | + |? | )
. |vN(z)||??t(??z)(t, ??)|2 d??dz
. ? ?t?(t, ?) ?2 2L4(dx)??(t, ?) ?L4(dx).
Lemma 2.19. Let s0a be the solution to
( )
1 ? ?? 0x ??y sa(t, x, y) = 2mN(t, x, y), s0a(0, x, y) = 0.i ?t
45
Then it follows
?? ?s0 ?a(t, ?) 2 . 1 (2.14)L (dxdy)
where the estimate only depends on ??0 ?Wk,1.
Proof. Using Duhamel?s principle, we have
?? ? ??? t ??s0a(t, ?)? = 2 ?2 ei(t?s)?m (s, ?) ds?L (dxdy) ?? ? N ?? 0 L2(dxdy??)t
. ??P ei(t?s)?? |???|?1 ? mN(s, ?) ds
??
? 0 L??2(dxdy)?? t+ P ei(t?s)?|???|>1 mN(s, ?) ds?? .
0 L2(dxdy)
For the first term we shall directly apply Minkowski?s inequality to get
???? ? ??tP|???|?1 ?e
i(t?s)?
[? mN(s, ?) ds
??
0 L2(dxdy)
t ]1/2
. 2? [? |m?(s, ?, ?)| d?d? ds0 |???|?1t ? ]1/2
. ? |vN(z)| |???
?
z(s, ? )|2 d??d??dz ds
0 |??|?1
t
. ??(s, ?) ?2L4(dx) ds.
0
Using the decay estimate, we have that the first term is bounded. For the second
46
term we have
???? ? t ??P|???|>1 ei(t?s)?mN(s, ?) ds??
??
0 2
?? ?
L (dxdy) ?
t
? i(t?s)(|?|2+|?|2) m?(s, ?, ?)
?
= |???|>1 ?se ds
?
|?|2 + |?|2 ?
? 0 L2(dxdy)??? m?(0, ?, ?). ???? ???? ?m?(t, ?, ?) ?+ ?|?|2 + |?|2 L2(d?d?) |?|2 + |?|2 ?? 2? L (d?d?)??? ?t ?+ ? ei(t?s)(|?|2+|?|2)?sm?(s, ?, ?)|???|>1 ds? .|?|20 + |?|2 ?L2(d?d?)
It?s clear the first two terms are bounded by the previous lemma. For the last term,
using Minkowski?s and the previous lemma we have
???? ? ?t ?? i(t?s)(|?|2+|?|2)?sm?(s, ?, ?) ?|???|>1 e ds2 2 ?
?0
|?| + |?| L2(d?d?)
t
. ? ?t?(s, ?) ?L4(dx)??(s, ?) ?L4(dx) ds.
0
Again by the decay estimate, the second term is also bounded.
Lemma 2.20. Let sa be a solution to
Sold(sa) = 2mN(t, x, y), sa(t, ?) = 0.
47
Then
? sa(t, ?) ?L2(dxdy) . 1
where the estimate depends only on the ??0 ?Wk,1.
Sketch of the Proof. The essential idea is to decompose the solution into a two parts
sa = s
0
a + s
1
a
where s0a satisfies the equation in the previous lemma and s1a solves
S 1 0old(sa) = ?V (sa(t, ?)).
By the previous lemma, we know the L2-norm of s0a is uniformly bounded in time.
Next, we shall cite [GM13b], Lemma 4.5, for the proof that the L2-norm of s1a is
also uniformly bounded in time.
Sketch of the Proof of Proposition 2.17. The proof is the same as the proof of The-
orem 4.1 in [GM13b]. Again, the only difference comes from the replacement of the
estimate to the solution of Sold(sa) = 2mN by the result of the previous lemma.
Remark 2.21. Following Remark 2.11, if we consider the subcritical uniform in N
well-posedness forH1x data, we would obtain the estimate supt?[0,T ](? s2(t, ?) ?L2(dxdy)+
? p2(t, ?) ?L2(dxdy)) . 1 for some small time T and independent of N .
48
2.4 Proof of Theorem 2.1
The proof of Theorem 2.1 is essentially the same as the proof given in [GM13b,
Kuz15a] provided we have established the decay estimate for ?. For the sake of
simplicity, we shall only provide a complete proof of the first part of Theorem 2.1
since the second part of the theorem is significantly lengthier to present. We shall
refer the interested reader to [Kuz15a] for a complete proof of the second part of
Theorem 2.1.
2.4.1 List of Error Terms
For convenience, we shall include the list of error terms which were explicitly
computed in ?5 of [GM13b].
Recall the error terms are defined to be
E(t) = eB([A,V ] +N?1/2V)e?B (2.15)
where [A,V ] and N?1/2V are cubic and quartic polynomials in (a , a?x x) respectively.
Using the conjugation formulae
?
eBa e?Bx =? dy {ch(k)(y, x)ay + sh(k)(y, x)a
?
y} (2.16a)
eBa?e?Bx = dy {sh(k)(y, x)ay + ch(k)(y, x)a?y} (2.16b)
we could further expand the error terms into another fourth-order polynomial in
49
(a?x, ax).
The following is the result of expanding E(t). First, let us list all the error
terms of N?1/2eBVe?B which is a fourth-order polynomial in (a?x, ax) with no linear
nor cubic terms. The quartic term is given by
?
1 {
dy1dy2dy3dy4
2N
v?N(y1 ? y2) sh(k)(y3, y1) sh(k)(y2, y4) + (2.17a)
? dx {p?(y2, x)vN(y1 ? x) sh(k)(x, y4)} sh(k)(y3, y1) + (2.17b)
? dx {p?(y1, x)vN(x? y2) sh(k)(y3, x)} sh(k)(y2, y4) + (2.17c)
} dx1dx2{p?(y1, x1)p(x2, y2)vN(x1 ? x2) sh(k)(y3, x1) sh(k)(x2, x4)} (2.17d)
a? ? ? ?y ay a1 2 y a3 y .4
The quadratic term is given by
?
1 {
dy1dy2dx1dx2
2N
ch(k)(y1, x2) sh(k)(x2, y2)(sh(k) ? sh(k))(x1, x1)vN(x1 ? x2) + (2.18a)
ch(k)(y1, x2) sh(k)(x1, y2)(sh(k) ? sh(k))(x1, x2)vN(x1 ? x2) + (2.18b)
ch(k)(y1, x1) sh(k)(x2, y2)(sh(k) ? sh(k))(x1, x2)vN(x1 ? x2) + (2.18c)
ch(k)(y1, x1) sh(k)(x1, y2)(sh(k) ? sh(k))(x2, x2)vN(x1 ? x2) + (2.18d)
sh(k)(y1, x1) sh(k)(x2, y2)(sh(k) ? sh(k))(x1, x2)vN(x1 ? x2)}+ (2.18e)
ch(k)(y1, x1) ch(k)(x
? ?
2, y2)(ch(k) ? sh(k))(x1, x2)vN(x1 ? x2) ay ay (2.18f)1 2
50
The zeroth-order term is given by
?
1 {
dx1dx2
2N
(sh(k) ? sh(k))(x1, x2)vN(x1 ? x2)(sh(k) ? sh(k))(x1, x2) + (2.19a)
(sh(k) ? sh(k))(x1, x1)vN(x1 ? x2)(sh(k) ? sh(k))(x2, x2)}+ (2.19b)
(sh(k) ? ch(k))(x1, x2)vN(x1 ? x2)(ch(k) ? sh(k))(x1, x2) . (2.19c)
In the case of eB[A,V ]e?B we have a cubic polynomial in (a?x, ax) with no
quadratic nor zeroth-order terms. The cubic term is given by
?
?1
{
? dy1dy2dy3 vN(y1 ? y2)?(y2) sh(k)(y3, y1) + (2.20a)N
? dx {vN(y1 ? x)??(x) sh(k)(x, y3)} sh(k)(y2, y1) + (2.20b)
? dx {p?(y1, x)vN(x? y2) sh(k)(y3, x)}?(y2) + (2.20c)
? dx {p?(y2, x)vN(y1 ? x)?(x)} sh(k)(y3, y1) + (2.20d)
? dx1dx2 {p?(y1, x1)vN(x1 ? x2)??(x2) sh(k)(y2, x1) sh(k)(x2,}y3)} + (2.20e)
dx1dx2 {p?(y1, x1)p(x ? ? ?2, y2)vN(x1 ? x2)?(x2) sh(k)(y3, x1)} ay a1 y a2 y .3
(2.20f)
51
Lastly the linear term is given by
?
?1
{
dydx1dx2
N
sh(k)(y, x2)(sh(k) ? sh(k))(x1, x1)??(x2)vN(x1 ? x2) + (2.21a)
sh(k)(y, x1)(sh(k) ? sh(k))(x1, x2)??(x2)vN(x1 ? x2) + (2.21b)
ch(k)(y, x1)(ch(k) ? sh(k))(x1, x2)??(x2)vN(x1 ? x2) + (2.21c)
ch(k)(y, x1)(sh(k) ? sh(k))(x1, x2)?(x2)vN(x1 ? x2) + (2.21d)
ch(k)(y, x2)(sh(k) ? sh(k))(x1, x1)?(x2)vN(x1 ? x2) +} (2.21e)
sh(k)(y, x1)(sh(k) ? ch(k))(x1, x2)?(x2)vN(x1 ? x ?2) ay. (2.21f)
2.4.2 Estimates for the Error Terms
To prove theorem 2.1 it suffices to establish the following estimates on E(t).
Proposition 2.22. For the two error terms we have the following estimates
3??1
2
?1 ? NeB[A,V ]e?B? ?F . (2.22)
N 1 + t3/2
and
1 ? eBVe?B? ? . N3??1F . (2.23)
N
Proof. Since many of the terms are similar, without loss of generality, we shall pick
representatives in each category and prove the bound holds for the representatives.
52
First, let us look at the quartic term. The two representatives are (2.17a) and
(2.17d) since (2.17b) and (2.17c) could be handled similarly by the techniques in
bounding (2.17d). In the case of (2.17a), we see that
1 ? vN(y1 ? y2) sh(k)(y3, y1) sh(k)(y2, y4) ?L2N (dy1dy2dy3dy4)
1
. ? vN ?L?(dx) ? sh(k) ?2 3??1
N L
2(dxdy) . N
where we have used Proposition 2.17. For (2.17d), we have
????? ?1 ?dx1dx2{p?(y1, x1)p(x2, y2)vN(x1 ? x2) sh(k)(y3, x1) sh(k)(x ?2, x4)N ?L2(dy1dy2dy3dy4)
1
. ? vN ?L?(dx)? p(k) ?2 2 3??1
N L
2(dxdy)? sh(k) ?L2(dxdy) . N .
For the quadratic term, the worse term is given by (2.18f) due to the ? function
contribution. Looking term with the most ? function contribution, we have
1 ? 1sh(k)(y1, y2)vN(y1 ? y2) ?L2(dy dy ) . ? vN ?L?(dx)? sh(k) ?
3??1
1 2 L
2(dxdy) . N .
N N
For the cubic term, we shall consider (2.20a) and (2.20f). In the case of (2.20a), we
have
?1 ? vN(y1 ? y2)?(y2) sh(k)(y3, y1) ?L2(dy1dy2dy3)
N
1 N (3??1)/2
. ? ?? ?L?(dx)? vN ?L2(dx)? sh(k) ?L2(dxdy) . .
N 1 + t3/2
53
And for (2.20f), it follows
?1 ????? ??dx1dx2 {p?(y1, x1)p(x2, y2)vN(x1 ? x2)?(x2) sh(k)(y3, x1)}?N ?L2(dy1dy2dy3)
(3??1)/2
. ?1 ?? ? 2 NL?(dx)? p(k) ?L2(dxdy)? sh(k) ?L2(dxdy)? vN ?L2(dx) . .
N 1 + t3/2
Lastly, for the linear term, we shall consider (2.21c). Again, consider the term with
the ? contribution, we have
?1 ????? ??dx2 sh(k)(y, x2)??(x2)vN(y ? x ?2)N ?L2(dy)
1 N (3??1)/2
. ? ?? ?L?(dx)? vN ?L2(dx)? sh(k) ?L2(dxdy) . .
N 1 + t3/2
2.5 Application: Derivation of The Focusing NLS in R3
We provide two derivation of the focusing nonlinear Schr?dinger equation.
For the first derivation we will use the method of pair excitation developed in the
previous sections and the second derivation will be via a method introduced by Pickl
in [Pic11,Pic10].
2.5.1 Pair Excitation Method
In this section, we provide the Fock space method1 for analyzing the rate of
convergence of the one-particle marginal toward mean field. However, in the next
1The pair excitation method is also referred to as the Fock space method.
54
subsection, we shall provide Pickl?s method which offers an error bound which will
be independent of time. Nevertheless, the purpose of this section is to show that
one could still derive the focusing NLS from the pair excitation method developed
thus far in the chapter.
Let us recall a couple results proven in [Kuz15b]:
Lemma 2.23. Let k(x, y) ? L2(R3 ? R3) symmetric in (x, y). Then the following
operator inequality holds
eB(k)N e?B(k) . N + 1 (2.24)
uniformly in time.
Lemma 2.24. We define the reduced dynamics, denoted by ?red, to be
? ?
?red(t) := e
B(kt)e NA(?t)eitHN e? NA(?0)?. (2.25)
Then we have the following estimates
??N e?B? ?? ? ? ?. N ? (N + 1)1/2red F ?red ?F and (2.26)
?
?N 1/2?red ?F . N??exact ? ?approx ?F . (2.27)
Following [RS09] and [Kuz15b], we rewrite the one-particle marginal density
55
as follows
(1)
?N,t(t, x, y)
1 ( ? ? )
= ?eitHP e? NA?, a?a eitHe? NAN y ?c2NN x ?
1 ?? ? ? ?? ? ? ? ? ?= eitHP e? NA?, e? NA e NAa???e? NA? e? NAa??e? NA? e NAeitH ?N x y e? NA?2 ?cNN
1 (
?
a?x+ N?? ay+
? ? ? ? )
?
N?
= eitHP ? NANe ?, e
? NAa?xa e
NA itH
y e e
? NA?
c2NN
?(t, x) ( ? ? ? ? )
+ ? eitHP e? NA?, e? NAa e NAeitHe? NA?
c2N N
??(t, y) (
N y
? ? ? ? )
+ ? eitHP e? NA?, e? NAa? NA itH ? NA
2 N x
e e e ? + ?(t, x)??(t, y).
cN N
Here PN is the projection operator onto the Nth sector of the Fock space. Moreover,
the identities
? ? ?
e NAa?e? NA = a?x x + N?? (2.28a)
? ? ?
e NAaxe
? NA = ax + N? (2.28b)
are direct consequences of the Lie-type identity used in [GM13b].
Using the above calculation, we have
?? ?? 1 ?(1) ? ? ?? ? ?? NA itH ? NA ???N (t, x, y) ?N(t, x)??N(t, y) axa?ye e PNe ??cNN F|? (t, y)| ?? ? ? ?N+ ? a e NAeitH ?x PNe? NA??
cN N F
|?N(?t, x)| ?+ ?? ? ? ?a e NAeitHP e? NA ?y N ??
cN N F
56
which means
? ? ?
(1) 2
dxdy ??N (t, x?, y)? ?N(t, x)??N(t, y)
?
1 ?? ? ? ?2 ? ? ? ?2. N e NAeitH ?e? NA?? 1 ? ?+ ?N 1/2e NAeitHe? NA?
c2
?
N2N
1 ? F c
2
? ? ? N
N
2 1 ? F
= N 2e?B? 1/2 ?B
2 2 red
? ?
F + N e ? ? .c N c2 red FN NN
Applying lemma (2.23) and (2.24), we get
? ?? ?(1) 2dxdy ?N (t, x, y)? ?N(t, x)??N(t, y)?
1 ??N ?B ??2 ? ?. e ?red F + ?1 ?N 1/2 2?3/2 red ?N N F
?
. N?? 2exact ? ?approx ?F .
Finally, by the appendix in [Kuz15b] and remark 1.4 in [RS09], we have provided
both a derivation of the focusing Schr?dinger equation and a rate of convergence
of the N body interacting bosonic system toward mean field for ? in the range
0 < ? < 1 .
6
Remark 2.25. One should note we could only use part one of Theorem 2.1 for our
derivation of the focusing NLS since we are considering evolution of coherent states,
i.e. k(0, ?) = 0.
2.5.2 Pickl?s Method
Following closely the presentation in [Pic10], we consider the quantities
57
Definition 2.26. Let ? ? L2(R3)
(a) For each 1 ? j ? N we define the projectors p?j : L2(R3N) ? L2(R3N) and
q?j : L
2(R3N)? L2(R3N) given by
?
p?? (x , . . . , x ) = ?(x ) ??(x?j N 1 N j j)?N(x1, . . . , x
?
j, . . . , xN)dxj
and q?j = 1? p
?
j respectively.
(b) Furthermore, for any 1 ? k ? N we defined P ?k : L2(R3N)? L2(R3N) given by
??N
P ?k := (p
?
` )
1?a`(q?` )
a`
a?Ak `=1
where
?N
Ak = {(a1, . . . , aN) | ai ? {0, 1} and ai = k}
i=1
(c) Assume 0 < ? ? 1. Let us define the function m? : {1, . . . , N} ? R?0 given by
?????k/N?, for k ? N?,
m?(k) := ????1, otherwise
58
and a corresponding functional ??N : L2(R3N)? L2(R3)? R?0 given by
?N
??N(?N , ?) := ??N , m?(k)P
?
j ?N?
k=1
= ?? , m??,?? ? = ? (m??,?)1/2N N ?N ?2L2 .x
For convenience, we shall use the notation ?N instead of ?1N .
As a direct consequence of the definitions, one could verify the following
?N(?N , ?) = ? q?1 ? 2 ?N ?L2 ? ?N(?N , ?)x
for 0 < ? < 1. Again, by the definition, we could derive an error bound for the rate
of convergence of the one particle density towards the mean field limit
? ?
? (1)?N ? |????| ? ? ?op ? p?? ?21 N L2 ? 1?? |????| ?opx
+ 2? q?? ? ? p?? 1 N L2 ? 1? ?
? 2
N L2 + ? q ? ? 2x x 1 N Lx
? ?? p? 2 ? ? ?1?N ?L2 ? 1 + 2? q1 ?N ?L2? p1?N ?x L2x x
+ ? q? 21 ?N ?L2x
. ? q?? ?21 N L2 + ? q
?
1 ?N ?L2 .x x
Since |????| is a rank one projection operator, by remark 1.4 in [R?S09] the trac?e
norm is two times the operator norm, i.e., ? (1) ? | ?? | ? ? (1) ?2 ?N ? ? op = Tr ??N ? |????|?.
59
Then it follows from the above estimates
??? ? ?(1)Tr ?N,t ? | ??t???t|? . ??N(?N , ?t) + ??N(?N , ?t). (2.29)
Thus, to obtain a rate of convergence for the error it suffices to prove an estimate
for ??N(?N , ?). Let us now state the main theorem in [Pic10] which we will use to
derive the focusing NLS:
Theorem 2.27. Assume 0 < ?, ? < 1 and vN satisfies the same conditions as before.
Assume for every N ? N there exists a solution to the linear N-body Schr?dinger
equation ?N(t, x) and a L? solution of the mean field equation ?t on some interval
[0, T ) with T ? R>0 ? {?}. Then for any t ? [0, T )
(? t )
??N(?N,t, ?t) ? exp C 2 ?[ ( v???s ?L?(dx) ds ?N)(?N,00 t ]
, ?0)
+ exp C ?? ?2 ?s ??v s L?(dx) ds ? 1 sup K N
0 0?s?t
where ? = 1? max{1? ?? 4?, 3? ? ?,?1 + ?+ 3?}, Cv is some constant depending2
only on v and
K? := Cv(??|?|2 ?L2(dx) + ?? ?L?(dx) + 1)?? ?L?(dx).
Proof of Theorem 2.5. Note if ?N(0, x) = ??N then ?? ?NN(? , ?) = 0. Hence com-
60
bining with our above decay result for ? satisfying the Hartree equation
?
1
?t????+ ( v)|?|2? = 0
i
we have that
[ ( ? t ) ]
??N(?N,t, ?t) ? exp Cv ?? ?2 ds ? 1 sup K?sN ??s L?(dx)
0 0?s?t
where
K?t = Cv(??|?t|2 ?L2(dx) + ??t ?L?(dx) + 1)??t ?L?(dx)
. (? |?x?t|2 ?L2(dx) + ??t???t ?L2(dx) + ??t ?L?(dx) + 1)??t ?L?(dx)
. (??x?t ?L?(dx)??x?t ?L2(dx) + ??t ?L?(dx)??2x?t ?L2(dx) + ??t ?L?(dx) + 1)??t ?L?(dx)
1
. .
1 + t3/2
Thus, it follows
??? ? ?(1)Tr ?N,t ? |?t???t|?? . ?? (? , ? ) . N ??/2N N,t t .
By remark 1 in [Pic10], we see there is a choice of ? such that ?? < 0 when 0 < ? <
1 .
6
61
Chapter 3: Uniform in N Global Well-posed of the Time-Dependent
HFB system in R1+1
Based and the discussion in the introduction and (1.6), it is heuristically clear
that there is no critical scaling when d = 1, 2. To be more specific, for d ? 2, the
coupling constant for the interaction of the rescaled system is inversely proportional
to the number of particles which means the mean-field scaling is more prominent
than the short-range scaling effect. Thus, we do not expect to see any short scale
correlation effects. One of the purposes of this chapter is to offer a preliminary
step to a rigorous demonstration of the fact that there is no development of short
scale correlation structure when d = 1 for the effective description by showing the
effective equations are well-posed for all ? > 0. The case d = 2 for all ? > 0 is still
open.
Another reason to consider the entire range of ? in R1+1 is inspired by the
Lieb-Liniger model [LL63, Lie63] which is a 1D model for a system of ultradcold
Bose particles inside the torus endowed with a pairwise interaction given by the
repulsive ?-function, i.e. the Lieb-Liniger Hamiltonian for the N -particle Bose gas,
62
in appropriate units, is
?N 2 ?
HN = ?
?
+ 2c ?(xi ? xj) (3.1)
?x2
i=1 i 1?i<j?N
where c ? 0 denotes the repulsion strength. More specifically, one can view the
Lieb-Liniger model on R as a heuristic endpoint case of our analysis of the dynamics
generated by (1.2) in the weak-coupling limit regime, c? 0.
Our interest in the model is twofold. From a phsyics point of view, the model
has an important feature of being exactly solvable in the ground state with com-
putable spectrum. Moreover, the recent advancement in the techniques of trapping
and cooling atoms has opened up a variety of possible experimental studies for ul-
tracold Bose gases that are effectively one-dimensional; for a comprehensive survey
on the subject, we refer the reader to [BDZ08]. Hence a firm mathematical under-
standing of the dynamics generated by the Hamiltonian (3.1) is an indispensable
theoretical tool to suggest further experimental investigation of certain 1D proper-
ties for ultracold Bose gases. In particular, an effective description of the dynamics
generated by the Lieb-Liniger model would provide a simplified way to analyze the
dynamics of these effectively one-dimensional Bose gases. From a mathematical
perspective, the Lieb-Liniger model on R is the simplest instance of a many-body
quantum mechanical model with interaction given by the ?-potential. Up to date,
there is no rigorous results on the effective description of the evolution of any quan-
tum system with ?-interaction.
63
3.1 Notations and Main Statements
Let us indicate some of the notations adopted by this chapter.
Notations. Following [GM17], we use the notations
1 ? 1 ?
S? := ??x + ?y and S := ??x ??y
i ?t i ?t
to denote the two Schr?dinger-type differential operators. Moreover, unless spec-
ified, x, y ? R, which means ?x = ?xx and, similarly, ?y = ?yy. The two types
of semilinear equations, corresponding to the above operators, considered are the
inhomogeneous von-Neumann Schr?dinger equation
S?? = F (3.2)
and the inhomogeneous Schr?dinger equation
( )
1
S + vN(x? y) ? = F (3.3)
N
where vN(x) = N?v(N?x) and v ? L1(R) ? C?(R).
Remark 3.1. We assume v is non-negative and even in our presentation since our
prime interest is in studying vN ? c?. However, it should be noted that v can
be asymmetric and negative when we study the local well-posedness of the time-
dependent HFB; whether these facts have interesting physical consequences will not
be explored in this chapter.
64
Next, let us define the space for the initial data. For every s > 0, we define
the space
X s = {(?,?,?) ? Hs ?HsHerm ?Hssym}
with Hs being the Sobolev space Hs(R), HsHerm the Sobolev space Hs(R2) restricted
to functions ? such that ?(x, y) = ?(y, x), and Hssym the Sobolev space space Hs(R2)
restricted to functions ? such that ?(x, y) = ?(y, x). More specifically, X s is en-
dowed with the norm
? (?,?,?) ?X s := ? ??x?s? ? 2 2L2(R) + ? (??x? ? 1 + 1? ??y? )s/2? ?L2(R2)
+ ? (?? ?2 ? 1 + 1? ?? ?2)s/2x y ? ?L2(R2).
When the context is clear, we use the symbol ?? s 2x,y? in place of (??x? ? 1 + 1 ?
?? ?2)s/2y . Furthermore, we study the local well-posedness of our equations in some
Strichartz spaces, which are mixed Lp spaces endowed with the norm
(? )1/qT (? )q/r
?u ? rLq [0,T ]Lr(dx)Ls(dy) := dt dx ?u(t, x, ? ?Ls(dy)
0
where the triplet (q, r, s) satisfies some Strichartz admissible conditions, which will
be made clear in the following sections. We also adopt the equivalent notation
Lq(dt)Lr(dx)Ls(dy), with the implicit assumption that it depends on T , in place of
Lq([0, T ])Lr(dx)Ls(dy).
65
The hyperbolic trigonometric integral operators introduced in ?1 are defined
as follows
1 1
sh(k) := k + k ? k? ? k + k ? k? ? k ? k? ? k + . . .
3! 5!
1 1
ch(k) := ? + p(k) := ? + k? ? k + k? ? k ? k? ? k + . . .
2! 4!
where ? indicates composition of operators. The symmetric kernel of k, k(t, x, y) =
k(t, y, x), is called the pair excitation function. The following are some useful
trigonometric identities
sh(2k) = 2 sh(k) ? ch(k), ch(2k) = ? + 2sh(k) ? sh(k) (3.4a)
ch(k) ? ch(k)? sh(k) ? sh(k) = ?. (3.4b)
Lastly, we use the usual conventional notation
??(t, x) := ?(t, x, x)
to define the restriction of ? to the diagonal of the plane.
Remark 3.2. We adopt the usual convention of identifying the collection of Hilbert-
Schmidt integral operators on L2(Rd), denoted by L2, with their integral kernels in
L2(Rd ? Rd).
Main Statement and Structure. Let us state the main results of the chapter
Theorem 3.3 (Uniform in N Local Well-Posedness of the time-dependent HFB in
66
R1+1). Suppose ? > 0 and R > 0. Then there exist T = T (?,R) > 0, ? = ?(?),
both independent of N , and a corresponding spacetime function space XT , depending
only on T and ?, such that for any given
(? ,? ,? ) ? {(?,?,?) ? X ?0 0 0 | ? (?,?,?) ?X? < R},
there exists a unique solution to the time-dependent HFB equations (1.33) with initial
data (?0,?0,?0) satisfying (?t,?t,?t) ? C([0, T ]? X ?) ?XT .
Remark 3.4. The proof is based on Picard-Lindel?f theorem or sometimes known
as the Banach fixed-point method. We refer the reader to ?3.7 for the definition of
the function space XT and Theorem 3.36 for the a-priori estimates involved in the
proof of Theorem 3.3.
Remark 3.5. Given ? > 0, we will later see that the choice of ? must satisfy the
conditions 1??? > 0 and 0 < ? < 1 ; see Remark 3.15 and Remark 4.10. Informally,
2
this means when ? is large we can only have uniform control of low Sobolev norms.
Ideally, we would like to choose ? = 0, but the nonlinearity requires us to choose
? > 0; see Remark 3.16. Hence an interesting point to observe is the competition
between large ? > 0, which requires low regularity of the initial condition, and the
non-linearity, which requires some regularity.
Remark 3.6. The choice of the Banach space XT is sufficient, maybe necessary, for
our analysis of the time-dependent HFB equations. Heuristically, the space XT is
an intersection of Strichartz spaces, which capture evolution due to the Schr?dinger-
type operators, plus a trace-type space, which captures the interactions coming from
67
the nonlinearity of the coupled equations.
Corollary 3.7 (Uniform in N Global Well-Posedness of the time-dependent HFB
in R1+1). Suppose ? > 0 and R > 0. Then for any
(?0,?0,?0) ? {(?,?,?) ? X ? | ? (?,?,?) ?X? + ? (?x?,?x,y?,?x,y?) ?X? < R},
the corresponding local solution to the time-dependent HFB equations (1.33) given
by Theorem 3.3 extends globally with (?t,?t,?t) ? C([0,?)? X ?)?X?,loc (See ?8
for definition of X?,loc).
Remark 3.8. To prove the global well-posedness it suffices to prove that the following
estimates
? ??x???(t, ?) ?L2(dx) . 1
? ?? ?x,y? ?(t, ?) ?L2(dxdy) . 1
? ?? ?x,y? ?(t, ?) ?L2(dxdy) . 1
hold uniformly in t and N , which is a consequence of the conservation laws proved
in [GM13a]. See ?3.7.3.
Remark 3.9. Our result does not require the condition V 2 ? C(I ? ?) which is a
standard assumption used to treat the multiplicative operator V as a perturbation
of the non-interacting case. More precisely, since we are working with V (x) =
68
N??1v(N?x), then we see that
?
N??2 dx |v(x)|2|f(N??x)|2 = ?V f ?2L2(R) . ? f ? ?2 + ? f ?2L2(R) L2(R)
can only be true uniformly in N provided ? < 2. Nevertheless, in the one di-
mensional setting, V can still be considered as a perturbation even without the
condition.
Now let us explain a bit the structure of the paper. In ?3.2 and ?3.3, we develop
estimates that are essential for closing the iteration scheme of the ? equation. The
main results of those two sections necessary for the proof of Theorem 3.36 are
Proposition 3.19, Proposition 3.20 and Propostion 3.21. Likewise, from ?6 and ?7,
we will need Proposition 3.32, Corollary 3.33, Proposition 3.34 and Remark 4.10 to
close the estimate for the ? equation. Finally, in ?3.7 we prove a-priori estimates
that are necessary for us to establish the local well-posedness theory for the time-
dependent HFB equations then extend the result to a global well-posedness result
under further assumption on the initial data.
3.2 Estimates for the Homogeneous ? Equation
The main purpose of this section is to prove (3.12) for the von-Neumann
Schr?dinger equation
1 ?
? + [??,?] = 0 (3.5)
i ?t
69
for arbitrarily smooth initial condition ?(0, x, y) = ?0(x, y). The two key ingredients
involved in the proof of Corollary 3.14 are the collapsing estimate and the sharp trace
theorem1.
Let us adopt the following convention for our spacetime Fourier transform:
the spacetime Fourier transform of a Schwartz function f ? S(R?Rd), denoted by
f? , is defined to be
?
f?(?, ?) = dtdx e?i(?t+??x)f(t, x). (3.6)
Likewise, the Fourier transform f? of some function f ? S(Rd) is defined by
?
f?(?) = dx e?i??xf(x) (3.7)
with corresponding inversion formula
?
1
f(x) = d? ei??xf?(?). (3.8)
(2?)d
Remark 3.10. The reader should be aware of our attempt to keep track of the values
of the fractional derivatives in this section. Keeping a record of these values allows
us to show that the mapping used when implementing the fixed-point argument is
indeed a self map.
Now, using the spacetime Fourier transform, we can establish the following
1Here, sharp trace theorem refers to the statement: for any hyperplane ? ? Rd and s > 12 , the
1
trace operator T : Hs(Rd)? Hs? 2 (?) is bounded.
70
collapsing estimate for the solution to (3.5).
Proposition 3.11 (Collapsing Estimate). Suppose ? is a solution to S?? = 0, then
1
?? 2x??(t, x) ?L2(dtdx) . ??0 ?L2(dxdy). (3.9)
Proof. Taking the spacetime Fourier transform of ? yields
? ?
?? (t, x) = dtdx e?i?t?i??x? ? ??(t, x) = dtdxdy e
?i?t?i??x
? ?(x? y)?(t, x, y)
1 1
= d?dt e?i?t??(t, ? ? ?, ?) = d?dt eit(???|???|2+|?|2)? ??2 2 0(? ? ?, ?)(2?) (2?)
1
= d? ?(? + |? ? ?|2 ? |?|2)??0(? ? ?, ?)
(2?)2 ( )
1 ?2 ? ? ?2 + ?
= ?? , .
8?2|?| 0 2? 2?
?1
Taking the L2?,?(R ? R) norm of ? 2x?? and applying Cauchy-Schwarz gives us the
estimate
?
?1
d?d? ||? 2x| 2??(t, x)(?, ?)| . ?? ?20 L2(dxdy)
since
?
sup d? ?(? + |? ? ?|2 ? |?|2)|?| . 1.
?,|?|
Utilizing the above collapsing estimate, we prove a couple perturbed version
71
of the collapsing estimate which will be crucial for the chapter.
Lemma 3.12. Suppose ? is a solution to S?? = 0. Then for any ? > 0 we have
the estimate
1
??? ? ?? +?x??(t, x) L?(dt)L2(dx) . 2x,y ?0 ?L2(dxdy). (3.10)
Proof. For any fixed t, it follows from the sharp trace theorem and the conservation
of mass we have that
1
??? +?2x??(t, x) ?L2(dx) . ??x,y ?0 ?L2(dxdy).
Proposition 3.13. Suppose ? is a solution to S?? = 0. Then for any 0 < ? < 12
and 0 < ?? < 1?? there exist q = q(?) and ? = ?(?) such that the following estimate
2
holds
1 ?
?? ??2x ??(t, x) ?Lq(dt)L2(dx) . ???x,y?0 ?L2(dxdy). (3.11)
Proof. Interpolating2 estimates (3.9) and (3.10), we obtain the estimate
1 ?
?? ??2x ?? ? ?Lq(dt)L2(dx) . ??x,y?0 ?L2(dxdy)
2c.f. Chapter V ?4 in [SW71].
72
with ? given by
( 1 )+ ?
? = 2 ?1 ? .? ?
2
Moreover, checking the arithmetic, we see that
1? 2?
q = 1 ? 2? ?? ? ?
2
since ?? < 1 ? ?.
2
Corollary 3.14. Suppose ? is a solution to S?? = 0. Then for any 0 < ? < 1 there2
exists q = q(?) such that the following estimate holds
1 3
? |? | ?? ( ??)?x 2 ??(t, x) ? 2Lq(dt)L2(dx) . ??x,y ?0 ?L2(dxdy). (3.12)
Proof. Fix ?. Choose ? to be
1? 2? 1 + ?? 2 3? = =
2(5? 2?) 1
= ? ?.
? ? 2
2
To avoid confusion, the reader should note that ? and ? here correspond to ?? and ?
in Proposition 3.4. Hence by the previous proposition, there exists q(?) given by
2
q =
(2? ?)(1 ? ?)
2
such that estimate (3.12) holds.
73
Remark 3.15. For convenience, we shall henceforth denote the quantity (3 ? ?)? by
2
?.
Remark 3.16. Heuristically, we want the estimate
1
? ??(t, x) ? 2L2(dt)L?(dx) . ??x??(t, x) ?L2(dtdx) . ??0 ?L2(dxdy)
but the estimate is a false endpoint of the Gagliardo-Nirenberg estimate. However,
by using the above corollary and the fact that we are working on a finite interval
[0, T ], we get that
1
? ??(t, x) ?L2(dt)Lp(dx) . ? |? | ??x 2 ??(t, x) ?L2(dtdx)
1
. T some power? |? | ??x 2 ??(t, x) ?Lq(dt)L2(dx)
. T some power? |? ?x,y| ?0 ?L2(dxdy).
We will elaborate more on this point in the next section.
Next, let us establish the homogeneous Strichartz estimate for the linear op-
erator S?.
Proposition 3.17 (Non-Endpoint Strichartz). Suppose ? is a solution to S?? = 0
with initial condition ?0 and (k, `) is an admissible pair, i.e.
2 1 1
+ = (3.13)
k ` 2
74
where (k, `) ? (2,?]? [2,?]. Then it follows
? eit(?x??y)?0 ?Lk(dt)L`(dx)L2(dy) . ??0 ?L2(dxdy). (3.14)
Proof. The proof is essentially the same as the standard non-endpoint Strichartz
estimate using both the TT ? principle and Christ-Kiselev lemma. See ?2.3 in
[Tao06].
3.3 Estimates for the Inhomogeneous ? Equation
Let us now consider the inhomogeneous ? equation
S?? = F (3.15)
where F is smooth. The main purpose of this section is to obtain collapsing esti-
mates similar to estimates proven in Proposition 6.1 and Corollary 3.14 but for that
inhomogeneous equation. The main results of this section are Proposition 3.19 and
Proposition 3.20.
Remark 3.18. For the purpose of obtaining estimates for (3.15), we do not need
to assume F to have any symmetry. That being said, in order for our iteration
scheme to preserve the symmetry ?(x, y) = ?(y, x), i.e. stay in the designated space
which we have specified in Theorem 3.3, it is wise to assume F is skewed symmetric,
i.e. F (x, y) = ?F (y, x). Likewise, the forcing term with respect to the ? equation
should also satisfy F (x, y) = F (y, x). Henceforth, we assume the forcing F for each
75
of the three equations has the correct symmetry.
Observe the solution to the inhomogeneous equation can be written as
? t
?(t, x, y) = eit(?x??y)? (x, y) + i ei(t?s)(?x??y)0 F (s, x, y) ds (3.16)
0
which then yields
? t
? (t, x) = [eit(?x??y)? ](x, x) + i [ei(t?s)(?x??y)? 0 F ](s, x, x) ds (3.17)
0
Then it follows from the estimate (3.9) that
? T
1 1
? |? | ? ? i(t?s)(?x?? )x 2 y? L2(dtdx) . ??0 ?L2(dxdy) + ? |?x| 2 [e F ](s, x, x) ?L2(dtdx) ds
0
. ??0 ?L2(dxdy) + ?F ?L1[0,T ]L2(dxdy).
Hence we have obtained the following proposition
Proposition 3.19. Suppose ? solves S?? = F , then we have
1
? |?x| 2?? ?L2(dtdx) . ??0 ?L2(dxdy) + ?F ?L1[0,T ]L2(dxdy). (3.18)
The following is a perturbed version of the above proposition.
76
Proposition 3.20. Suppose ? solves S?? = F , then for every 0 < ? < 1 we have2
(
1
? |? | ??? ? . T some powerx 2 ? L2(dtdx) ? |?x,y|??(t, x,)y) ?L?[0,T ]L2(dxdy)
+ ? |? ?x,y| F ?L1[0,T ]L2(dxdy) . (3.19)
Proof. Applying Corollary 3.14 to (3.17) yields
1
? |? | ??x 2 ??(t, x) ?L2((dtdx)
. T some power? ? |?
?
x,y| ?(t, x, y) ?L?[0,T ]L2(dxdy)
T )
1
+ ds ? |? | ??[ei(t?s)(?x??2 y)x F ](s, x, x) ?Lq [0,T ]L2(dx)
0
. T some power
( )
? |? |?x,y ?(t, x, y) ? ?L?[0,T ]L2(dxdy) + ??x,yF ?L1[0,T ]L2(dxdy)
where q = 2? 1? . Then by Remark 3.16 we obtain the desired estimate.(2 ?)( ?)
2
To conclude this section, let us state the inhomogeneous Strichartz estimate.
Proposition 3.21. Suppose ? is a solution to S?? = F with initial condition ?0
and (k, `) and (k?, ??) are an admissible pairs (see (3.13)). Then it follows
??(t, x, y) ?Lk(dt)L`(dx)L2(dy) . ??0 ?L2(dxdy) + ?F ? k?? (3.20)L (dt)L??? (dx)L2(dy)
77
and
? |? ?x,y| ?(t, x, y) ? ? ?Lk(dt)L`(dx)L2(dy) . ? |?x,y| ?0 ?L2(dxdy) + ? |?x,y| F ?Lk?? (dt)L??? (dx)L2(dy)
(3.21)
where (k??, ???) denotes the H?lder conjugates of (k?, ??).
3.4 Application of the Inhomogeneous ? Estimates
The purpose of this section is to develop estimates which we will later use
in the proof of our main theorem in ?3.7. However, as an immediate application
of the previous two sections, we are now ready to consider the uniform in N local
well-posedness of the following Hartree-Fock equation
1 ?
? = [?? vN ? ??,?] (3.22)
i ?t
or equivalently
S??(t, x, y) = [vN ? ??(t, x)? vN ? ??(t, y)]?(t, x, y) = F (3.23)
in some Strichartz-type space X equipped with the norm
1 1
?? ?X := ? |? | ??x 2 ??(t, x) ?L2[0,T ]L2(dx) + ? |?x| 2??(t, x) ?L2[0,T ]L2(dx) (3.24)
+ ? ?? ??x,y ?(t, x, y) ? ?L?[0,T ]L2(dxdy) + ? ??x,y? ?(t, x, y) ?L4[0,T ]L?(dx)L2(dy)
78
where ? is sufficiently small, say ? < 1 .
5
The uniform in N local well-posedness is proven using the standard Banach
fixed-point argument. More precisely, we close the estimate for (3.22) in X. In
particular, we close the estimate for each of the three norms indicated in (3.24).
However, by Proposition 3.19 and Proposition 3.20, it suffices to consider estimates
for the corresponding forcing terms.
First, let us estimate ?F ?L1[0,T ]L2(dxdy). By H?lder?s inequality, we see that
?F ?L1[0,T ]L2(dxdy) . ? vN ? ??(t, x)?(t, x, y) ?L1[0,T ]L2(dxdy)
. ? vN ? ??(t, x) ?L2(dt)Lp(dx)??(t, x, y) ?L2(dt)Lr(dx)L2(dy)
where we made the choice p = 1/? and r = 2(1 ? 2?)?1. Then by Gagliardo-
Nirenberg-Sobolev inequality, Young?s convolution inequality and H?lder inequality,
in the time variable, we obtain the estimate
1
?F ? ??2L1[0,T ]L2(dxdy) . ? vN ? ?x ??(t, x) ?L2(dtdx)??(t, x, y) ?L2(dt)Lr(dx)L2(dy)
1
. T some power?? ??2x ??(t, x) ?L2(dtdx)??(t, x, y) ?Lq(dt)Lr(dx)L2(dy)
where q = 2??1. Note that q is chosen so that (q, r) is a 1D Strichartz admissible
pair. Hence by interpolation, we see that
?F ? some power 2L1[0,T ]L2(dxdy) . T ?? ?X .
79
1
Likewise, we can show that ? |?x,y| 2F ?L1[0,T ]L2(dxdy) also closes.
Next, let us estimate ? |? ?x,y| F ?L1[0,T ]L2(dxdy). Using the classical Kato-Ponce
inequality, sometimes refers to as ?fractional Leibniz rule", we see that
? |? ?x,y| F ?L1[0,T ]L2(dxdy) . ? |?x,y|?[vN ? ??(t, x)?(t, x, y)] ?L1[0,T ]L2(dxdy)
. ? vN ? ??(t, x) ?L2(dt)Lp(dx)? |?y|??(t, x, y) ?L2(dt)Lr(dx)L2(dy)
+ ? v ? |? |?N x ??(t, x) ?L2(dt)Lp?(dx)??(t, x, y) ?L2(dt)Lr?(dx)L2(dy)
+ ? vN ? ??(t, x) ? ?L2(dt)Lp(dx)? |?x| ?(t, x, y) ?L2[0,T ]Lr(dx)L2(dy)
where p? = 2[(5 ? 2?)?]?1, r? = [?2 ? 5? + 1 ]?1 and p, r as defined above. Hence by
2 2
the same argument as above with q? = 2[(5 ? ?)?]?1 we see that
2
? |? |?x,y [vN ? ??(t, x)?(t, x, y)] ?L1[0,T ]L2(dxdy)
. T some power
1
? |? | ??x 2 ??(t, x) ? ?L2(dtdx)? |?y| ?(t, x, y) ?Lq(dt)Lr(dx)L2(dy)
+ T some power
1
? |?x| ??2 ??(t, x) ?L2(dtdx)??(t, x, y) ?Lq?(dt)Lr?(dx)L2(dy)
1
+ T some power? |?x| ??2 ??(t, x) ? ?L2(dtdx)? |?x| ?(t, x, y) ?Lq(dt)Lr(dx)L2(dy)
Again, note that (q?, r?) is an admissible pair which means the desired estimate holds
by interpolation.
Remark 3.22. The belove estimate is included in this section purely for the author?s
own organizational purposes. Hence the reader may skip it for now and refer back
to it in ?8.
80
Lastly, observe we have
??? ?x+yF ?L1[0,T ]L2(dxdy) . ??x+y[vN ? ??(t, x)?(t, x, y)] ?L1[0,T ]L2(dxdy)
. ? vN ? |? ?x| ??(t, x) ?L2(dt)Lp?(dx)??(t, x, y) ?L2(dt)Lr?(dx)L2(dy)
+ ? vN ? ??(t, x) ? ?L2(dt)Lp(dx)? |?x+y| ?(t, x, y) ?L2[0,T ]Lr(dx)L2(dy)
. T some power
1
? |? ??x| 2 ??(t, x) ?L2(dtdx)??(t, x, y) ?Lq? [0,T ]Lr?(dx)L2(dy)
+ T some power
1
? |? | ??x 2 ??(t, x) ? ?L2(dtdx)? |?x| ?(t, x, y) ?Lq [0,T ]Lr(dx)L2(dy)
where ?x+yF := 1(?xF +?yF ).2
Remark 3.23. Since similar calculations will be performed in ?8, then for convenience
we shall fix the values of p, q, r, p?, q?, r? as indicated above for a given ? in the remaining
of this chapter.
As a result of the above calculation, we obtain the following proposition
Proposition 3.24. Suppose ? solves (3.22) with Schwartz initial condition ?0 and
v ? L1(R). Then the following estimate holds
?? ?X . ? ?? ? some power 2x,y? ?0 ?L2(dxdy) + T ?? ?X .
Thus, there exists T0 > 0 such that for all 0 < T ? T0
?? ?X . ? ??x,y???0 ?L2(dxdy).
81
Similarly, we can show that
? ?t? ?X . ? ?? ?x,y? ?t?0 ?L2(dxdy) + T some power?? ?X? ?t? ?X
which again means there exists T0 > 0 such that
? ?t? ?X . ? ??x,y???t?0 ?L2(dxdy).
3.5 Homogeneous ? Equation
In this section we prove collapsing estimates for the linear Schr?dinger equa-
tions
1 ?
???x???y? = 0 (3.25)
i ?t
which we will need later. As mentioned in the introduction, one of the main dif-
ficulties in the analysis of equation (3.3) is that the Lp-norms of the potential
N?1vN(x ? y) are not uniformly bounded in N when p > 1 and ? arbitrarily
large since N?1? v (x ? y) ? ? N?1+?(1?
1 )
p
N p . More precisely, from Proposition
3.26, we see that the natural space to put the nonlinearity of equation (3.35) is
in L1[0, T ]L2(dxdy). In particular, when handling the term N?1vN(x? y)?(t, x, y)
from equation (3.3) in L1([0, T ]?L2(R2)), we see there is no way (at least no simple
way) to put the term N?1vN(x? y) in L1(d(x? y)). Thus, the purposes of ?6 and
?7 are to develop sufficient amount of tools to handle N?1vN(x?y)?(t, x, y) and all
82
the nonlinearity coming from the TDHBF equations.
One of the crucial tools for our analysis is the Xs,b spaces (sometimes called
the Bourgain spaces or dispersive Sobolev spaces) which is defined to be the closure
of the Schwartz class, St,x(R? R? R) with respect to the norm
?u ?Xs,b = ? (1 + |?|
2 + |?|2)s(1 + |? + |?|2 + |?|2|)bu?(?, ?, ?) ?L2(d?)L2(d?d?).
S
For this paper, s is always zero and we are only interested in defining the Xs,b
spaces for the operator S. Hence we dropped both the s and S labels from the
norm to simplify the notation. For instance, we have ?u ?Xb = ?u ?X0,b . We referS
the interested reader to ?2.6 in [Tao06] for a more complete introduction to these
spaces.
Same as the von-Neumann Schr?dinger equation, we first obtain a collapsing
estimate for the above equation.
Proposition 3.25. Suppose S? = 0 with Schwartz initial condition ?(0, x, y) =
?0(x, y) then
? p (?t,?x) ?(t, x, x) ?L2(dtdx) . ??0 ?L2(dxdy). (3.26)
where p(?t,?x) is a pseudodifferential operator with symbol p?(?, ?) = |? + |?|2|1/4.
Proof. Let us begin by taking the spacetime Fourier transform of the trace of ? to
83
get
? ?
?(?t, x, x) = ? dtdx e
?i(?t+??x)?(t, x, x) = dtdxdy e?i(?t+??x)? ?(x? y)?(t, x, y)
= d?dtdxdy e?i(?t+(???)?x+??y)? ?(t, x, y) = d?dt e
?i?t??(t, ? ? ?, ?)
1
= d? ?(? + |? ? ?|2 + |?|2)??0(? ? ?, ?).
(2?)2
Applying Cauchy-Schwarz inequality yields the following estimate
?
d?d? |(? + |?|2)1/4?(?t, x, x)(?, ?)|2 . sup |I(?, ?)|??0 ?2L2(dxdy)
?,?
where
? ?
I(?, ?) := ? + |?|2 d? ?(? + |? ? ?|2 + |? + ?|2).
Observe, we have the identity
? ? ? ?
?(? ? ?? ? |?|2) + ?(? + ?? ? |?|2)
d? ?(? + |? ? ?|2 + |? + ?|2) = ? d?
R 4 ?? ? |?|2
? 1= .
2 ?? ? |?|2
Thus, it follows
?
d?d? ||? + |?|2|1/4?(?t, x, x)(?, ?)|2 . ?? 20 ?L2(dxdy).
84
Unfortunately, the homogeneous derivative p(?t,?x) of the restriction of ? to
the diagonal is not of any immediate use to our studies of the nonlinear coupled
equations. Since the nonlinearity in time-dependent HFB involves trace of ?, we
need estimates that will allow us to control the restricted ? by the spacetime deriva-
tive p(?t,?x) of the restriction of ?(t, x, y) to the diagonal. One such estimate is
given by the following proposition.
Proposition 3.26. Suppose S? = 0, then we have
??(t, x, x) ?L4(dt)L2(dx) . ? p(?t,?x)?(t, x, x) ?L2(dtdx) (3.27)
Proof. We prove the above estimate using a TT ? argument. Consider T : L2t,x ?
L4 2tLx defined by
( )? ?
F? 1 F? (?, ?)
(TF )(t, x) = := d? ei(??x+?t)
|? + |?|2|1/4 2? |? + |?|2|1/4
then we see that 4/3TT ? : L 2 4 2t Lx ? LtLx is given by
( )? ( )?
F?
TT ?F = = F ? 1 =: F ?K.
|? + |?|2| 12 | 1? + |?|2| 2
By triangle inequality and Plancherel, we obtain the estimate
? ?
?K ? F (t, ?) ?L2(dx) ? ds ? ? ?
1
K?(t s, ?)F? (s, ?) L2(d?) . ds ? F? (s, ?) ?L2(d?)
| 1t? s| 2
85
since we have
??? 2??? ? i|?| (t?s) ?
?
|K?(t? s, ?)| = ei?(t?s) e 1 d? ??
?? |? | 2 ? 1. |t? s| 12
which is independent of ?. Thus, it follows
???? ??? ? F? (s, ?) ? ?2?TT F ? L (d?) ?L4(dt)L2(dx) . ?? ds 1 ? .?? |t? s| 2 ?
L4(dt)
Now, apply Hardy-Littlewood-Sobolev inequality n = n?n+?, with n = 1, p = 4/3
p q
and q = 4 we have that
????? ?? ? ? F? (s, ?) ? ?L2(d?) ?ds | ? | 1 ?? . ?F ?L4/3(dt)L2(dx)?? t s 2
L4(dt)
which means TT ? is a bounded operator. Hence it follows from the TT ? principle
that T is also a bounded operator, i.e.
?TF ?L4(dt)L2(dx) . ?F ?L2(dtdx)
or equivalently
?F ? . ? |? + |?|2|1/4L4(dt)L2(dx) F? (?, ?) ?L2(d?d?).
As an immediate corollary of Proposition 3.26, we have that
86
Corollary 3.27. Suppose ? solves S? = 0, then for every 0 < ? < 1 we have
???x?(t, x, x) ? ?L4(dt)L2(dx) . ??x+y?0 ?L2(dxdy) (3.28)
where ?x+y? := 1(?x? +?y?).2
Proof. If S? = 0, then S?x+y? = 0. Applying the previous estimate, we obtain the
estimate
? (?x+y?)(t, x, x) ?L4(dt)L2(dx) . ? p(?t,?x)(?x+y?)(t, x, x) ?L2(dtdx)
. ??x+y?0 ?L2(dxdy).
Noting the identity
(? 1x+y?)(t, x, x) = ?x (?(t, x, x)) , (3.29)
2
we get the estimate
??x?(t, x, x) ?L4(dt)L2(dx) . ??x+y?0 ?L2(dxdy).
Interpolating the above estimate with the estimate
??(t, x, x) ?L4(dt)L2(dx) . ??0 ?L2(dxdy)
yields the desired result.
87
Let us also record the following non-endpoint Strichartz estimate for the ho-
mogeneous ? equation:
Proposition 3.28 (Non-endpoint Strichartz). Suppose ? is a solution to S? = 0
with initial condition ?0 and (k, `) is an admissible pair as defined in Proposition
3.17. Then it follows
? eit?x?0eit?y ?Lk(dt)L`(dx)L2(dy) . ??0 ?L2(dxdy). (3.30)
Proposition 3.29. For any number 1+ > 1 and arbitrarily close to 1 there exists
? > 0 such that the following estimate holds
?F ? . T some power? 1+? ?F ?X 2 L2[0,T ]L1+ (dx)L2(dy). (3.31)
Proof. By Proposition 3.28 and Lemma 2.9 in [Tao06], we have the estimate
?F ?L4[0,T ]L?(dx)L2(dy) . ?F ? 1+? (3.32)X 2
for all ? > 0. Moreover, from (3.32) we also get the dual estimate
?F ? ? 1?? . ?F ? 1/4L4/3[0,T ]L1(dx)L2(dy) . T ?F ?X 2 L2[0,T ]L1(dx)L2(dy). (3.33)
By linearly interpolating (3.33) with
?F ? ? 1+1 = ?F ?X 2 2 L2[0,T ]L2(dx)L2(dy)
88
yields
?F ? . T some power? 1+? ?F ?L2[0,T ]L1+X 2 (dx)L2(dy)
for ?? < ? < 1 and some number 1+ depending on ?. In particular, for any number
2
1+ arbitrarily close to 1 we can choose ? sufficiently small such that (3.31) holds.
Remark 3.30. Let us make the observation: since
|? + ?|2 + |? ? ?|2 = 2|?|2 + 2|?|2 (3.34)
then we also have the estimate
?F ? . T some power? 1+? ?F ?X 2 L2[0,T ]L1+ (d(x?y))L2(d(x+y)).
3.6 Inhomogeneous ? Equation
The main result in this section is Corollary 3.33 which allows us to obtain
a collapsing-type estimate for equation (3.3) and essentially show that N?1vN?,
mentioned in the previous section, can be viewed as a uniformly in N perturbation
of equation (3.35).
Consider the inhomogeneous equation
S? = F (3.35)
89
then it follows from the Xs,b energy estimate3 and Proposition 3.29 that we have
??(t, x, x) ?L4(dt)L2(dx) . ??0 ?L2(dxdy) + ?F ?X? 12+?
. ?? ? + T some power0 L2(dxdy) ?F ? 2 1+L (dt)L (d(x?y))L2(d(x+y)).
Summarizing the above result we obtain the following proposition:
Proposition 3.31. Suppose ? solves S? = F , then we have
??(t, x, x) ? some powerL4(dt)L2(dx) . ??0 ?L2(dxdy) + T ?F ?L2(dt)L1+ (d(x?y))L2(d(x+y)).
(3.36)
Using the above proposition, we establish the following proposition:
Proposition 3.32. Suppose ? solves (3.3) with initial condition ?0. Then we have
??(t, x, x) ?L4(dt)L2(dx) . ??0 ?L2(dxdy) + ?F ? 1 . (3.37)X? 2+?
Proof. Since by Proposition 3.26 we have
??? ? ?eit?x? eit?y ? ??0 . ??0 ?L2(dxdy),x=y L2(dtdx)
then it follows from Lemma 2.9 in [Tao06]
?F ?L4(dt)L2(dx) . ?F ? 1X 2+?
3cf. [Tao06] section 2.6
90
for any ? > 0. In particular, applying the Xs,b energy estimate we get that
?? 1(t)? ? 1+? . ? vN?(t)? ? ? 1+? + ?F ? 1X 2 X 2 X? 2+? + ??0 ?L2(dxdy)N
where ?(t) is a time localization bump function. Applying Proposition 3.29, we see
that
1 ? 1v (x? y)? ? . T some powerN 1X? +? ? vN?(t)? ?N 2 N L2(dt)L1+ (d(x?y))L2(d(x+y))
1
. T some power? vN ? 1+
N L (d(x?y))
??(t)? ?L2(dt)L?(d(x?y))L2(d(x+y))
1
. some power
N1??+?/(1+
T ??(t)? ?
) L
4(dt)L?(d(x?y))L2(d(x+y))
1
. T some power? + ??(t)? ? 1 .N1 ?+?/(1 ) X 2+?
Hence for 1+ sufficiently close to 1 we are in the perturbative regime. This allows
us to absorb the contribution from the potential term 1 vN(x ? y)? when N isN
sufficiently large.
Using the above proposition we could show that
Corollary 3.33. Suppose ? solves (3.3) with initial condition ?0. Then for every
0 < ? < 1 we have
2
? |?x|??(t, x, x) ?L4(dt)L2(dx) . ? |? ?x+y| ?0 ?L2(dxdy) + ? |? ?x+y| F ?X? 12+? . (3.38)
91
Proof. Taking the spatial derivative ?x+y of (3.3) yields
( )
1
S + vN(x? y) ?x+y? = ?x+yF (3.39)
N
since [?x+y, N?1vN(x? y)] = 0. Hence by Proposition 3.32, we obtain the estimate
? (?x+y?)(t, x, x) ?L4(dt)L2(dx) . ??x+y?0 ?L2(dxdy) + ??x+yF ?X? 1 .2+?
Again, noting the identity (3.29), we obtain the estimate
??x?(t, x, x) ?L4(dt)L2(dx) . ??x+y?0 ?L2(dxdy) + ??x+yF ? ? 1+? . (3.40)X 2
Interpolating (3.37) with (3.40) yields the desired result.
Now, let us record some Strichartz estimates:
Proposition 3.34. Suppose ? is a solution to S? = F with initial condition ?0
and (k, `), (k?, ??) are Strichartz admissible pairs. Then it follows
??(t, x, y) ?Lk(dt)L`(dx)L2(dy) . ??0 ?L2(dxdy) + ?F ?Lk?? . (3.41)(dt)L??? (dx)L2(dy)
In particular, it follows
? |?x,y|??(t, x, y) ? ? ?Lk(dt)L`(dx)L2(dy) . ? |?x,y| ?0 ?L2(dxdy) + ? |?x,y| F ?Lk?? (dt)L??? .(dx)L2(dy)
(3.42)
92
Remark 3.35. Let us note that Proposition 3.34 also holds for solution to (3.3) when
N is sufficiently large. More specifically, by interpolation, we can show
1
1 2? |?x|?[vN(x?
T
y)]? ?L4/3[0,T ]L1(d(x?y))L2(d(x+y)) . ? ?? ? 41 ?? L [0,T ]L?(d(x?y))L2(d(x+y)).N N
(3.43)
Thus, for any ? > 0, we can choose ? = ?(?) so that 1? ?? > 0.
3.7 The time-dependent HFB System in 1D
In this section we prove the local well-posedness of our system of nonlinear
equations addressed in the introduction. First, let us recall the time-dependent HFB
equations in 1D
{ } ?
1 ? ??x1 ?t(x1) = ? ? dy {vN(x1 ? y)??(t, y)} ? ?t(x1) (3.44a)i ?t
? ? dy {vN(x1 ? y)(?t(y, x1)? ?t(y)?t(x1))?t(y)}
{ } ? dy {vN(x1 ? y)(?t(x1, y)? ?t(y)?t(x1))?t(y)}
1 ? ??x1 + ?x2 ?t(x1,?x2) (3.44b)i ?t
= ? ? dy {(vN(x1 ? y)? vN(x2 ? y))?t(x1, y)?t(y, x2)}
? ? dy {(vN(x1 ? y)? vN(x2 ? y))?t(x1, y)?t(y, x2)}
? ?dy {(vN(x1 ? y)? vN(x2 ? y))??(t, y)?t(x1, x2)}
+ 2 dy {(vN(x1 ? y)? vN(x2 ? y))|?t(y)|2?t(x1)?t(x2)}
93
{ }
1 ? ??x1 ?
1
?x2 + ? vN(x1 ? x2) ?t(x1, x2) (3.44c)i ?t N
= ? ? dy {(vN(x1 ? y) + vN(x2 ? y))??(t, y)?t(x1, x2)}
? ? dy {(vN(x1 ? y) + vN(x2y))?t(x1, y)?t(y, x2)}
? ?dy {(vN(x1 ? y) + vN(x2 ? y))?t(x1, y)?t(y, x2)}
+ 2 dy {(vN(x1 ? y) + vN(x ? y))|? (y)|22 t ?t(x1)?t(x2)}
The space XT is a Strichartz-type space equipped with a norm which is the sum of
the following norms
NT (?) := ? ??x???(t, x) ? ?L4[0,T ]L?(dx) + ? ??x? ?(t, x) ?L?[0,T ]L2(dx) (3.45a)
NT (?) := ? ?? ??x,y ?(t, x, y) ?L4[0,T ]L?(dx)L2(dy) (3.45b)
+ ? ?? ?x,y? ?(t, x, y) ?L?[0,T ]L2(dxdy)
1 1
+ sup ? |? | ?(t, x+ z, x) ? ??x 2 L2(dtdx) + sup ? |?x| 2 ?(t, x+ z, x) ?L2(dtdx)
z z
NT (?) := ? ?? ?x,y? ?(t, x, y) ?L4[0,T ]L?(dx)L2(dy) (3.45c)
+ ? ?? ?x,y? ?(t, x, y) ? ?L?[0,T ]L2(dxdy) + sup ? ??x? ?(t, x+ z, x) ?L4[0,T ]L2(dx).
z
Moreover, let us denote the space of functions (?t,?t,?t) where the above norms
are finite for any 0 ? T <? by X?,loc.
Let us present the main a-priori estimates of the chapter
Theorem 3.36. Suppose ?,? and ? solve (3.44a), (3.44b), and (3.44c) respectively
94
with Schwartz initial condition (?0,?0,?0). Then we have the following estimates
( )
NT (?) . ? ?? ??? ? + T some powerx 0 L2(dx) N (?)2T + NT (?) + NT (?) NT (?)
(3.46a)
N (?) . ? ?? ??? ? + T some power
( )
T x,y 0 L2(dxdy) (NT (?)
2 + NT (?)
2 + NT)(?)
4 (3.46b)
NT (?) . ? ?? ?x,y? ?0 ? some powerL2(dxdy) + T NT (?)N (?) + N (?)4T T . (3.46c)
In particular, there exists T0 such that for all T ? T0 we have that
NT (X) := N
2
T (?) + NT (?) + NT (?) . 1.
Similarly, the following estimates hold for the time derivative of ?,?,?, i.e.
?? ?? ??
N (? ?T t?) . ? ??x? ?t?? ? (3.47a)
t=0 L2(dx)
+ T some power? (?NT?(X)NT (?t?) + NT (?tX)NT (?))? ? ?
NT (?
?
t?) . ? ??x,y? ?t?? ? (3.47b)
t=0 L2(dxdy)
+ T some power? N? T (X? ) (NT (?tX) + NT (?)NT (?t?))
NT (?t?) .
?? ??x,y?? ??t?? ?? (3.47c)
t=0 L2(dxdy)
+ T some powerNT (X) (NT (?tX) + NT (?)NT (?t?))
95
which again means there exists T0 such that for all T < T0 we have
NT (?tX) := NT (? ?)
2
t + NT (?t?) + NT (?t?) . 1.
Indeed, for any i, j ? N we have the estimates
( )
N ?i?jT t x+yX . 1. (3.48)
Remark 3.37. The reader should note that the solution obtained from the Banach
fixed-point theorem is smooth if the initial data (?0,?0,?0) is sufficiently smooth.
Indeed, for each fixed N , one can show
( )
N ?i?j?kX . N some non-negative powerT t x y .
Despite the fact that the higher Sobolev norms are not uniformly bounded in N ,
each of the solutions has sufficient smoothness for us to apply the conservation laws
which we will state later in the section.
We split the presentation of the proof of the theorem into two subsections.
3.7.1 Proofs of Estimates (3.46b) and (3.46c)
Let us first consider equation (3.44b). Since the term (vN ? ??) ?? has already
been handled in ?5, it suffices to consider only the terms (vN ??) ? ? and (vN?) ? ?.
In particular, it suffices to consider just the derivative of the terms since any com-
96
putation for the derivatives will encompass the computation for the non-derivative
terms.
Let us first handle the term (vN ? ??) ? ?. By a direct change of variables, we
can rewrite the kernel composition as follows
? ?
(vN? ? ?)(x, y) = dw vN(x? w)?(x,w)?(w, y) = dz vN(z)?(x, x? z)?(x? z, y).
Then by Kato-Ponce inequality we obtain the following
? |? ?x,y| [(?vN?) ? ?] ?L1[0,T ]L2(dxdy)
? ? dz |vN(z)|? |? |
?
x,y [?(x, x? z)?(x? z, y)] ?L1[0,T ]L2(dxdy)
. ?dz |vN(z)|? |?
?
x| ?(x, x? z) ?L2[0,T ]Lp?(dx)??(x, y) ?L2[0,T ]Lr?(dx)L2(dy)
+ dz |vN(z)|??(x, x? z) ? ?L2[0,T ]Lp(dx)? |?x| ?(x, y) ?L2[0,T ]Lr(dx)L2(dy)
where p, r, p?, r? are the values stated in Remark 3.23. Applying Cauchy-Schwarz
inequality in the time variable gives us
? |? ?x,y| [(vN?) ? ?] ??L1[0,T ]L2(dxdy)
. T some power
1
?dz |vN(z)|? |? |
??
x 2 ?(x, x? z) ?L2(dtdx)??(x, y) ?Lq?(dt)Lr?(dx)L2(dy)
some power 1+ T dz |vN(z)|? |? | ??x 2 ?(x, x? z) ? ?L2(dtdx)? |?x,y| ?(x, y) ?Lq(dt)Lr(dx)L2(dy)
. T some powerNT (?)
2
97
where (q, r) and (q?, r?) are admissible pairs. Likewise, we have
? |? ?x,y| [(?vN ??) ? ?] ?L1[0,T ]L2(dxdy)
? ? dz |vN(z)|? |?
?
x,y| [??(x, x? z)?(x? z, y)] ?L1[0,T ]L2(dxdy)
. ?dz |vN(z)|? |?x|
??(x, x? z) ?L4[0,T ]L2(dx)??(x, y) ?L4/3[0,T ]L?(dx)L2(dy)
+ dz |vN(z)|??(x, x? z) ?L4[0,T ]L2(dx)? |?x|??(x, y) ?L4/3[0,T ]L?(dx)L2(dy)
1
. T 2NT (?)
2.
As for equation (3.44c), there are essentially three terms we need to estimate,
namely (vN ???)?, (vN?)?? and (vN?)??. Similar to the handling of the nonlinear
terms for the ? equation, it suffices to look at just the derivatives of the nonlinear
terms.
For the first term, observe we have
? |?x+y|?[(vN ? ??)?] ?L2[0,T ]L1+(d(x?y))L2(d(x+y))
. ? vN ? ?? ? ?L2[0,T ]L2+(dx)? |?x+y| ? ?L?[0,T ]L2(d(x?y))L2(d(x+y))
+ ? vN ? |? ?x| ?? ?L2[0,T ]L2+(dx)?? ?L?[0,T ]L2(d(x?y))L2(d(x+y))
1
. ? |? ??x| 2 ?? ? ?L2(dtdx)? |?x+y| ? ?L?[0,T ]L2(d(x?y))L2(d(x+y))
1
+ ? |?x| ??2 ?? ?L2(dtdx)?? ?L?[0,T ]L2(d(x?y))L2(d(x+y))
. NT (?)NT (?)
The terms F = (vN?) ? ? and ? ? (vN?) are handled similarly.
98
3.7.2 Proof of Estimate (3.46a)
Let us begin by stating the following Strichartz estimate
Proposition 3.38. Suppose ? is a solution to S? = F with initial condition ?0 and
let (k, `) be an admissible pair. Then it follows for all ? > 0 we have
? |? |?x ? ? ? ?Lk[0,T ]L`(dx) . ? |?x| ?0 ?L2(dx) + ? |?x| F ?L4/3[0,T ]L1(dx). (3.49)
It suffice to consider only (vN ? ??) ?? and (vN?) ?? since the method applies
word-for-word to the remaining nonlinear terms.
For the first nonlinearity, we apply Kato-Ponce inequality to get the estimate
? |?x|?[(vN ? ??) ? ?] ?L4/3[0,T ]L1(dx) . ? vN ? |?x|??? ?L4/3[0,T ]L2(dx)?? ?L?[0,T ]L2(dx)
+ ? vN ? ?? ? ?L4/3[0,T ]L2(dx)? |?x| ? ?L?[0,T ]L2(dx)
. T some powerNT (?)NT (?).
For the second nonlinear term, we have
? |?x|?[?(vN?) ? ?] ?L4/3[0,T ]L1(dx)
. d?z |vN(z)|? |?
?
x| ?(x, x? z) ?L4/3[0,T ]L2(dx)??(x? z) ?L?[0,T ]L2(dx)
+ dz |vN(z)|??(x, x? z) ?L4/3[0,T ]L2(dx)? |? |?x ?(x? z) ?L?[0,T ]L2(dx)
. T some powerNT (?)NT (?).
99
3.7.3 Global Well-Posedness of the Time-Dependent HFB Equations
In this subsection, we prove the global well-posedness of the time-dependent
HFB equations. Let us begin by recalling the number and energy conservation laws
derived in ?9 of [GM13a]4. Recall the total particle number is given by
?
N := N dx ??(t, x) (3.50)
and the energy is defined by
{? ?
E 1:= N 2 2? dx |?x?(x)| + dxdy |?x,y sh(k)(x, y)| (3.51)2N
1
+ ? dxdydz vN(x? y)|?(x) sh(k)(y, z) + ?(y) sh(k)(x, z)|
2
2N }
1 { }
+ dxdy vN(x? y) 2|?(x, y)|2 + |?(x, y)|2 + ?(x, x)?(y, y)
4
Note that we have suppress the dependence on t in E for the sake of compactness
of notation.
Theorem 3.39 (Conservation Laws). Suppose (?t,?t,?t) solves the time-dependent
HFB equations and v ? L1(R)?C?(R). Then the total particle number and energy
is conserved.
Proof. See ?8 in [GM13a].
As an immediate corollary of Theorem 3.39, we have
4cf. Corollary 2.7. and Theorem 2.8 in [BBC+18]
100
Corollary 3.40. Let (?t,?t,?t) be a solution to the time-dependent HFB equations.
Then there exists a constant C > 0 such that for any T > 0 and 0 < s < 1 we have
that
sup ? (?t,?t,?t) ?X s ? C, (3.52)
t?[0,T ]
independent of N .
Proof. The estimate for ?t follows immedately by interpolating between the conser-
vation of total particle number and conservation of energy. Next, applying Cauchy-
Schwarz and the conservation of total particle number, we obtain the estimate
? 1?(t, ?) ? 2 2L2(dxdy) ? ??t ?L2(dxdy) + ? sh(kt) ?L2(dx) . 1. (3.53)N
Similarly, using Cauchy-Schwarz and the conservation of energy, we obtain
??x?(t, ?) ?L2(dxdy) (3.54)
? ? 1?t ?L2(dx)??x?t ?L2(dx) + ? sh(kt) ?L2(dxdy)??x sh(kt) ?L2(dxdy) . 1.
N
Interpolating (3.53) and (3.54) yields a desired bound for ?t.
To uniformly bound ?t, we use the trig identity (3.4a) to get the estimate
??(t, ?) ?L2(dxdy) ? ? ?2
1
?t L2(dx) + ? sh(kt) ?L2(dxdy) (3.55)N
1
+ ? sh(kt) ?L2(dxdy)? p(kt) ?L2(dxdy).
N
101
By identity (3.4b), we see that p ? p+ 2p = sh ? sh which means
? p(k) ?2 2L2(dx) ? ? p ? p+ 2p ?Tr = ? sh(k) ?L2(dxdy)
since p(k)(x, x) ? 0. Hence by the conservation of total particle number we have
that
??(t, ?) ?L2(dxdy) . 1.
Similarly, we can show that ??x?(t, ?) ?L2(dxdy) . 1.
102
Chapter 4: Global Well-posed of the Time-Dependent HFB system in
R1+3 and Fock Space Estimate
4.1 Main Result
The main goal of this chapter is to extend Theorem 1.5 to obtain a global in
time result. Let us state the main result of this chapter
Theorem 4.1. Let 1 ? ? < 2 and v ? S a nonnegative interaction potential
3 3
satisfying the condition that |v?| ? w? for some w ? S. Suppose (?t,?t,?t) are
solutions to the time-dependent HFB equations with some smooth initial conditions
(?0,?0,?0) satisfying the following regularity condition uniformly in N : for some
? > 0 and 0 ? i ? 1, 0 ? j ? 2
?? ? ?
? ??x?
1/2+??it?jx?(t, ?)? ? . 1t=0 L2(dx)
? ? ??? ?1/2+??? ?1/2+??i?j ?(t, ?)? ?? x y t x+y ? . 1t=0 ?L2(dxdy)? ?? ?1/2+??? ?1/2+? i j ? ?? x y ??t?x+y?(t, ?) . 1t=0 L2(dxdy)??jx+y sh(2k)(0, x, y)? . 1.L2(dxdy)
Then there exists constants ? = ?(?), ? = ?(?), C = C(?, ?) and a phase function
103
?(t), depending on N , such that we have the Fock space estimate
??? ? ? ? 5+?eitHe? NA(?0)e?B(k0)?? ei?(t)e? N(?t)e?B(k ) ?? ? C exp (?T )t ?
F N1/6
for all 0 ? t ? T .
As remarked in ?2 of [GM17], we need to first prove the following a-priori
estimates
? ?? ? 1+??? ? 1+?x 2 y 2 ?(t) ?L2(dxdy) ? C(t)
? ?? ? 1+?x 2 ?? ?
1+?
y 2 ?(t) ?L2(dxdy) ? C
? ?? ? 1+?x 2 ?(t) ?L2(dx) ? C
and use them to obtain appropriate norm bounds on the solutions of the time-
dependent HFB equations; see Proposition 4.14, 4.16, 4.19, 4.20. Afterward, by
replicating the proof of Theorem 1.5 in ?9 and 10 of [GM17], one can obtain the
desired Fock space estimate.
Remark 4.2. Recently, Grillakis and Machedon extended the local well-posedness
result of the time-dependent HFB system to the range 0 < ? < 1 for more general
initial data in [GM18]. By a similar argument as in the proof of Theorem 4.1, we
could also extend result of Theorem 4.1 to the range 0 < ? < 1.
104
4.2 Global Estimates for the Time-Dependent HFB Equations
In this section we prove, for a sufficiently small ? > 0, the following estimates
? ?? ? 1+?x 2 ?? ?
1+?
y 2 ?(t) ?L2(dxdy) ? C(t) (4.1a)
? ?? ? 1+??? ? 1+?x 2 y 2 ?(t) ?L2(dxdy) ? C (4.1b)
? ?? ? 1+?x 2 ?(t) ?L2(dx) ? C (4.1c)
hold uniformly in N for any fixed time t. The proof of estimates (4.1a)-(4.1c) relies
on the conservation laws established in [GM13a]. For the reader?s convenience, we
restate the conservation laws for the time-dependent HFB system in the following
proposition. Let us recall the total particle number and energy, which we denote by
N and E respectively, can be evaluated explicitly as follows:
{? ? }
N 1= N dx |?(x)|2 + dxdy | sh(k)(x, y)|2 (4.2a)
N
and
{? ?
E = N dx |??(x)|2 1? + dxdy |?x,y sh(k)(x, y)|
2 (4.2b)
2N
1
+ ? dxdydz vN(x? y)|?(x) sh(k)(y, z) + ?(y) sh(k)(x, z)|
2
2N { }}1
+ dxdy v 2 2N(x? y) 2|?(x, y)| + |?(x, y)| + ?(x, x)?(y, y) .
4
105
For the sake of compactness of notation, we have suppressed the dependence on the
time variable since it only plays a passive role in our studies of the equations.
Proposition 4.3 (Conservation Quantities). Suppose (?(t),?(t),?(t)) is a smooth
solution to the time-dependent HFB system with v ? L1(R)?C?(R). Then the total
particle number and energy for the system are conserved.
Remark 4.4. The reader should be aware of the fact that we are assuming that the
energy per particle is constant and independent of N . More precisely, we make the
assumption that N and E are proportional to N for some fixed N . In fact, we have
that N = N and E ? N or, equivalently, N?1E ? 1.
As an immediate corollary of the conservation quantities, we prove estimate
(4.1b) and (4.1c).
Corollary 4.5. Let ?(t) and ?(t) be smooth solutions to the time-dependent HFB
equations. Then, for any 0 < ? ? 1 , we have the estimates
2
? ?? ? 1+?x 2 ??
1+?
y? 2 ?(t) ?L2(dxdy) . 1
? ?? ? 1+?x 2 ?(t) ?L2(dx) . 1
which hold uniformly in N and independent of t.
Proof. It suffices to prove estimate (4.1b) since the prove of (4.1c) is similar. By
106
Proposition 4.3 and Cauchy-Schwarz inequality, we obtain the estimate1
??(t) ?L2(dxdy) ? ??(t) ?
2 ?1
L2(dx) +N ? sh(kt) ? sh(kt) ?L2(dxdy) (4.3a)
? ??(t) ?2 ?1 2L2(dx) +N ? sh(kt) ?L2(dxdy) = 1
independent of t and N . Likewise, we see that
??x?y?(t) ? 2 ?1 2L2(dxdy) ? ??x?(t) ?L2 +N ??x sh(kt) ?L2 . 1. (4.3b)
Hence interpolating (4.3a) and (4.3b) yields the desired result.
In the remainder of the section, we shall prove estimate (4.1a) holds for some
sub-linear function C(t). To this end, let us begin by making the observation that
proving estimate (4.1a) is equivalent to establishing the estimate
N?1
1 1
? ??x? +?2 ??y? +?2 sh(2kt) ?L2(dxdy) . C(t) (4.4)
for some sufficiently small ? > 0. Furthermore, to aid us in proving estimate (4.4),
we apply the operator identity
sh(2k) = 2 sh(k) ? ch(k) = 2 sh(k) + 2 sh(k) ? p (4.5)
1Here, we abused the notation by identifying the composition operator Tk = Tf ? Tg, where Tf
and Tg are integral operators with kernel f(?x, y) and g(x, y), with its kernel
k(x, y) = dz f(x, z)g(z, y).
107
and the triangle inequality to obtain a preliminary estimate
1 1
? ??x? +?2 ?? +?y? 2 sh(2kt) ?L2(dxdy)
1
. ? ?? ? +?
1+? 1 1
x 2 ??y? 2 sh(kt) ?L2 + ? ?? ? +?x 2 ?? ? +?y 2 sh(kt) ? pt ?L2
=: I1(t) + I2(t).
Hence it remains to show N?1Ii(t) . C(t) for i = 1, 2.
To estimate I2(t), we use the following lemma
Lemma 4.6. We have the following estimates
? ?
N?1
?? 1 1|?x| ?2 |?y| 2 sh(kt) ? pt ? . 1 (4.6a)
L2(dxdy)
and
N?1 ??x sh(kt) ? ?ypt ?L2(dxdy) . 1 (4.6b)
where both are independent of time t. In particular, by interpolating estimates (4.6a)
and (4.6b), we obtain the estimate
? ?
N?1
?? 1 1|?x| +?|? | +? ?2 y 2 sh(kt) ? pt ? . 1 (4.7)
L2(dxdy)
for any 0 ? ? ? 1 .
2
Proof. Using Plancherel identity and Cauchy-Schwarz inequality, we establish the
108
estimate
??? ?1 1|?x| 2 |?y| ?2 sh(kt) ? pt ? (4.8)
L2(dxdy)
. ??x sh(kt) ?L2(dxdy) ? pt ?L2(dxdy) + ? sh(kt) ?L2(dxdy) ??ypt ?L2(dxdy) .
Next, taking derivatives of the kernel of the operator identity
sh(k) ? sh(k) = p ? p+ 2p
yields the operator identity
?x sh(k) ? ?y sh(k) = ?xp ? ?yp+ 2?x?yp.
In particular, we have that
??x sh(k) ?2L2(dxdy) = ??xp ? ?yp+ 2?x?yp ? ? ?? p ?
2
tr x L2(dxdy)
since both ?x?y(p ? p+ 2p) and 2?x?yp are positive trace class operators. Hence
combining estimate (4.8) with the conservation laws, we obtain the estimate
??? ??1 1 1N |?x| 2 |? ?y| 2 sh(kt) ? pt ?
L2(dxdy)
. N?1 ??x sh(kt) ?L2(dxdy) ? sh(kt) ?L2(dxdy) . 1.
109
Likewise, we have shown
N?1 ??x sh(kt) ? ?ypt ?L2(dxdy) . N
?1??x sh(kt) ?2L2 . 1.
Next, to estimate I1(t), we first prove a couple preliminary lemmas.
Lemma 4.7. Let ?(t) and ?(t) be solutions the time-dependent HFB equations.
Then it follows we have the estimates
??x,y?(t) ?L2(dxdy) . 1 (4.9a)
and
??x,y?(t) ?L2(dxdy) . 1 (4.9b)
which holds uniformly in N and independent of time t.
Proof. This is an immediate corollary of Lemma 4.6.
Lemma 4.8. Let ?(t) be a solution to the time-dependent HFB equations. Then
we have the following energy estimate
??x?y?(t) ?L2(dxdy) . ??x?y?0 ?L2(dxdy) +N3?t. (4.10)
110
Proof. For convenience, let us restate the equation for ?(t) which is
(S + V) ? =? (vN?) ? ?? ?? ? (vN?)? (vN ??) ? ?? ? ? (vN?) (4.11)
+ 2(vN ? |?|2)(x)?(x)?(y) + 2(v 2N ? |?| )(y)?(y)?(x) =: F
where vN? = vN(x? y)?(x, y) and
1
V = vN + (vN ? diag ?)(x) + (vN ? diag ?)(y).
N
Differentiating equation (4.11) by ?x?y gives us the following equation
(S + V)(?x?y?) = [S + V,?x?y] ? +?x?yF
and
[S + V,?x?y] = N?1(?x? v )? +N?1y N ? ?1yvN?x? +N ?xvN?y?
+ [(?yvN) ? diag ?(y)]?x? + [(?xvN) ? diag ?(x)]?y?.
111
Using the energy method, we obtain the estimate
d ?? ? 2x y?(t) ? 2
dt L (dxdy)
= 2 Re??t?x?y?(t),?x?y?(t)?
= 2 Re?(S + V)(?x?y?(t)),?x?y?(t)?
? 2 ? [S + V,?x?y] ?(t) +?x?yF ?L2(dxdy) ??x?y?(t) ?L2(dxdy)
which leads to the energy estimate
???x?y?(t) ?L2(dxdy) ? ??x?y?0 ?L2(dxdy) (4.12)t
+ ds ? [S + V,?x?y] ?(s) +?x?yF (s) ?L2(dxdy) .
0
We are now ready to estimate the forcing terms. First, for the commutator term we
have the estimate
? [S + V,?x?y]?(t) ?L2(dxdy)
? N?1? (?x? ?1yvN)?(t) ?L2(dxdy) + 2N ??yvN?x?(t) ?L2(dxdy)
+ 2 ? [(?(yvN) ? diag ?(y)]?x?(t) ?L2(dxdy) )
. N4??1 ??x?(t) ? 4??1L2(dxdy) + ? diag ?(t) ?L1(dx) ??x?(t) ?L2(dxdy) . N .
The other forcing term in estimate (4.12) can be handled in a similar fashion. We
shall estimate only one of the terms since the proof is exactly the same for the other
112
terms. Observe, for the (vN? ? ?) we have that
??y(vN?) ? ?y? ?L2(dxdy)
? ??x(vN?(t)) ? 3?L2(dxdy) ??y?(t) ?L2(dxdy) . N .
Hence combining all the estimates yields the desired estimate.
Lemma 4.9. There exists ?0 > 0 such that
N?1
1 1
? |?x| +?2 |?y| +?2 sh(kt) ?L2(dxdy) . C(t) (4.13)
for 0 < ? ? ?0 where C(t) is a sub-linear function independent of N .
Proof. Applying Lemma 4.8 and (4.5) give us the estimate
N?1??x?y sh(kt) ?L2(dxdy)
. N?1??x?y sh(2kt) ? ?1L2(dxdy) +N ??x sh(kt) ? ?ypt ?L2(dxdy)
. ??x?(t) ?2 ?1L2(dx) + ??x?y?(t) ?L2(dxdy) +N ??x sh(kt) ? ?ypt ?L2(dxdy)
. 1 +N3?t.
Interpolating the above inequality with the estimate
?1 1 1N ? |?x| 2 |? | ?1
1
y 2 sh(kt) ?L2(dxdy) . N ??x sh(kt) ?L2(dxdy) . ?
N
113
we have shown that there exists ?0 > 0 such that
N?1
1
? |? +?
1
0 +?0
x| 2 |?y| 2 sh(kt) ?L2(dxdy) . C(t)
where C(t) is a sub-linear function independent of N . In fact, we see that for all
0 < ? ? ?0 = 1 we have the estimate2(6?+1)
N?1
1
? |? | +?
1 1
x 2 |?y| +?2 sh(k ) ? ? +(6?+1)? 2?t L2(dxdy) . N 2 (1 + t) . (4.14)
Remark 4.10. With an eye for 0 < ? < 1, we need 2?(1 + ?) < 1, as assumed in
2
Theorem 3.4 of [GM18], then it follows we need the assumption that 0 < ? ? ?0 =
min(1?? , 1 ) in our proof of the global well-posedness of the time-dependent
2? 2(6?+1)
HFB system.
Let us summarize our findings.
Proposition 4.11. Let ?(t) be a solution of the time-dependent HFB equation for
0 < ? < 2 . Then for any 0 < ? ? ? = 10 we have the estimate3 2(6?+1)
1 1
? ?? ? +??? ? +? ?
1+(6?+1)? 2?
x 2 y 2 ?(t) ?L2(dxdy) . 1 +N 2 (1 + t) .
114
4.3 Global Well-posedness of the Time-Dependent HFB System
Let us define the norms which we shall use in the proof of the uniform in
N global well-posedness of the time-dependent HFB equations for 0 < ? < 2 ,
3
c.f. [GM17].
Fix ? > 0 as in Remark 4.10 and define the norms
1
N +?[T0,T1](?) := sup ? ??x? 2 ?(t, x, x+ z) ?L2([T0,T1])L2(dx) (4.15a)
z
1 1
+ ? ?? ? +?x 2 ?? +?y? 2 ? ?L?([T0,T1])L2(dxdy)
1
N?[T ,T ](?) := sup ? |?x| ?20 1 ??x? ?(t, x, x+ z) ?L2([T0,T ])L2(dx) (4.15b)1
z
1
+ ? ?? ? +?
1
x 2 ?? +?y? 2 ? ?L?([T0,T1])L2(dxdy)
1 1
N[T ,T ](?) := ? ?? ? +? +?x 2 ? ?L?([T ,T ])L2(dx) + ? ??x? 2 ? ?L2([T ,T ])L60 1 (dx). (4.15c)0 1 0 1
For convenience, we denote the sum of the three norms by
N[T0,T1](X) := N[T0,T1](?) + N?[T0,T1](?) + N[T0,T1](?).
If [T0, T1] = [0, T ] then we denote NT (X) := N[0,T ](X) (similarly for the other
norms). Moreover, we adopt the notation
N[T0,T1](DX) := N[T0,T1](D?) + N[T0,T1](D?) + N[T0,T1](D?)
where D is some differential operator.
115
The goal of this section is to prove the uniform in N global well-posedness of
solutions for the time-dependent HFB equations. However, it suffices to prove an
a-priori estimate of the form
NT (DX) . F (T ) (4.16)
for some positive real-valued function F defined on all of [0,?).
We begin by proving a couple lemmas to aid us in establishing (4.16).
Lemma 4.12. Let (?(t),?(t),?(t)) be a solution to the time-dependent HFB system.
Then there exists ? > 0 such that we have the following estimates
N[T0,T1](X) . C0(T0) + (T1 ? T )?0 C0(T1)N[T0,T1](X) (4.17)
where
1
C0(T ) := ? ??x? +?2 ?(T, ?) ?L2(dx)
1 1
+ ? ?? ? +?x 2 ??y? +?2 ?(T, ?) ?L2(dxdy)
1
+ ? ?? ? +?
1
x 2 ??y? +?2 ?(T, ?) ?L2(dxdy).
Proof. It suffices to consider the proof of estimate (4.17) for ? and ? since the proof
116
for ? is similar. Recall the equation for ? is given by
S??? =? (vN?) ? ?? + ? ? (vN ??)? (vN ??) ? ?? + ?? ? (vN ??)
? (vN ? diag ?) ? ?? + ?? ? (vN ? diag ?) (4.18)
+ 2(vN ? |?|2) ? |????| ? 2|????| ? (v ? |?|2N ) =: F
Then, by Proposition 5.8 in [GM17] and Lemma 4.2 in [GM18], we have the estimate
1 1
N?[T0,T1](?) . ? ?? ? +? +?? x 2 ??y? 2 ?(T0?, ?) ?L2(dxdy)? 1
+ ? ?? ? +? 1?? ? +? ?x 2 y 2 F ?
X?
1
2+?
1+? 1? ?? +?x? 2 ??y? 2 ?(T0, ?) ?L2(dxdy)
??2? 1+? 1+ (T1 ? T +?0) 2 ? ??x? 2 ??y? 2 F ? 6 .L2([T0,T1])L 5+(dx)L2(dy)
where we choose 0 < ? < ? . Here, the symbol 6+ denotes a fixed number slightly
2 5
bigger than 6 with dependence on ?, in fact, 6+ = 6 .
5 5 5?2?
For the forcing term ? ? (vN ?diag ?), we apply Young?s convolution inequality,
Sobolev inequality, and Corollary 4.5 to obtain the estimate
1 1
? ?? ? +??? ? +?x 2 y 2 [?? ? (vN ? diag ?)] ? 6L2([T0,T1])L 5+(dx)L2(dy)
1
. ? ?? ? +?
1
y 2 ? ? +?L2([T ,T ])L3+(dx)L2(dy)? ??x? 2 diag ? ?0 1 L2([T0,T 21])L (dx)
1 1
+ ? ?? ? +??? ? +?x 2 y 2 ? ?L2([T0,T1])L2(dxdy)? diag ? ?L2([T0,T1])L3+(dx)
. N?[T0,T1](?).
117
where 3+ = 3? . Next, for the forcing term ? ? (vN ??), we apply Kato-Ponce,1 ?
Sobolev, and estimate (4.1c) from the previous section to obtain the estimate
1
? ??? ? +?
1
x 2 ??y? +?2 [? ? (vN ??)] ? 6L2([T0,T1])L 5+(dx)L2(dy)
1 1
. ?dz vN(z)? ?? ?
+?
x 2 ??(x, x? z)?? ? +?y 2 ?(x? z, y) ? 6L2([T0,T1])L 5+(dx)L2(dy)
1 1
+? dz vN(z)? ??(x, x? z)??
+? +?
x? 2 ??y? 2 ?(x? z, y) ? 6L2([T0,T ])L 5+(dx)L21 (dy)
1 1
. ?dz vN(z)? ??x?
+?
2 ?(x, x? z) ? +?L2(dtdx)? ??y? 2 ? ?L?([T0,T1])L3+(dx)L2(dy)
1 1
+ dz vN(z)??(x, x? z) ? +? +?L2(dt)L3+(dx)? ??x? 2 ??y? 2 ? ?L?([T0,T1])L2(dxdy)
. C0(T1)N[T0,T1](?)
The remaining nonlinear terms (vN?) ? ? and (v 2N ? |?| ) ? |????| can be handled in
a similar manner. Thus, we have shown
??2?
N?[T0,T1](?) . C0(T0) + (T1 ? T0) 2 C0(T1)N[T0,T1](X).
Next, let us recall the equation for ? given by
1
S? =? vN?? (vN ? diag ?) ? ?? ? ? (vN ? diag ?)
N
? (vN?) ? ?? ?? ? (vN?)? (vN ??) ? ?? ? ? (v ?) (4.19)N
+ 2(vN ? |?|2) ? ?? ?? 2?? ? ? (vN ? |?|2) =: G
To estimate ?, we employ Proposition 5.9 of [GM17] and Lemma 4.2 of [GM18] to
118
get the estimate
1+? 1N (?) . ? ?? ? ?? ? +?[T 20,T1] x y 2 ?(T0, ?) ?L2(dxdy)
??2? 1 1
+ (T1 ? T0) 2 ? ?? ? +? +?x 2 ??y? 2 G ? 6L2([T0,T1])L 5+(dx)L2(dy)
Following the same argument as for the ? equation, we arrive at the desired result.
To obtain a-priori estimates of the form (4.16) we need to employ the following
elementary lemma.
Lemma 4.13. Let ?1, ?2 > 0 and C > 0. Then there exists a monotone sequence of
positive real numbers Tk such that
1
lim Tk =? and (T ?1 ?2k+1 ? Tk) Tk+1 ? ? k ? N.
k?? C
Proof. Consider the sequence Tk defined by
( )
(1? ?)? 1 1 1
T 1??k := 1 + + . . .+ ? k (4.20)
C?/?2 2? k? (1? ?)1??C?/?2
where ? = ?2 . It is clear that {Tk} is monotone increasing and tends to infinity?1+?2
as k ??. Moreover, by estimate (4.20) we immediately see that
(Tk+1 ?
1
T )?1T ?2k k+1 ? C
119
which completes the proof.
Proposition 4.14. Let T > 0. Assume (?(t),?(t),?(t)) is a solution to the time-
dependent HFB system, then we have the following a-priori estimate
NT (X) . 1 + T
5+3?. (4.21)
Proof. Define Tk as in the previous lemma with ? = ??2?
2
1 , ?2 = 2?, ? =
? , and
2 8+2?
C > 0 sufficiently large. Using estimate (4.17), we obtain the estimate
N[T ,T (X) . C
2?
k k+1] 0(Tk) . Tk (4.22)
which means
NT (X) ? NT (X) + N[T ,T ](X) ? N 2?k+1 k k k+1 T (X) + CTk k
. (data) + T 2?1 + . . .+ T
2?
k .
Switching to continuous T -variable yields the desired estimate
? T 1/(1??)
2?(1??) 4?NT (X) . (data) + x dx . (data) + T ? +1+2?? ? 2? .
0
Lemma 4.15. Let (?(t),?(t),?(t)) be a solution to the time-dependent HFB system.
120
Then there exists ? > 0 such that we have the following estimates
N[T0,T1](?tX) . C1(T0) + (T1 ? T0)?N[T0,T1](X)N[T0,T1](?tX), (4.23a)
N ?[T0,T1](?x+yX) . C2(T0) + (T1 ? T0) N[T0,T1](X)N[T0,T1](?x+yX) (4.23b)
where
1
C1(T ) := ? ?? ? +?x 2 ?t?(T, ?) ?L2(dx)
1 1
+ ? ?? +? +?x? 2 ??y? 2 ?t?(T, ?) ?L2(dxdy)
1
+ ? ?? ? +?
1
x 2 ?? +?y? 2 ?t?(T, ?) ?L2(dxdy)
and
1
C2(T ) := ? ?? ? +?x 2 ?x?(T, ?) ?L2(dx)
1+? 1+ ? ??x? 2 ?? ? +?y 2 (?x+y?)(T, ?) ?L2(dxdy)
1 1
+ ? ?? ? +??? ? +?x 2 y 2 (?x+y?)(T, ?) ?L2(dxdy).
Proof. Taking the time derivative of (4.11) yields
( )
S +N?1vN ?t? =? (vN?) ? ?t?? (vN ? diag ?) ? ?t?
+ similar terms =: F
121
By the same argument as in Lemma 4.12, we have the estimate
N[T0,T1](?t?)
??2? 1 1
. C1(T0) + (T1 ? T ) ? ?? +? +?0 2 x? 2 ??y? 2 F ? 6 .L2([T ,T + 20 1])L 5 (dx)L (dy)
We shall look at two generic cases, as stated above, to deduce (4.23). In the first
case, we estimate the term (vN?) ? ?t?, which goes as follow
1 1
? ??? ? +?x 2 ?? +?y? 2 (vN?) ? ?t? ? 6L2([T0,T1])L 5+(dx)L2(dy)
1 1
. dz v (z)? ?? ? +??(x, x? z)?? ? +?? N x 2 y 2 ?t?(x? z, y) ? 6L2([T0,T1])L 5+(dx)L2(dy)
1 1
+? dz vN(z)??(x, x? z)?? ?
+?
x 2 ?? ? +?y 2 ?t?(x? z, y) ? 6L2([T0,T1])L 5+(dx)L2(dy)
1 1 1
. ?dz vN(z)? ??x?
+?
2 ?(x, x? z) ? +? +?L2(dtdx)? ??x? 2 ??y? 2 ?t? ?L?([T0,T1])L2(dxdy)
1 1
+ dz vN(z)??(x, x? z) ?L2(dt)L3+(dx)? ?? ? +?x 2 ?? +?y? 2 ?t? ?L?([T0,T1])L2(dxdy)
. N[T0,T1](?)N?[T0,T1](?t?).
In the second case, we estimate the term (vN ? diag ?) ? ?t? as follows
1 1
? ?? ? +??? ? +?x 2 y 2 [(vN ? diag ?) ? ?t?] ? 6L2([T0,T1])L 5+(dx)L2(dy)
1
. ? (v ? ?? ? +?
1
N x 2 diag ?) ? ??y? +?2 ?t? ? 6L2([T ,T ])L 5+0 1 (dx)L2(dy)
1 1
+ ? (vN ? ?? +? +?x? 2 diag ?) ? ??y? 2 ?t? ? 6L2([T0,T1])L 5+(dx)L2(dy)
. N?[T0,T1](?)N[T0,T1](?t?).
122
Hence combining the above estimates yields
??2?
N[T0,T 21](?t?) . C1(T0) + (T1 ? T0) {N[T0,T1](?)N?[T0,T1](?t?)
+ N[T0,T1](?t?)N?[T0,T1](?) + N[T0,T1](?t?)N[T0,T1](?)}
??2?
. C1(T0) + (T1 ? T0) 2 N[T0,T1](X)N[T0,T1](?tX).
Similarly, we can show
??2?
N?[T0,T1](?t?) . C1(T0) + (T1 ? T0) 2 N[T0,T1](X)N[T0,T1](?tX)
and
??2?
N[T0,T1](?t?) . C1(T0) + (T1 ? T0) 2 N[T0,T1](X)N[T0,T1](?tX).
Therefore, summing up the three inequalities yields (4.23a). Moreover, the prove of
(4.23b) is exactly the same since?x+y commutes withN?1vN(x?y), i.e. [? , N?1x+y vN(x?
y)] = 0.
Using the above lemma we could again prove some a-priori estimates for both
the norm of ?tX and ?x+yX.
Proposition 4.16. Let T > 0. Suppose (?(t),?(t), ?(t)) is a solution to the time-
dependent HFB system, then we have the following uniform in N a-priori estimates
123
( )
N (? X) . exp ?T 5+?T t ( ), (4.24a)
NT (? X) . exp ??T 5+?x+y (4.24b)
for some ?, ?? > 0, which are independent of T .
Proof. Again we choose the sequence Tk defined by (4.20) for some sufficiently large
C > 0. Applying Lemma 4.15 and estimate (4.22) yield the estimate
??2?
N[Tk,T (? X) . Ck+1] t 1(Tk) + (Tk+1 ? Tk) 2 N[Tk,T (X)Nk+1] [T ,T ](?tX)k k+1
??2?
. C 2?1(Tk) + (Tk+1 ? Tk) 2 Tk+1N[T ,T ](?tX).k k+1
Then it follows
N[T ,T ](?tX) . C1(T ).k k+1 k
In particular, we have the estimate
NT (?tX) ? NT (?k+1 k tX) + N[T ,T (? X)k k+1] t ?k
. NT (?tX) + C1(Tk) . (data) + C1(Tk j)
j=1
124
By switching to the continuous T -variable and set ?2? = , we obtain the estimate
8+2?
? T
NT (?tX) . C1(T0) + ? d? C1(?)?
4+?
0
T
. C1(T0) + d? N? (?tX)?
4+?.
0
Finally, applying Gronwall?s inequality yields
( )
N (? 5+?T tX) . C1(T0) exp ?T .
The proof for (4.24b) is exactly the same.
Remark 4.17. Note, from the a-priori estimate (4.24a), we could deduce
( )
? ?t? ? 5+?L1([0,T ]?L2(R3)) ? T? ?t? ?L?([0,T ]?L2(R3)) . T exp (?T )
? ?t? ? 5+?L1([0,T ]?L2(R6)) ? T? ?t? ?L?([0,T ]?L2(R6)) . T exp (?T )
? ? ? ? ? T? ? ? ? . T exp ?T 5+?t L1([0,T ]?L2(R6)) t L?([0,T ]?L2(R6)) .
Then by the 1D Sobolev inequality we have that ? ? C([0, T ] ? L2(R3)) and
?,? ? C([0, T ]? L2(R6)), that is, ?,?, and ? are strong solutions to the nonlinear
equations.
Let us conclude this section with some a-priori estimates for the higher order
derivatives of (?,?,?) which we will later use to estimate sh(2k).
Lemma 4.18. Suppose ?(t),?(t), and ?(t) are solutions to the time-dependent HFB
125
equations, then we have the following estimates
N[T0,T1](?t?x+yX) (4.25a)
??2?
. C3(T0) + (T1 ? T0) 2 N[T0,T1](X)N[T0,T1](?t?x+yX)
??2?
+ (T1 ? T0) 2 N[T0,T1](?tX)N[T0,T1](?x+yX)
N 2[T0,T1](?x+yX) (4.25b)
??2?
. C4(T0) + (T1 ? T0) 2 N[T0,T1](X)N (?2[T0,T1] x+yX)
??2?
+ (T ? T ) N21 0 2 [T0,T (? X)1] x+y
N[T0,T1](? ?2t x+yX) (4.25c)
??2?
. C5(T0) + (T
2
1 ? T0) 2 N[T0,T1](X)N[T0,T1](?t?x+yX)
??2?
+ (T1 ? T0) 2 N[T0,T1](?x+yX)N[T0,T1](?t?x+yX)
??2?
+ (T ? T ) 21 0 2 N[T0,T1](?x+yX)N[T0,T1](?tX)
126
where
1
C3(T ) = ? ?? ? +?x 2 ?t?x?(T, ?) ?L2(dx)
1 1
+ ? ?? +? +?x? 2 ??y? 2 (?t?x+y?)(T, ?) ?L2(dxdy)
1
+ ? ?? ? +?
1
x 2 ?? +?y? 2 (?t?x+y?)(T, ?) ?L2(dxdy)
1
C4(T ) = ? ?? ? +??2x 2 x?(T, ?) ?L2(dx)
1 1
+ ? ?? +? +?x? 2 ??y? 2 (?2x+y?)(T, ?) ?L2(dxdy)
1 1
+ ? ?? ? +??? ? +?(?2x 2 y 2 x+y?)(T, ?) ?L2(dxdy)
1
C +? 25(T ) = ? ??x? 2 ?t?x?(T, ?) ?L2(dx)
1 1
+ ? ?? ? +?x 2 ?? ? +?y 2 (?t?2x+y?)(T, ?) ?L2(dxdy)
1 1
+ ? ?? ? +?x 2 ?? ? +?y 2 (?t?2x+y?)(T, ?) ?L2(dxdy).
Proof. The proof is similar to the proof of Lemma 4.15.
Proposition 4.19. Let T > 0. Suppose (?(t),?(t),?(t)) is a solution to the time-
dependent HFB system, then we have the following uniform in N a-priori estimates
( )
N (? ? X) . exp ? T 5+?T t x+y ( 1 ) , (4.26a)
NT (?2x+yX) . exp 5+?(?2T ) , (4.26b)
NT (?
2
t?x+yX) . exp ? T 5+?3 (4.26c)
for some constants ?1, ?2, ?3 > 0, which are independent of T .
127
Proof. Let us begin by choosing the same sequence Tk defined by (4.20) for some
sufficiently large C > 0. By Lemma 4.18 and (4.22), we obtain the estimate
N[T ,T ](?t?x+yX)k k+1
??2?
. C3(Tk) + (Tk+1 ? Tk) 2 N[Tk,T (?k+1] tX)N[Tk,Tk+1](?x+yX).
In particular, if we set ? = ?2 then we have the estimate
8+2?
C1(Tk)C2(Tk)
N[T ,T ](?t?x+yX) . C3(Tk) + .k k+1 T 2?k+1
Hence it follows
NT (?t?x+yX) ? NT (?t?x+yX) + Nk+1 k [T (? ? X)k,Tk+1] t x+y
C1(Tk)C (T ). 2 kNT (?t?x+yX[) + C3(Tk k) + T 2?? k+1k ]C1(Tj)C2(T ). jC3(T0) + C3(Tj) + .
T 2?
j=1 j+1
Switching to continuous T -variable yields the estimate
? T
NT (?t?x+yX) . C3?(T0) + d? N? (?x+yX)N? (?tX)?
4??
0
T
+ d? N? (?t? 4+?x+yX)?
0
. C ?(T ) + T 5?
( )
? exp kT 5+?3 0
T
+ d? N? (?t?x+yX)? 4+?.
0
128
Using Gronwall?s inequality, we obtain the estimate
? ( ) ( )N (? ? X) . (C (T ) + T 5 ? exp kT 5+? ) exp cT 5+?T t x+y 3 ( 0 )
. exp ?T 5+?
for some ? > 0. The proofs for the other two estimates are similar.
4.4 Estimates for sh(2k)
The purpose of this section is to obtain estimates for sh(2k), which will be
used to obtain Fock space estimates. Recall the equation for sh(2k) is given by
S(sh(2k)) =? 2vN?? (vN?) ? p2 ? p?2 ? (vN?)
? ((vN ? diag ?)(x) + (vN ? diag ?)(y)) sh(2k) (4.27)
? (vN?) ? sh(2k)? sh(2k) ? (vN?)
where S = 1?t ??R6 .i
Proposition 4.20. Let sh(2k) satisfy (4.27) with some initial conditions. Then for
any fixed T > 0 and 0 ? j ? 2 we have that
( )
??jx+y sh(2k)(t, ?) ?L2(dxdy) . exp 5+3?(?jT ) (4.28a)
sup ? sh(2k)(t, x, ?) ? . exp ?T 5+3?L2(dy) . (4.28b)
x
for some ?j, ? > 0.
129
To prove the above proposition we will need a couple lemmas
Lemma 4.21. Let s0a be the solution to
Ss0a =? 2vN(x? y)?
s0a(0, x, y) = sh(2k)(0, x, y)
on the interval [0, T ]. Then there exists ?j > 0 for 0 ? j ? 2 such that
( )
??j s0x+y a(t, ?, ?) ?L2(dxdy) . exp ? T 5+?j (4.29)
for all t ? [0, T ].
Proof. Observe we could write the solution as
? t
s0(t, x, y) = eit?x,ya sh(2k)(0, x, y) + i e
i(t?s)?x,yvN(x? y)?(s) ds
0
then it follows
? s0a(t, ?) ?L2(dxdy) ??? t ??
? ? sh(2k ) ? i(t?s)?x,y2 ? ?0 L (dxdy) + ? e vN(x? y)?(s) ds ? .
0 L2(dxdy)
130
Let us focus on the nonlinear term. By a change of variables, we get
????? ?t ei(t?s)?x,yv (x? y)?(s) ds??N ?
??0 L2(dxdy)
. ?? ? t ( ) ??P ei(t?s)? x+ y x? yx,y?|?|>1 ? vN(y)? s(, , )ds
?
? 0 2 2
?
L??2(dxdy)?? t x+ y x? y+ P i(t?s)?x,y|?|?1 e vN(y)? s, , ds?? .
0 2 2 L2(dxdy)
( )
Let us denote ? s, x+y , x?y by ??(s, x, y). For the first term we shall rewrite the
2 2
integral using integration by parts, i.e.
? t ? t
ei(t?s)?
?
x,yv (y)??(s) ds ?? ds ei(t?s)?x,y ?1N ? ?x,y(vN(y)??(s))0 0 ?st
= ei(t?s)?
?
x,y??1x,y(vN(y) ??(s))
0 ?s
+ eit???1(v (y)??(0))???1x,y N x,y(vN(y)??(t))
then it follows
???? ? ?t ?P ei(t?s)?x,y| ???|?1 0 ? vN(y)??(s, ?) ds?L2(dxdy)t
. ds ??? ???P|?|?1??1x,y(vN(y) ??(s, ?))??
0 ?s L2(dxdy)
+ ?P ?1 ?1|?|?1?x,y(vN(y)??(0, ?)) ?L2(dxdy) + ?P|?|?1?x,y(vN(y)??(t, ?)) ?L2(dxdy).
131
Next, by Plancherel, we obtain the estimate
? ???? ??t ?ds P ??1|??|?1 x,y(vN(y) ??(s, ?))?0 ? ?s ?L2(dxdy)t
. ds ??? ?
?
? v?N?s??(s, ?, ?) ??
? 0 ? |?|
2 + |?|2 ??
L2(|?|?1,d?)L2(d?)
t
. ? ds ? vN(y)?s??(s, x, y) ?L1(dy)L2(dx)0 t
? ds ? ?s?(s, x+ y, x) ?L?(dy)L2(dx)
0
? ( )
. t? ? ? ? . exp ? t5+?s L2([0,t])L?(dy)L2(dx) 0 .
For the second and third terms, we have the estimate
?P|?|?1??1x,y(vN(y)??(t, x, y)) ?L2(dxdy)
. ??(t, x, x? y) ?L?(dy)L2(dx)
. ??(s, x, x? y) ?L?([0,t])L?(dy)L2(dx) ( )
. ? ?s?(s, x, x? y) ?L2([0,t])L?(dy)L2(dx) . exp ?0t5+? .
Thus, we have shown
( )
? s0 ? 5+?a L?([0,T ])L2(dxdy) . exp ?0T .
132
In particular, it is easy to check
??jx+ys0a ?
j
L?([0,T ])L2(dxdy) . ? ?t?x+y? ?L2([0,T ])L?((dx)L
2(dy))
. NT (?t?jx+yX) . exp ? T 5+?j .
Lemma 4.22. Let sa be the solution to
S?sa = ? 2vN?
sa(0, x, y) = sh(2k)(0, x, y)
on the interval [0, T ]. Then there exists ?j > 0 for 0 ? j ? 2 such that
( )
??jx+ysa(t, ?, ?) ?L2(dxdy) . exp ? T 5+?j
for all t ? [0, T ].
Proof. Recall S? = S + V where
V (u) = ((vN ? diag ?)(x) + (vN ? diag ?)(y))u+ (vN?) ? u+ u ? (vN?).
Using the previous result, we see that
S?s1 = ?V (s0a a) s1a(0, x, y) = 0
133
where s 1a = sa + s0a. It?s not hard to see
(S + V )(?j s1) = [S + V,?jx+y a x+y]s1a ??
j 0
x+yV (sa)
= [V,?jx+y]sa ??
j
x+yV (s
0
a).
Using energy estimate, we have
? t
??j s1 ? ? ds ? [V,?j ]s1 ? + ??jx+y a L2 x+y a L2 x+yV (s0a) ?L2 .
0
Let us consider the case when j = 0. Observe we have
? t
? s1 ? ? ds ?V (s0a L2 a) ?L2 .
0
Observe
? t
d?s ? (vN ? diag ?)(x) ? s
0
a ?L2(dxdy)
0
t
? ? ds ? vN ? diag ?(x) ?
0
L2(dx)? sa ?L?(dx)L2(dy)
0
t
? ds ? vN ? diag ?(x) ? 0L2(dx)? ? sa(x, x+ y) ?L?(dx) ?L2(dy)
0
1
. t 2 sup ? ? (?j 0x+ysa)(s, x, y) ?L2(dy) ?L2(dx)
0(<s<t )
. exp ? 5+?0T
134
and
? t
ds ? (v ?) ? s0? ?N a ?L2(dxdy)0 t
? ? ds? dzvN(z)??(x, x? z)s
0
a(x? z, y) ?L2(dxdy)
0
t
. ds dzvN(z)??(x, x? z) ?L?(dx)? s0a ?L2(dxdy)
0 ( )
. exp ? 5+?0T .
The proof for j = 1, 2 is the same.
Proof of Proposition 4.20. The proof is exactly the same as the one given for The-
orem 7.1 in [GM17].
135
Chapter 5: Morawetz Estimates of the time-dependent HFB System
Consider the Bogoliubov state (quasifree state)
?
?HFB(t) =M? = e? NA(?)e?B(k)? (5.1)
where ? = (1, 0, 0, . . .) denote the vacuum vector in Fock space. Following [GM17],
we define the kernel of the L-matrices (operators)
L 1 (5.2)n,m(t, x1, . . . , xn; y1, . . . , ym) := ?M?,P(m+n)/2 n,mM??N
where Pn,m = a?x ? ? ? a? a ? ? ? a .1 xn y1 ym
5.1 Main Results
Let us recall the evolution equation for the L1,1-density matrix derived in
[GM17]
1 ? L1,1(t, x1;x? ) = (?? + ? )L (t, x ;x? ?1 x ?1 x1 1,1 1 1) +Bv(L2,2)(t, x1;x1) (5.3)i ?t
136
where
?
Bv(L2,2)(t, x ;x?1 1) = ?dx2 v(x1 ? x2)L2,2(t, x1, x2;x
?
1, x2)
? dx?2 v(x?1 ? x?2)L ? ? ?2,2(t, x1, x2;x1, x2).
To effectively describe the conservation law associated with (5.3), we introduce the
pseudo-stress-energy tensor T?? for ?, ? = 0, 1, 2, 3 of the one-particle Fock marginal
density matrix defined by
T00 = ? := L1,1(t, x;x), (5.4a)
1 ( )
Tj0 = T0j = pj?:= ?x L1,1 ? ?x?L1,1 (t, x;x), (5.4b)2i j j
Tjk = ?
?
jk := ? dx ?(x? x
?)(?x ?
?
x? + ?x? ?x )L1,1(t, x;x ) (5.4c)k j k j
1
+ ?jk dy v(x?
1
y)L2,2,(t, x, y, x, y)? ?jk?x[L1,1(t, x;x)]
2 2
where ?jk is the Kronecker delta. The quantity T00 is called the total-particle-number
density or mass density, the quantity T0j = Tj0 is called total-particle-number current
or, equivalently, momentum density, and the quantity Tjk is the stress tensor 1. For
1Since the Fock marginal density matrix is defined with a normalizing factor, then N ? T00 is
the true total-particle-number density. We lack a better name for Tjk.
137
a sufficiently smooth solution to (5.3), a short computation shows that
?
?
? T = dudu?ei(u?u
?)?xL? (t, u;u?t 00 ? 1,1 )?t
?
= i dudu? ei(u?u )?x(|u|2? ? |u
?|2)L? ?1,1(t, u;u )
?
+ i dudu? ei(u?u )?x? B?v(L2,2)(t, u;u
?)
?
= ? ? dudu? ei(u?u )?x(u+ u?x )L? ?1,1(t, u;u ) = ?2?x Tj 0j
where the last equality follows from Bv(L2,2)(x;x) = 0. Let us summarize the above
result and introduce some useful notation in the next proposition.
Proposition 5.1 (local conservation of total particle number). Let L1,1(t) be a
sufficiently smooth solution to (5.3). Then we have the following conservation law
??
+ 2?x ? p = 0 (5.5)
?t
where the total-particle-number current is given by
?
(u+ u?
p(t, x) = ? dudu? ei(u?u?)?x )L? ?1,1(t, u;u ). (5.6)
2
In fact, with a bit more work, we also obtain the following continuity equation
for the total-particle-number current.
Proposition 5.2. Let L1,1(t) be a sufficiently smooth solution to (5.3). Then we
138
have the following continuity equation
?
?tpj + ?k?kj + lj = 0 (5.7)
k
where the source term lj is given by
?
1
lj = dy v(x? y){?yL2,2(x, y;x, y)??xL2,2(x, y;x, y)}.
2
Since we will prove Proposition 5.2 directly from the Morawetz identity, let us
postponed the proof till a later section.
We define the Morawetz action associated to the observable a(x) ? C?(R3) of
the one-particle Fock marginal density to be2
?
Ma(t) := dx ?a(x) ? p(t, x). (5.8)
Then we have the following Morawetz identity for the one-particle Fock marginal
density matrix.
2The definition comes from looking at the evolution of the expectation value of an observable
a against our mass density?. More precisely, using (5.5)?and integration by parts, we get that
?t dx a(x)L1,1(t, x;x) dx = dx ?a(x) ? pt(x).
139
Theorem 5.3 (Morawetz identity). The following identity holds
?
1
?tMa = ? ? dx (?x?xa(x))L1,1(x;x) (5.9)2
1
+ ?dxdy ?xa(x)v(x? y)L? 2,2
(x, y;x, y)
2
+ 2< dxdx? ?(x? x?) (?x ?x a(x))?x ?x?L1,1(x;x?)
? ? j ` ` jj,`
1
+ dx ?xa(x) ? dy v(x? y){?xL2,2(x, y;x, y)??yL2,2(x, y;x, y)}
2
for the one-particle Fock marginal density.
Remark 5.4. The one-particle Morawetz identity for the Gross-Pitaevskii hierarchy
has been considered in Theorem 3.1 of [CPT12]. In fact our result coincides with
Theorem 3.1 when v = ? even though our contexts are different.
For the convenience of notations, we have suppressed all the t variable in our
calculation since it only plays a passive role in the proof of Theorem 5.3. The main
result of this chapter is to establish the following interaction Morawetz estimate for
the one-particle Fock density matrix.
Theorem 5.5 (Interaction Morawetz-Type Estimate). Let ?(t) be a sufficiently
smooth global solution to (5.3). Then we have the following estimate
?
dtdx |?(t, x;x)|2 . 1 (5.10)
which is uniform in N and depends only on the initial data.
140
5.2 Proof of Theorem 5.3
By direct calculation, we see that
?
1 (u+ u?? )
? p(x) = dudu? ei(u?u )?x (u2 ? (u?)2t ? )L?1,1(u, u
?)
i 2
1 ? i(u?u?)?x (u+ u
?)
+ du?du e B?v(L2,2)(u, u
?)
i 2
1 ?
= ? ? ? dudu? ei(u?u )?x(u+ u?)? (u+ u?? x )L?1,1(u, u
?)
2
1 ? i(u?u?)?x (u+ u
?)
+ dudu e B?v(L2,2)(u, u?)
i 2
=: J1 + J2.
For the J1 term, applyinf integration by parts yields
?
1
J = dxdudu? ei(u?u
?)?x{?2a(x)(u+ u?1 , u+ u?)}L? ?1,1(u;u ).
2
141
Next, expand the expression into components and apply Fourier inversion, we see
that
? { }
1 ?
J = dxdudu? ei(u?u
?)?x
1 (?x ?x a(x))(ui + u
? ?
i)(uj + uj) L?1,1(u;u?)2 ? i j
1 ?i,j
= ? dxdx? ?(x? x?) (?x ?x a(x))(? ?x ? ?x? )(?x ? ?x? )L1,1(x;x )
2 ? i j i i j j
1 ?i,j
= ? dxdx? ?(x? x?) (?x ?x a(x))(?x ?x + ?x??x? )L1,1(x;x?)
2 ? i j i j i ji,j
1 ?
+ dxdx? ?(x? x?) (?x ?x a(x))(? ? + ? ?x x? x??x )L1,1(x;x )
2 ? i j i j i j?i,j1
= ? dxdx? ?(x? x?) (?x ?x a(x))(?x ?x + ? ? ?? + ? ?? + ? ?? ? )L (x;x )
?2 i j i j
xi xj xi xj xi xj 1,1
? i,j
+ dxdx? ?(x? x?) (? ? ?xi x a(x))(?x ?x? + ?x??x )L1,1(x;x )? j i j i j? i,j1
= ? dx (?x ?x a(x))?x ?x [L1,1(x;x)]
2 ? i j i ji,j ?
+ 2< dxdx? ?(x? x?) (?x ?x a(x))?x ?x?L1,1(x;x?).i j i j
i,j
Now, let us handle J2 which contains Bv(L2,2). Assume v is radially symmetric.
Then consider the Fourier transform of Bv(L2,2). For convenience, we decompose
the operator into two parts
B = B+v v ?B?v (5.11)
142
where
?
B+v (L2,2)(x ;x?1 1) = dx2dx?2 v(x1 ? x2)?(x2 ? x?2)L2,2(x1, x2, x?1, x?2) (5.12a)
and
?
B?(L )(x ;x?v 2,2 1 1) = dx dx? ? ? ?2 2 v(x1 ? x2)?(x2 ? x2)L2,2(x1, x ? ?2, x1, x2). (5.12b)
By direct calculations, we see that
?
B?+v (L
? ?
2,2)(?; ?
?) = ? dx dx
? dx dx? e?i(x1???x1?? )1 1 2 2 v(x1 ? x2)?(x2 ? x?2)L2,2(x,x?)
? ? ?i(x ???x? ???= ?dx dx dx dx e ? 1 1
)
1 1 2 2 L2,2(x,x?)
? d?? e?i(x2?x? ?? 2
)?2 d? ei(x1?x2)?22 2 v?(?2)
= ? d?2d?
?
2 v?(?2)L?2,2(? ? ?2, ?2 + ?? , ??, ??2 2)
= d? d?? v?(? ? ?? )L? (? ? ? + ?? , ? , ??2 2 2 2 2,2 2 2 2 , ??2)
and, similarly, we also have
?
B??v (L2,2)(?, ??) = d? d??2 2 v?(??2 ? ? ? ? ?2)L?2,2(?, ?2, ? ? ?2 + ?2, ?2).
143
Then it follows
?
1 ?
dudu? ei(u?u
?)?x (u+ u )
? B?v(L
?
2,2)(u;u ) (5.13)
i 2
1 ?
= dudu?d? d?? ei(u?u
?)x (u+ u ) v?(? ? ?? )L? (u? ? + ?? , ? ;u? ?? 2 2 2 2 2,2 2 2 2 , ?i 2 2)?
? 1 dudu?d? d?? i(u?u?)x (u+ u ) ?2 2 e v?(?2 ? ?2)L?2,2(u, ? ;u? ? ?? + ? , ??2 2i 2 2 2
)
Now, apply the change of variables u 7? u ? ?2 + ?? ?2 and u ?7 u? ? ? + ??2 2 to the
second term on the RHS of (5.13) yields
RHS o?f (5.13)
1 ? ?= dudu d? d?? ei(u?u )?x(? ? ?? )v?(? ? ?? )L? (u? ? + ?? , ? ;u? ?2
i 2
2 2 2 2 2,2 2 2 2 , ?2).
Hence, let us focus on the term
? ?
1
dx ? ?a(x) ? dudu?d?2d?? ei(u?u )?x2 (?2 ? ??2)i (5.14)
? v?(? ? ? ? ?2 ? ?2)L?2,2(u? ?2 + ?2, ?2;u , ?2).
144
We begin by rewriting (5.14) in physical coordinates
? ? ?
1
(5.14) = dx ?a(x) ? ?dudu?d?2d??2 dXdY dX ?dY ?ei(u?u )?x(? ? ??2 2)i
? ?v?(? ? ?? )e?i(u??2+?
?
2)?X?i?2Y+iu?X?+i?? Y ??22 2 L2,2(X, Y ;X ?, Y ?)
1
= dXdY dX ?dY ?L2,2(X, Y ;X ?, Y ?) d? d??2
i 2
? ??(X ?X ?)? a(X) ? (? ? ?? )v?(? ? ?? )ei(X?Y )?2?i(X?Y
?)??2
X 2 2 2 2
= ? dXdY dY ?L ??2,2(X, Y ;X, Y )
?? ? ?? Xa(X) ? ? d? d?
? ei(X?Y )?2?i(X?Y )?2X 2 2 v?(? ? ??2 2)
= ? dXdY dY ?L[2,2(X, Y ;X, Y
?
() )]
?? ? Y + Y
?
? Xa(X) ? ?X ?(Y ? Y )v X ? 2
= ?? dXdY L2,2(X, Y ;X, Y )?Xa(X) ? ?Xv (X ? Y )
= dXdY v(X ? Y ){?Xa(X) +?Xa(X) ? ?X}L2,2(X, Y ;X, Y ).
Combining the calculation for J1 and J2, we obtain the Morawetz identity
?
1
?tMa =? ? dx (?x?xa(x))L1,1(x;x)2
+ d?xdy v(x? y) {?xa(x) +?? x
a(x) ? ?x}L2,2(x, y;x, y) (5.15)
+ 2< dxdx? ?(x? x?) (?x ?x a(x))?x ?x?L ?j ` ` j 1,1(x;x )
j,`
145
which completes the proof of Theorem 5.3. Moreover, if we define the pseudo stress-
energy tensor
?
T ? ? ?jk = dx ?(?x? x )(?x ?x? + ?x? ?x )L1,1(x;x )k j k j
1
+ ?jk { dy v(x? y)L2,2,(x, y;x, y)??[L1,1(x;x)]}.
2
then, by (5.9), we have the continuity equation
?
?tT0j + ?kTkj + lj = 0
k
where
?
1
lj = dy v(x? y){?yL2,2(x, y;x, y)??xL2,2(x, y;x, y)}.
2
This completes the proof of Proposition 5.2.
It is instructive to consider the special case v(x) = ?(x), c.f. equations (19)-
(20) in [BBC+18]. If v = ?, then we have the following one-particle Morawetz
identity for the L1,1 density matrix
?
?tMa = ?
1
? dx (?x?xa(x))L1,1(x;x)2
1
+ ? dx (?xa(x))L2,2(x, x;x, x)? (5.16)2
+ 2< dxdx? ?(x? x?) (?x ?x a(x))?x ?x?L1,1(x;x?).j ` ` j
j,`
Again, from the Morawetz identity we can read off the Tjk components of the pseudo
146
stress-energy tensor for j, k = 1, 2, 3, which are given by
1 1
?tp(x) = ??[?L1,1(x;x)]? ?[L2,2(x, x;x, x)]2 2 (5.17a)
? 2< dx? ?(x? x?)(?x??x +?x?x?)L1,1(x;x?)
or
1 1
?tpk(x) = ??x ?k x[L2 ?1,1(x;x)]? ?x [L2,2(x, x;x, x)]2 k (5.17b)
? ? dxdx?x ?(x? x?)(? ? + ? ? )L (x;x?j xk x? ?j x xj 1,1 ).k
j
Thus, we see that
?
Tjk = dx
? ?(x? x?)(?x ? ? + ? ?? ? )L (x;x )k xj x x 1,1k j
1
+ ?jk {L2,2(x, x;x, x)??[L1,1(x;x)]}
2
and
?
?tT0j + ?kTkj = 0.
k
Remark 5.6. The reader should note that when v(x) = ?(x) we have that lj ? 0
since L2,2(x, y;x, y) = L2,2(y, x; y, x). Hence, this shows that the Tjk are conserved
for every fixed k.
147
5.3 Morawetz Estimates
In this section we use (5.9) to prove the classical Morawetz estimates for the
time-dependent HFB system in 3D. Here, we consider the cases v = ? and v positive
radially symmetric. In particular, the main results of this section are Proposition
5.7 and Proposition 5.8.
Case 1: v(x) = ?(x)
Multiplying (5.5) by a(x) and integrate in space yields
?
?t ?dx a(x)L1,1(x;x) (5.18)
1
+ dxdx? ?(x? x?)a(x)(?x +?x?) ? (? ?x? ??x)L1,1(x;x ) = 0
2
Next, apply integration by parts, we get that
? ?
1
?t dx a(x)L1,1(x;x) = dx ?a(x) ? [(?x? ??x)L1,1](x;x).
2
Then, by (5.16), it follows we have that
? ?
1
?2t dx a(x)L1,1(x;x) = ? ? dx (?x?xa(x))L1,1(x;x) (5.19)2
1
+ dx (?xa(x))L2,2(x, x;x, x)
2 ? ?
+ 2< dxdx? ?(x? x?) (? ? a(x))? ? L (x;x?x x ? ).j ` x` xj 1,1
j,`
148
If a(x) = |x|2, then we have obtained the following viriel identity
?
?2 2t ?dx |x| L1,1(x;x) ?? (5.20)
= dx L2,2(x, x;x, x) + 4 dx [?x ?x?L1,1](x;x).j j
j=1
If a(x) = |x|, then by simple calculations using the Green?s identities we can show
that the relevant distributional derivatives of a(x) in R3 are given by
2
?|x| = and ?2|x| = ?8??(x).
|x|
Hence we have
?
?2t dx |x|L1,1(x;x)? (5.21a)
L L2,2(x, x;x, x)= 4?? 1,1(t, 0, 0() + dx? )? |x|?j` xjx`
+ dx ? dx? ?(x? x?){? ? ?
| | | |3 xj x
? + ?x? ?x }Lj ` 1,1(x;x ).x x `
j,`
Define the matrix
?
A = dx?ij ?(x? x?){? ?x ?x? + ?x? ?x }L (x;x )j ` j ` 1,1
149
then we can rewrite (5.21a) as follows
?
x
?t dx ? [(? ? ?? )L| | x ? x 1,1](x;x)x ? (5.21b)
L L
T
2,2(x, x;x, x) TrA? u Au
= 4? 1,1(t, 0, 0) + dx + 2 dx|x| |x|
where u = x| | . Integrate (5.21b) with respect to t yieldsx
? T ? T ?
L L2,2(t, x, x, x, x) TrA? u
TAu
4? ? 1,1(t, 0, 0) dt+ ? + 2 dxdt?T ?T |x| |x|
x x
= dx ? [(?x? ??x)L1,1](T, x, x)? dx ? [(?x? ??x)L1,1](?T, x, x).|x| |x|
Recall, by Lemma 4.4 in [GM17], we have
L 11,1(x;x?) = sh ? sh(x;x?) + ?(x)?(x?) = ?(x;x?) (5.22a)
N
and
L ? ? 1 12,2(t, x1, x2;x1, x2) = ?(x ;x? )?(x ;x? ) + ?(x ;x? )?(x ;x?1 1 2 2 1 2 2 1) (5.22b)2 2
1 1
+ ?(x2;x
?
1)?(x1;x
?
2) + ?(x2;x
?
2)?(x1;x
?
1)2 2
+ ?(x1, x2)?(x
?
1, x
?
2)? 2??(x1)??(x2)?(x? ?1)?(x2).
In particular, it follows that
L2,2(t, x, y;x, y)
(5.23a)
= |?(x, y)|2 + ?(x;x)?(y; y) + |?(x; y)|2 ? 2|?(x)|2|?(y)|2
150
and
L2,2(t, x, x;x, x) = |?(x, x)|2 + 2|?(x;x)|2 ? 2|?(x)|4. (5.23b)
Denote u := sh(k) and write
?
1
T00 = |?(t, x)|2 + dy |u(x, y)|2? (5.24a)( N ) 1
T0j = Im ??(x)?j?(x) + dy Im (u?(x, y)?ju(x, y)) . (5.24b)
N
Then observe we have
?
1
Tjk = dy {?x u?(x, y)?x u(x, y) + ?x u?(x, y)?x u(x, y)}
N j k k j
+ ??j(x)?k(x) + ??k(x)?j(x) (5.24c)
1
+ ?jk{L2,2(x, x;x, x)??x?(x;x)}.
2
By (5.24b) we have
?
x
?dx ? [(?(x? ??x)?](t, x;x))|x| ? ( )
xj xj
= dxdz Im u(x) ?x u?(x) + dx Im ?(x) ? ?(x) ,|x| j |x| xj
then, by Proposition A.10 in [Tao06] and the conservation law in [GM13a], we
establish the estimate
????? ?x ? 1dx ? [(?x? ??x)?](t, x;x)?? . ? |? 2x,y| 2 ?0 ?|x| L2(dxdy)
151
which means
?
L2,2(t, x, x;x, x)
dtdx . 1 (5.25)
|x|
since TrA ? uTAu ? 0. Finally, by (5.23b) and the fact that ?(x;x) ? |?(x)|2, we
obtain the estimate
?
|?(t, x;x)|2
dtdx . 1 (5.26a)
|x|
and
?
1 |(u? ? u)(t, x;x)|2
dtdx . 1 (5.26b)
N2 |x|
Hence we have proven the desired Morawetz estimate when v = ?.
Similarly, if we consider the observable a(x) = |x? z|, then by (5.19) we have
?
?2t dx |x? z|L1,1(x?;x) (5.27)
L L2,2(x, x;x, x)= 4?? 1,1(t, z, z() + dx? |x? z)| ??j` xjx`
+ dx ? dx? ?(x? x?){? ?x ?x? + ?x? ?x }L1,1(x;x ).|x? z| |x? z|3 j ` j `
j,`
which again leads to
?
|?(t, x;x)|2
sup dtdx . 1 (5.28a)
z |x? z|
152
and
?
1 |(u? ? u)(t, x;x)|2
sup dtdx . 1. (5.28b)
N2z |x? z|
Note, we also have
?
sup dt ?(t, z, z) . 1. (5.29)
z
Let us summarize our result in the following proposition.
Proposition 5.7. Let (?(t),?(t),?(t)) be a smooth solution to the time-dependent
HFB system corresponding to the case v(x) = ?(x). Then we have the global esti-
mates
?
|?(t, x;x)|2 + 2|?(t, x;x)|2 ? 2|?(t, x)|4
sup dtdx . 1
z |x? z|
and
?
sup dt ?(t, z, z) . 1.
z
Case 2: v(x) positive radial
Let us assume v(x) = v(|x|) and v is monotone decreasing away from the
origin.
153
Take a(x) = |x| in (5.15), we obtain the estimate
?
? v(x? y)L2,2(x, y;x, y)?tMa 4?L?1,1(0, 0) + ? dxdy |x|
1 x
+ dx ? dy v(x? y){?xL2,2(x, y, x, y)??yL2,2(x, y, x, y)}
2 |x|
Next, integrating by parts yields
?
1 ? xdx?dy v(x (y) (? {?xL2,2(x, y), x, y)?? L| | y 2,2(x, y,2 x )x, y)}
?1 x x= ? dxdy [ ?x ? v(x? y) +?x(v(x? y)]) L2,2(x, y, x, y)2 |x| |x|
= ? dxdy v? | (x? y) ? x 1( x? y|) + v(x? y) L
| ? || | | | 2,2
(x, y, x, y)
x y x x
since v is radial, that is, v(x? y) = v(|x? y|). By direct calculation, we see that
?
v(x? y)L2,2(x, y;x, y)
dx?dy ? |x|
1 x
+ ?dx ? dy v(x? y){?xL2,2(x, y, x, y)??yL2,2(x, y, x, y)}2 |x|
= ? [?dxdy
?
?v (|x]? y|
(x? y) ? x
) L2,2(x, y, x, y)|x? y||x|
? (x? y) ? x= ? + dxdy v
?
((|x? y|) L| ? || | 2,2(x), y, x, y)x<y x>y x y x
= ?? dxdy v
?(| ? | (x? y) ? x (y ? x) ? yx y )( + L (x, y, x, y)x>y |x? y||x| |x? y||y| 2,2)
? ? | ? | |x| ? |y| cos ? |y| ? |x| cos ?= dxdy v ( x y ) + L (x, y, x, y) ? 0
x>y |x? y| |
2,2
x? y|
154
Thus, by conservation of energy, we obtain the estimate
? ?
d? ?(?, 0, 0) . |M(T )|+ |M(?T )| . 1.
??
Taking a(x) = |x? z|, one can deduce
?
sup dt ?(t, z, z) . 1. (5.30)
z
Let us summarize our findings
Proposition 5.8. Let (?(t),?(t),?(t)) be a smooth solution to the time-dependent
HFB system corresponding to a positive radially symmetric interaction potential v.
Then we have the global estimate
?
sup dt ?(t, z, z) . 1.
z
Unlike the case v = ?, we were not able to obtain Morawetz-type estimate for
the ? equation. The lack of localization of v introduces the source term lj in (5.7)
which make the the previous approach with of the ? infeasible in the current setting.
155
5.4 Proof of the Interaction Morawetz Estimate for ?
Case 1: v(x) = ?(x)
Let us write down the Virial interaction potential V a corresponding to a(x) as
?
V a(t) := dxdy T00(t, x)T00(t, y)a(x? y) (5.31)
and let us define the Morawetz interaction potential Ma := ?tV a or
Ma(?t) := ? atV (t)
= d?xdy [?tT00(t, x)T00(t, y)a(x? y) + T00(t, x)?tT00(t, y)]a(x? y)
= ??2 dxdy [?jT0j(t, x)T00(t, y)a(x? y) + T00(t, x)?jT0j(t, y)]a(x? y)
= 2 dxdy [T00(t, y)T0j(t, x)? T0j(t, y)T00(t, x)]aj(x? y)
Thus it follows
?
?tM
a(t) = 2 dxdy (T00(x)Tjk(y)? 4T0j(x)T0k(y) + Tjk(x)T00(y))ajk(x? y)
(5.32)
156
provided a(x) is even. First, let us consider the integrand of the off-diagonal terms
( [ ? ]
2
2 ?([x, x)? dzRe(?y u?(y, z)?y u(y, z)) + 2 Re(??j(]y)?k(y))N j k
? 14 [ ? dz Im(u?(x, z)?x u(x, z)) + Im(??(x)?j(x))N j ]
? 1 [dz ?Im(u?(y, z)?y u(y, z)) + Im(??(y)?k(y))N k ])
2
+ ?(y, y) dzRe(?x u?(x, z)?x u(x, z)) + 2 Re(??j(x)?k(x)) ajk(x? y)
N j k
which have good terms given by
?(x)??(x)(??j(y)?k(y) + ?j(y)??k(y)) (5.33a)
+ (??(x)?j(x)? ?(x)??j(x))(??(y)?k(y)? ?(x)??k(y)) (5.33b)
+ ?(y)???(y)(??j(x)?k(x) + ?j(x)??k(x)) (5.33c)
1
+ ? ? ?
2 ? dzdz {?y u?(y, z)?y u(y, z) + ?j k y u(y, z)?y u?(y, z)}u(x, z )u?(x, z ) (5.33d)N j k
1
+ dzdz? {u?(x, z?)? ? ?x u(x, z )? u(x, z )?x u?(x, z?)}
N2 j j
? {?u?(y, z)?y u(y, z)? u(y, z)?y u?(y, z)} (5.33e)k k
1
+ dzdz? {? ? ?x u?(x, z )?x u(x, z ) + ?x u(x, z?)? u?(x, z?x )}u(y, z)u?(y, z)
N2 j k j k
(5.33f)
157
and troublesome terms given by
?
1
(??j(y)?k(y) + ??j(y)??k(y)) dz u?(x, z)u(x, z) (5.34a)N
1
+ ?(x)??(x) dz {?y u?(y, z?)?y u(y, z) + ?y u(y, z)? u?(y, z)} (5.34b)N j k j yk
+ (??(x)?j(x)?
1
?(x)??j(x)) ? dz {u?(y, z)?y u(y, z)? u(y, z)?y u?(y, z)} (5.34c)N k k
1
+ (??(y)?k(y)? ?(y)??k(y)) ?dz {u?(x, z)?x u(x, z)? u(x, z)?x u?(x, z)} (5.34d)N j j
1
+ (??j(x)?k(x)?+ ?j(x)??k(x)) dz u?(y, z)u(y, z) (5.34e)N
1
+ ?(y)??(y) dz {?x u?(x, z)?x u(x, z) + ?x u(x, z)?j k j x u?(x, z)}. (5.34f)N k
For the good terms, the ? terms, (5.33a)-(5.33c), could be rewritten as follows
(5.33a) + (5.33b) + (5.33c) = Pj(x, y)Pk(x, y) +Qj(x, y)Qk(x, y) (5.35a)
where
Pj(x, y) := ?(x)??j(y) + ?j(x)??(y)
and
Qj(x, y) := ?(x)?j(y)? ?j(x)?(y).
158
Likewise, the u terms, (5.33d)-(5.33f), could also be rewritten as
(5.33d)?+ (5.33e) + (5.33f) (5.35b)
1
= dzdz? {R (z, z?, x, y)R (z, z?, x, y) + S (z, z?, x, y)S (z, z?j k j k , x, y)}
N2
where
Rj(z, z
?, x, y) := u(x, z?)?y u?(y, z) + ?x u(x, z
?)u?(y, z)
j j
and
Sj(z, z
?, x, y) := u(x, z?)?y u(y, z)? ?x u(x, z?)u(y, z).j j
For the troublesome (mixed) terms, let us consider their integrand, that is
(??j(y)?k(y) + ?j(y)??k(y))u?(x, z)u(x, z) (5.36a)
+ ?(x)??(x){?y u?(y, z)?y u(y, z) + ?y u(y, z)?y u?(y, z)} (5.36b)j k j k
1
+ (??(x)?j(x)? ?(x)??j(x)){u?(y, z)?y u(y, z)? u(y, z)?y u?(y, z)} (5.36c)
2 k k
1
+ (??(y)?k(y)? ?(y)??k(y)){u?(x, z)?x u(x, z)? u(x, z)?x u?(x, z)} (5.36d)
2 j j
1
+ (??(y)?j(y)? ?(y)??j(y)){u?(x, z)?x u(x, z)? u(x, z)?x u?(x, z)} (5.36e)
2 k k
1
+ (??(x)?k(x)? ?(x)??k(x)){u?(y, z)?y u(y, z)? u(y, z)?y u?(y, z)} (5.36f)
2 j j
+ (??j(x)?k(x) + ?j(x)??k(x))u?(y, z)u(y, z) (5.36g)
+ ?(y)??(y){?x u?(x, z)?x u(x, z) + ?x u(x, z)?x u?(x, z)}. (5.36h)j k j k
159
Define the terms
A?j (x, y) := u(x)??j(y)? u?(x)?j(y) (5.37a)
B? ?j (x, y) := Aj (y, x) (5.37b)
C?j (x, y) := ??(x)?y u(y)? ?(x)?y u?(y) (5.37c)j j
D?j (x, y) := C
?
j (y, x) (5.37d)
E?j (x, y) := u?(x)??j(y)? u(x)?j(y) (5.37e)
F?j (x, y) := E
?
j (y, x) (5.37f)
G?j (x, y) := ?x u?(x)??(y)? ?j x u(x)?(y) (5.37g)j
H?j (x, y) := G
?
j (y, x). (5.37h)
By direct computation, one can check that (5.36) can be written as follows
1 + + + + 1(5.36) = (Aj +D
? ? ? ?
4 j
)(Ak +Dk ) + (A4 j
+Dj )(Ak +Dk ) (5.38a)
1 + + + + 1+ (Bj + Cj )(Bk + Ck ) + (B
?
j + C
?
j )(B
? ?
4 4 k
+ Ck ) (5.38b)
1
+ (E+j ?G+ +j )(Ek ?G
+ 1
k ) + (E
?
j ?G?j )(E? ?4 4 k
?Gk ) (5.38c)
1
+ (F+ ?H+ 1j j )(F+ ?H+k k ) + (F
? ?
j ?Hj )(F?k ?H
?
k ). (5.38d)4 4
For the strictly-diagonal terms, we have that the L2,2 term is given by
?a(x? y){?(x, x)L2,2(y, y, y, y) + ?(y, y)L2,2(x, x, x, x)} (5.39a)
160
and the Laplacian term
??a(x? y){?(x, x)?y?(y, y) + ?(y, y)?x?(x, x)}. (5.39b)
Hence combining (5.35), (5.38), and(5.39) yields the following Morawetz Identity
?
? Mat (t) = 2 ?dxdy ?(t, x, x)?(t, y, y)(???a(x? y)) (5.40)
+ ?dxdy {?(t, x, x)L2,2(t, y, y, y, y) + ?(t, y, y)L2,2(t, x, x, x, x)}?a(x? y)
+ 2 ?dxdy {Pj(t, x, y)Pk(t, x, y) +Qj(t, x, y)Qk(t, x, y)}ajk(x? y)
2
+ dxdydzdz? {Rj(z, z?, x, y)Rk(z, z?, x, y)
N2
+ Sj(z?, z?, x, y)S (z, z?k , x, y)}ajk(x? y)
1
+ dxdydz {(A+ +D+j j )(A+ +k +Dk ) + (A
? ?
j +Dj )(A
? ?
2N k
+Dk )
+ (B+j + C
+
j )(B
+ + C+k k ) + (B
?
j + C
?
j )(B
?
k + C
?
k )
+ (E+ + + + ?j ?Gj )(Ek ?Gk ) + (Ej ?G
?)(E?j k ?G
?
k )
+ (F+ ?H+ + + ? ? ? ?j j )(Fk ?Hk ) + (Fj ?Hj )(Fk ?Hk )}ajk(x? y).
Take a(x) = |x|. Then we obtain the estimate
? ?
?(t, x, x)L2,2(t, y, y, y, y)
?tM(t) & dx |?(t, x, x)|2 + dxdy (5.41)|x? y|
161
In particular, we have that
? T ?
dt dx |?(t, x, x)|2 .M(T )?M(?T )
?T
and
? T ? ?(t, x, x)L2,2(t, y, y, y, y)
dt dxdy .M(T )?M(?T ).
?T |x? y|
Finally, observe by Proposition A.10 in [Tao06] we have
??? ( )?
| 4 xj ? yj
?
M(t)| ? ???? dxdydz ?(y, y) Im( u?(x, z) ?ju)(x? , z) ?N ? |x? y| ? ??? xj ? yj+? 4 dxdy ?(y{, y) I?m ??(x) ? ?(x) ?|x? y| j ? }
1
. dy ?(t, y, y) dz ?u(t, ?, z) ?2 + ??(t) ?21/2 1/2
N H?x H?x
which means
? T ? { ? }
dt dx |?(t, x, x)|2 . sup ??(t) ? 1L1 dz ?u(t, ?, z) ?2 + ??(t) ?2x 1/2 1/2
?T t=T,?T N H?x H?x
By the conservation laws in [GM13a], we see that indeed
??(t, x, x) ?2L2(dtdx) . 1. (5.42)
162
Likewise, we also have
?
?(t, x, x)L2,2(t, y, y, y, y)
dtdxdy . 1, (5.43a)
|x? y|
equivalently,
?
?(t, x, x){|?(t, y, y)|2 + 2|?(t, y, y)|2 ? 2|?(t, y)|4}
dtdxdy . 1. (5.43b)
|x? y|
Case 2: v(x) positive radial
As in the previous case, we write down the Morawetz interaction potential
?
Ma(t) = 2 dxdy (T00(t, y)T0j(t, x)? T0j(t, y)T00(t, x))aj(x? y)
then observe
?
?tM
a(t) = 2 dxdy (?tT00(t, y)T0,j(t, x) + T00(t, y)?tT0j(t, x)
???tT0j(t, y)T00(t, x)? T0j(t, x)?tT00(t, x))aj(x? y)
= 2 dxdy (?2?kT0k(t, y)T0j(t, x)? T00(t, y)(?kTkj(t, x) + lj(t, x))
+ (?kTkj(t, y) + lj(t, y))T00(t, x) + 2T0j(t, x)?kT0k(t, x))aj(x? y).
163
Hence we obtain the following identity
?
? Mat (t) = 2 dxdy {T00(x)Tjk(y)? 2T0j(x)T0k(y)
? 2T?0j(y)T0k(x) + Tjk(x)T00(y)}ajk(x? y) (main term)
+ 2 dxdy {lj(t, y)T00(t, x)? lj(t, x)T00(t, y)}aj(x? y) (error term)
(5.44)
Applying a similar calculation as in the case when v(x) = ?(x), we see that
?
main term = 2 ?dxdy ?(t, x, x)?(t?, y, y)(???a(x? y))
+ dxdy {??(t, x, x) dz v(y ? z)L2,2(t, y, z, y, z)
+ ?(?t, y, y) dz v(x? z)L2,2(t, x, z, x, z)}?a(x? y)
+ 2 ?dxdy {Pj(t, x, y)Pk(t, x, y) +Qj(t, x, y)Qk(t, x, y)}ajk(x? y)
2
+ dxdydzdz? {Rj(z, z?, x, y)Rk(z, z?, x, y)
N2
+ Sj(z?, z?, x, y)Sk(z, z?, x, y)}ajk(x? y)
1
+ dxdydz {(A+ +D+)(A+ +D+) + (A? +D?)(A? +D?)
2N j j k k j j k k
+ (B+j + C
+
j )(B
+
k + C
+
k ) + (B
?
j + C
?
j )(B
?
k + C
?
k )
+ (E+ ?G+j j )(E+k ?G
+) + (E? ?G?)(E?k j j k ?G
?
k )
+ (F+ ?H+)(F+ ?H+) + (F? ?H?)(F? ?H?j j k k j j k k )}ajk(x? y).
(5.45)
Therefore, it suffices to focus on the error term for the remaining of the section.
The treatment of the error term will follow that of [GM13b].
164
Observe
?
error term = ? 4? dxdy lj(t, x)T00(t, y)aj(x? y)
= ? 2 dxdydz v(|x? z|){?z Lj 2,2(x, z;x, z)? ?x Lj 2,2(x, z;x, z)}
? T0?0(y)aj(x? y)
= ? xj ? zj4? dxdydz v
?(|x? z|) L2,2(x, z;x, z)T| ? | 00
(y)aj(x? y)
x z
? 2 dxdydz v(x? z)L2,2(x, z;x, z)?(t, y; y)?x aj(x? y)j
To further the computation let us take a(x) = |x|, then it follows
?
x
error term ? ? | ? | j ? zj xj ? yj= 4? dxdydz v ( x z ) L2,2(x, z, x, z)T| ? | | ? | 00(y)x z x y
? 2 ? dxdydz v(y ? z)L2,2((x, z, x, z)T00(y)?|x? y| )
= ? 2 dxdydz v?(|x? z| xj ? zj xj ? yj zj ? xj zj ? yj) +
|x? z| |x? y| |z ? x| |z ? y|
? L?2,2(x, z, x, z)T00(y)
? 2 dxdydz v(y ? z)L2,2(x, z, x, z)T00(y)?|x? y|.
Thus, we have the following Morawetz estimate
? Mat (t) = (mai?n term) + (error term) (5.46)
? 16? dx |?(t, x, x)|2
165
which again means
{ ? }
? 1?(t, x, x) ?2 2L2(dtdx) . ??0 ?L1 dz ? sh(k0)(?, z) ? 1 + ??0 ?2 1 .x N H? 2 (dx) H? 2 (dx)
In particular, we also have
?? ?L4(dxdy) ? C (5.47a)
and
? ???1 ??2
dxdt ?
N2 ? dz | sh(k)(x, z)|2?? ? C. (5.47b)
Also we have that
|?(x, y)| ? ?(x, x)1/2?(y, y)1/2
which means
? (? )1/2 ?
dt dxdy |?(t, x, y)|4 ? dtdx ?(t, x, x)2 . 1. (5.48)
166
Chapter 6: Collapsing Estimates on Closed Manifolds
6.1 Main Results
Let (M, g) be a closed Riemannian manifold with dimension d ? 3 and denote
?g the corresponding Laplace-Beltrami operator associated to the metric g. Since
we only consider closed manifold with a fixed metric g, it is convenient to write ?
in place of ?g when the context is clear. We begin by considering both the linear
Schr?dinger equation
Sg?(t, x1,?x2) = (i?t + ?1 + ?2) ?(t, x1, x2) = 0,? (6.1)? = ? ? C?(M ?M)
t=0 0
and the von-Neumann Schr?dinger equation
S?g ?(t, x1,?x2) = (i?t + ?1 ??2) ?(t, x1, x2) = 0,? (6.2)? = ?0 ? C?(M ?M)t=0
defined on the product manifold (M ?M,G = g ? g).
The first goal of the chapter is to establish collapsing estimates, which are
natural generalization of bilinear Strichartz-type estimates to the case of arbitrary
167
tensor products, for both (6.1) and (6.2) on general closed manifolds.
Theorem 6.1. Assume d ? 3. Suppose ?(t),?(t) are solutions to (6.1) and (6.2),
respectively. Then there exists ? > 0 such that for all ? > ? = d?1? we have the2
estimates
? diag ?(???L
2([??,?]?H?1(M))
? ?
(6.3a)
1 ?
. ? ? ?? ? )? 1 ?min ( ?1) 2 (??2) 2 ?0 ? , ? (??1) 2 (??2) 2 ?0 ?
L2(M?M) L2(M?M)
and
? diag ?(???L
2([??,?]?H?1(M)) (6.3b)
. min ? ? ? ? )1 ? ? 1(? ? ??1) 2 (??2) 2 ?0 ? ,? (??1) 2 (??2) 2 ?0 ?? .
L2(M?M) L2(M?M)
Here, diagF denotes the restriction of F to the diagonal subset {(x, y) ? M ?M |
x = y}.
The proof of Theorem 6.1 is based on ideas introduced in [Sog93b] to prove
the local well-posedness of the nonlinear wave equations with variable coefficients
and the semiclassical techniques used in [BGT04] to prove the Strichartz estimates
for the Schr?dinger equation on closed manifold. Moreover, the collapsing estimates
for (6.1) can be viewed as a generalization of the bilinear Strichartz estimates on
closed manifold as proved in [Han12].
168
6.2 Collapsing Estimates
This section is devoted to the proof of Theorem 6.1. To begin, we follow the
ideas used in [BGT04] by first describing the effect of spectral localization relative to
the elliptic operator ?g on local coordinate patches of the product manifold. Then
we prove the spectral localized versions of Theorem 6.1. Finally, we use spectral
dyadic techniques to sum up the different range of the spectrum to obtain the
results of Theorem 6.1. Moreover, since the underlying geometries of (6.1) and (6.2)
are different, we will treat the equations separately.
The author would like to begin by apologizing to the reader for the fact that
this section will not be self-contained. In fact, we borrow many tools from [BGT04,
Han12]. Nevertheless, we will provide the reader with detailed reference to the
relevant section or statement of these papers when necessary.
We adopt the Kohn-Nirenberg pseudodifferential quantization rule, that is
?
1
a(x,D)u = eix??a(x, ?)u?(?) d?
(2?)d Rd
for every smooth symbol a and u ? C? d0 (R ). See [Sog93a,H?r94].
Let us state the following proposition which says the spectral localization op-
erator on product manifold is well-approximated by pseudodifferential operators.
Lemma 6.2. Let ? ? C?c (R) and supp? ? [1 , 1], ??? := ?? ? ?? : U? ? U2 ? ?
Rd ? V? ? V? ? M ?M a coordinate patch, and ?1, ?2 ? C?0 (V1 ? V2) such that
?2 = 1 near the support of ?1. Then there exist sequences of symbols (?kj )j?0 of
169
C?c (U
d
k?R ), k = 1, 2, such that for every N ? N, for every h,m ? (0, 1], for every
s ? [0, N ], and for every f ? C?(M ?M), we have the estimate
???? ?N?1? ? ??? (? P 1P 2 f)? hj ` ??? 1 1 1 m ?1j (x, hD 2x)?` (y,mD ?y)???(?2f)?h m ?
? j,`=0 H?s(R2d)
.N h
N?s1mN?s2? f ?L2(M?M)
s1+s2=s
where ?? denotes the standard pullback, P j 21 = ?(h ?j), and D = ?i?. Applying
h
the sharp trace theorem, we immediately get the estimate
???? ? ???
N??1 ? ??
?? (? P 1 2 i j 1
?
?? 1 1P 1 f)? h m ?i (x, hDx)?2j (y,mD )??y ??(?2f)? ??
h m ?
? i,j=0
?
x=y L2(Rd)
. hN?s1mN?s2N ? f ?L2(M?M).
s1+s2=s
Proof. This follows immediately from Proposition 2.1 in [BGT04].
6.2.1 Estimates for (6.1)
1
For brevity of notation, let us adopt the convention Di = (??i) 2 . We are
ready to prove the following proposition
Proposition 6.3. Let h ? (0, 1], ? ? C?(R) with supp? ? [1c , 1] and define2
P 1 = ?(h2?). Assume P 1 21P 1 ?0 = ?0. Then we have the estimate
h h h
? ?
1
? ? ?diag ? ? 2 ??L2([??h,?h],H?1(M)) . ?D1 D2 ?0 ? (6.4a)
L2(M?M)
170
of some ? > 0, independent of h, or equivalently
? diag ? ? ??L2([??,?],H?1(M)) . ?D1D2 ?0 ?L2(M?M) (6.4b)
where ?? is defined in Theorem 6.1.
Remark 6.4. The reader should note that the choice of derivatives in Proposition
6.3 is superficial. In fact, the spectral localization allows us to rewrite (6.4a) as
d
? diag ? ? ? 2L2([??h,?h],H?1(M)) . h ??0 ?L2(M?M) .
However, our choice of derivatives will be more transparent later in Section 6.2.3
when we discuss the Bourgain refinement estimates. Moreover, following our proof
of Proposition 6.3, one could also prove
d?2
? diag ? ? ? ? . h
?
2
L2([ ?h,?h] M) ??0 ?L2(M?M) .
Following closely the presentation of [BGT04], we start by proving Proposition
6.3 on local coordinate charts.
Proposition 6.5 (Local Collapsing Estimate for (6.1)). Let U1 ? V1 ? R2d be a
product of open balls U d1, V1 ? R , en?dowed with?Riemannian metrics g?, g?, respec-
tively, such that U1 ? V1 6= ? and g ? ?? U ? = gV ? U ? . Let U b UV 2 1 and V2 b V1 be1 1 1 1
open balls, again with U2 ? V2 6= ?, ? ? C? ? 2d 10 0 (U2 ? V2), ? ? C0 (R \{|?|, |?| < }).2
Then there exists ? > 0 such that for every ? > 0, h ? (0, 1], and w0 ? C?0 (U1?V1),
171
we can find a w? ? C?([??, ?]? U2 ? V2), compactly supported, satisfying
ih? 2sw? + h ?Gw? = r, w?(0, x, y) = ?0(x, y)?(hDx, hDy)w0(x, y),
such that we have the estimate
? ?
?? w?(s, x, x) ? . h?
1 ? 1 ?? ?
x 2 2L2([??,?]?U ?V ) ? |?2 2 x| |?y| w0 ? .L2(U1?V1)
Moreover, if w is a solution to (6.1) written in the above local coordinate with the
microlocalized initial data w?(0) then we have w(s, x, y) = w?(s, x, y)+R(s, x, y) where
?R(s, x, x) ? . hsome positive powerL2([??,?]?U2?V ) ?w2 0 ?L2(U1?V .1)
Proof of Proposition 6.5. The proof employs the WKB approximation to the solu-
tion of (6.1). More precisely, we seek an approximation w? given by the oscillatory
integral1
? ( )
i
w?(s, x, y) = e ?(s,x,y,?,?)
? ? d?d?
h a(s, x, y, ?, ?, h)w?0 , (6.5)
R2d h h (2?h)
2d
where
?N
a(s, x, y, ?, ?, h) = hjaj(s, x, y, ?, ?).
j=0
1Since w0 is compactly supported then we could trivially extend w0 to all of R2d. Also, w?0 is
the Fourier transform of the microlocalized data w?0 not w0. Moreover, the variables ? and ? are
scaled to have length on the order of 1, that is 12 ? |?|, |?| ? 2.
172
Here N is chosen to be sufficiently large, aj ? C?0 ([??, ?] ? U2 ? V2 ? R2d) are
solutions to some transport equations satisfying the initial data
a0(0, x, y, ?, ?) = ?0(x, y)?(?, ?) and aj(0, x, y, ?, ?) = 0 for all j ? 1,
and the phase function ? ? C?([??, ?] ? U2 ? V2 ? A), where the annulus A =
{1 ? |?|2 + |?|2 ? 4} contains the support of ?, is a real-valued smooth function on
4
the support of a satisfying the eikonal equation2
?
? ?+ Gjks ?j??k? = 0 (6.6)
1?j,k?2d
where Gjk is the dual metric with initial condition ?(0, x, y, ?, ?) = x ??+y ??. Since
the Riemannian metric is given by G = g ? g, then the eikonal equation reduces to
?s?+ g(?x?,?x?) + g(?y?,?y?) = 0, ?(0) = x ? ? + y ? ?. (6.7)
One could make a further observation, if ? is a solution to
?s?+ g(?x?,?x?) = 0, ?(0, x, ?) = x ? ? (6.8)
2By the standard Hamilton-Jacobi theory, we see that the eikonal equation is well-posed on
some small time interval [??, ?] (c.f. Chapter 9 of [Arn97]). Moreover, It is also convenient to
write (6.8) in the form ?s?+ |?x?|2g = 0 where |?|2 := gijg (x)?i?j .
173
then ?(s, x, ?) + ?(s, y, ?) uniquely solves (6.7). Rewrite (6.5), we get
? ( )
i
w?(s, x, x) = e {?(s,x,?)+?(s,x,?)}
? ? d?d?
h a?(s, x, ?, ?, h)w?0 ,
h h (2?h)2d
where a? is the collapsed function of a. Then it follows
?
1 i
? w?(s, x, x) ? e {?(s,x,?)+?(s,x,?)}j h (?(j?(s, x), ?) + ?j?(s, x, ?))i
? ? ? d?d?a?(s, x, ?, ?, h)w?0 ,
h h (2?h)2d+1
+ lower order term
for j = 1, . . . , d.
To complete the proof of Proposition 6.5 it suffices to prove the following
proposition.
Proposition 6.6. Consider T : L2(R2d)? L2(R1+d) defined by
?
i d?d?
(TF )(s, x) = e ?(s,x,?,?)h qh(s, x, ?, ?)F (?, ?) (6.9)
(2?h)2d+1
where ? is defined as above and qh vanishes either on the complement of 1 <4
|?|2 + |?|2 < 4 or (s, x) 6? [??, ?] ? X, X compact, and |??s,xqh| ? C?|? + ?|. If
? is sufficiently small then there exists a constant C, depending on finitely many
derivatives of qh, so that the following holds
? ?
? 3d+1? ? 1TF ? ?? ?L2(R 2 21+d) ? Ch ? |?| |?| F ? . (6.10)
L2(R2d)
174
As a preliminary to the proof of Proposition 6.6, let us recall some properties
of the phase function which will also be useful for the proof and later on in the
section.
Taylor expanding the solution of (6.8) about t = 0 yields
?(t, x, ?) = ?(0, x, ?) + t(?t?)(0, x, ?) +O(t2)
(6.11)
= x ? ? ? t|?x?(0, x, ?)|2 2g +O(t )
for |t| < ?. In many of the proofs of the collapsing estimates, we will need to handle
the phase function
??(t, x, ?, ?, ??, ??) = ?(t, x, ?)? ?(t, x, ?)? ?(t, x, ??)? ?(t, x, ??)
which by (6.11) has the form
( )
x ? {(? ? ??) + (? ? ??)} ? t |?|2g ? |?|2g ? |??|2g ? |??|2g +O(t2)
when |t| < ?. Making the change of variables (?, ?, ??, ??) 7?? 1(? + ?, ? ? ?, ?? +
2
??, ?? ? ??) yields
x ? (? ? ??)? tp?(x, ?, ?, ??, ??) +R(t, x, ?, ?, ??, ??) (6.12)
where p+(x, ?, ?, ??, ??) = 1(|?|2g + |?|2g ? |??|2? |??|2g g) and p?(x, ?, ?, ??, ??) = g(?, ?)?2
175
g(??, ??) and the remainder R satisfies
R(0, ?, ?, ??, ??) = (?tR)(0, ?, ?, ?
?, ??) = 0.
Then we have the following lemmas.
Lemma 6.7. Let (t, x) ? [??, ?] ? X, X ? Rd compact. If ? > 0 is sufficiently
small then we have the estimate
|? | ? 1t,x?(t, x) (|? ? ??|+ |p(x, ?, ?, ??, ??)|) . (6.13)
2
Proof. By the Taylor expansion (6.12), we see that
?? ? ? ? ? ??p(?, ?, ??, ??)? ? 0 ? ? ?tR?
? ? ? ? ? ? ?t,x? =? ?? t? ?+? ? .
? ? ?? ? ? ?xp(?, ?, ? , ? ) ?xR
Thus, we have that
|?t,x?| = |? ? ??|+ |p(x, ?, ?, ??, ??)|+O(t).
Hence when t is sufficiently small the O(t) error term will be dominated by 1 |?? ??|
2
which yield the desired result.
Lemma 6.8. Let N > 0 and (t, x) ? [??, ?] ? X, X ? Rd compact. If ? > 0 is
176
sufficiently small then there exists C > 0 such that
|?m???(t, x)| ? C sup sup |??p(x, ?, ?, ??, ??t x x )| (6.14)
m,|?|?N x?X
where m ? N>1, ? = (?1, . . . , ?d) ? Nd>1.
Proof. The proof is essentially the same as the proof of Lemma 6.7.
Let us continue with the proof of the proposition.
Proof of Proposition 6.6. Expanding the L2 norm of TF and making the change of
variables3 (?, ?) 7? (? + ?, ? ? ?), we get
? ? ? ? ?
?TF ?2 ? ? ? K(?, ?, ? , ? )F? (?, ?)F? (? , ? )2 d?d?d? d? | | 1 | ? | d?1? + ? 2 ? ? 2 |?? ?| 1 d?1+ ? 2 |?? ? ??| 2
1 d?1
where F? (?, ?) := |? + ?| 2 |? ? ?| 2 F (? + ?, ? ? ?) and
?
K(?, ?, ??, ??
1 i
) := e ?(s,x)h q?h(s, x, ?, ?, ?
?, ??) dxds
(2?h)4d+2
with phase function ?(s, x) = ?(s, x, ?, ?, ??, ??) given by
?(s, x, ?, ?, ??, ??) = ?(s, x, ? ? ?, ? + ?)? ?(s, x, ?? ? ??, ?? + ??)
and |q? (s, x, ?, ?, ??, ??h )| ? C|?||??|. Next, we employ the technique of non-stationary
3Here, we abused notation. More accurately, we should have (??, ??) maps to ? = ?? + ?? and
?? = ??? + ???...etc.
177
phase to estimate the kernel K. Define the operator
L(s, x,D ) = i?1s,x ?? ?2s,x?,?s,x?|?s,x?|
then by applying integration by parts yields
?
? ? 1 iK(?, ?, ? , ? ) = dxds e ?(s,x)h (L?)N q?h(s, x, ?, ?, ??, ??).
(2?h)4d+2
Here we note that there are essentially two types of terms which we need to handle,
namely
?? ?s,xq?h(?s,x?)
, |?| = |?| = N (6.15)
|? 2Ns,x?|
and
?
(??0 q? m ?k ?s,x h ? k=1 ?s,x?)(?s,x?) , |?|,m ? N,?0 + ? ? ?+ ?m = N (6.16)|?s,x?|2N+m
since the general terms of (L?)N q?h are linear combination of (6.15) and (6.16).
Applying Lemma 6.7 and 6.8 and the fact that q?N vanishes on the complement
of 1 < |?|2 + ?2 < 4 then it follows
4
| L? N | ? |?||??|( ) q?h . ? .
(h?1|? ? ??|+ h?1?|?|2g + |?|2 ? 2g ? |? |g ? |??|2?)Ng
? ?
Note, when |????|+?|?|2 + |?|2?|??|2?|??|2?g g g g ? h we will not perform any integration
178
by parts. Moreover, since | ? | and | ? |g are comparable, then by change of variables,
independent of h, we could estimate the kernel uniformly on [??, ?]?X as follows
| ? ? | 1 ??|?||??|K(?, ?, ? , ? ) . ? .h4d+2 (1 + h?1|? ? ??|+ h?1 |?|2 + |?|2 ? |??|2 ? |??|2?)N
Next, using polar coordinates, ? = ?? where ? > 0, ? ? Sd?1, we have
?
? ?2 1TF . d??d?1d?d??(??)d?1d??d?d?? F? (?, ??)F? (??, ?? ?2 ? )h4d+2
| || ?|| |? 1 | ? |? d???1 | ? ? ?|? 1? ? ? + ?? ? ?? ? + ? ? |?? ? ??? d?1
(6.17)
2 2 2
? ? ?
?|? 2
(1 + h?1|? ? ??|+ h?1 |?|2 + ?2 ? |??|2 ? ??2 )N
To further estimate the RHS of (6.17), we begin by applying Cauchy-Schwarz in-
equality in the angular variables to get
????? ? ?? ?? F? (?, ??)d? ??? ??? F? (??, ????)d?? ??1 d?1 1 d?1 ?Sd?1 |? + ??| 2 |? ? ??| 2 ? ? Sd?1 |?? + ????| 2 |?? ? ????| 2 ?
h(?, ?)h(??, ??)
.
(|?| d | d2 + ?2) 4 ( ??|2 + ??2) 4
(? ) 1
where 2h(?, ?) := d? |F? (?, ??)|2 . Note we have used the fact that
? ?
d? 1 ??? sind?2? ?d?
| ?Sd?1 ? + ??||? ? ??|d?1 d d?2(|?|2 + ?2) 2 ? 1? ?2 cos2 ?(1? ? cos ?) 2
1
. ,
(|?| d2 + ?2) 2
for some 0 < ? < ? where 0 < ? < 1 since 1 ? |? + ?|, |? ? ?| ? 2.
2
179
Finally, let us estimate
?
1 d?d?d??d?? |?||??|?d?1(??)d?1h(?, ?)h(??, ??)
(6.18)
h4d+2 (| d?|2 + ?2) 4 (|??| d2 + ??2) 4 (1 + h?2|? ? N??|2 + h?2D2) 2
where D = |?|2 + ?2 ? |??|2 ? ??2. We begin by making the change of variables
(?, ?) 7? (?, ?) = (?, |?|2 + ?2). Then (6.18) becomes
?
| || ?| ? | |2 d?2 d?21 ? ? ? ? (? ? ) 2 (? ? ? |??|2) 2 h?(?, ?)h?(??, ? ?)d?d?d? d? (6.19)
h4d+2 d d N? ? ?4 4 (1 + h?2|? ? ??|2 + h?2(? ? ? ?)2) 2
where the integration take place over the region ? ? |?|2 and ? ? ? |??|2. By Young?s
inequality, we have that
?
1 |?|2(? ? |?|2)d?2
(6.19) . ? d?d? d |h(?, ?)|
2
h3d+1 ? 2
1 |?|2?d?2? ? d??
d?1d? d |h(?, ?)|2h3d+1 (|?|2 + ?2) 2
1
. d?d? |?|2|?|d?2|F (?, ?)|2.
h3d+1
This completes the proof of Proposition 5.
Now, let us conclude the proof of Proposition 6.5. First note, by rescaling
? and ? in estimate (6.31) and applying Plancherel and boundedness of projection
operator, we get the desired estimate
?
?? ? ?
1 ? ?1 d?1 ?
xw?(s, x, x) 2 2 2L2([??,?]?U ?V . h ? |?x| |?y| w0 ? .2 2 L2(U1?V1)
180
Finally, for the error term we see that
? s
R(s, x, x) = [ei(s??)?Gr](?, x, x) d? ? O(hN+1)
0
with
? ( )
i ? ? d?d?
r(?, x, y) = hN+2 e ?(?,x,y)h b(?, x, y, ?, ?)w?0 , .
h h (2?h)2d
where b ? C?0 ([??, ?]?U2?V2?B). By a straightforward application of the trace
theorem, we see that
?
?? R(s, x, x) ? . sup ? [ei(s??)?Gx L2(I?L2(U ?V )) I ? r](s, x, x) ?H1 d?2 2 xs I
by trace theorem on R2d . sup ? [ei(s??)?GI r](s, x, y) ?H?? d?x,y
s I
by Strichartz ineq. .I ? r ?L?H?? .
N+2??
? h ? w?0 ?L2s x,y ?,?
where ? > d + 1. This completes the proof of Proposition 6.5.
2
Remark 6.9. To get higher derivative Strichartz estimates on closed manifold, we
1
used the fact Dg = (??g) 2 commutes with the Schr?dinger operator to get the
Strichartz estimates
?Dsg? ?Lp(I?Lq(M)) . ?Dsg?0 ? 1 .H p (M)
Finally, using the fact that ?Dsg? ?Lq(M) ? ?? ?W? s,q(M) we get the desired result.
181
Proof of P?roposition 6.3. Suppose {??} is a partition of unity subordinate to a finite
covering ?? = M , then we have that
??? ?? ?2diag[P 1 2?1P 1 ??]? = dt ? di?ag[P
1
1P 2 21 ?] ? 1
? H (M)h h L2? (I,H
1(M)) I ? h h2
. dt ????(?? ? diag[P 11P 21 ?(t)])
??
?
I ? h h ?H1(Rd)?
? ? 2dt ???(?2 ? 1 2 ??? ? diag[P 1P 1 ?(t)])
h h H1(Rd)
? I
where ? : ?? ? M ? Rd and ??(f) := f ? ? is the standard pullback. Hence it
suffices to prove estimate (6.4) on a single coordinate chart.
Furthermore, as in [BGT04], we begin by making the observation
w(s, ?) = eihs?GP 1 21P 1 ?0 (6.20)
h h
solves the semiclassical equation
ih?sw + h
2?Gw = 0, w(0) = w = P
1P 20 1 1 ?0.
h h
Applying boundedness of eit?G on H?, Lemma 6.2 and the trace theorem, we see
182
that
???? ?? ??(????) (?? ?
?
? 1 2? ? P 1P 1 ?(t))? ??
? h h x=y H1(Rd)?? ? ??. (1??
?
?(hDx, hDy))(???)
?(?? ? ? ? P 1 2? 1P 1 ?(t))? ??
? h h ? x? ??
=y H1(Rd)
+ ???(hD , hD )(? )?(? ? ? ? P 1 2x y ?? ? ? 1P 1 ?(t))? ??
h h x=y
. ??? ??
H1(Rd)
(1?? ?(hDx, hDy))(???)?(?? ? ? 1 2? ? P 1P 1 ?0?)?? h h ? H???(R2d)
+ ???(hD , hD )(? )?(? ? ? ? P 1 2 ?x y ?? ? ? 1P 1 ?(t))? ?
h h x=y H1(Rd)
. hN???? ??0 ?L2(M?M)
+ ?? ? ??(? )?(? ? ? )?(hD , hD )(? )?(P 1 ??? ? ? x y ?? 1P 21 ?(t))? ??
h h x=y H1(Rd)
+ lower order terms.
Note that ?(hDx, hDy) is a shorthand expression for the sum of product of pseu-
dodifferential operators in Lemma 6.2. Finally, apply Proposition 6.5 and sum up
the 1 number of small-time intervals completes the proof of Proposition 6.3.
h
6.2.2 Estimates for the ? Equation
In this subsection, we prove some estimates for (6.2) similar to ones in Propo-
sition 6.3.
Based on the Strichartz estimates for ?(t, x) in the Euclidean space setting
established in Theorem 3.3 of [CHP17], we prove the following proposition.
Proposition 6.10. Let h ? (0, 1], ? ? C?c (R) with supp? ? [1 , 1] and define2
183
P = ?(h2?). Assume P 1P 21 1 1 ?0 = ?0. Then we have the estimate
h h h
? ?
1
? diag ? ? ? 2 ?? ?L2([??h,?h],H1(M)) . ?D1 D2 ?0 ? (6.21a)
L2(M?M)
of some ? > 0, or equivalently
? diag ? ? ??L2([??,?],H1(M)) . ?D1D2 ?0 ?L2(M?M) (6.21b)
where ?? is as defined in Theorem 6.1.
Following the same strategy as in the proof of Proposition 6.3, it suffices to
prove the statement
Proposition 6.11 (Local Coordinate Collapsing Estimate for ?). Let U1?V1 ? R2d
be a product of open balls U1, V1 ? Rd, ?endowed w?ith Riemannian metrics g?, g?,
respectively, such that U ?V 6= ? and g ? = g ?1 1 ? ? ? ? . Let U2 b U1 and V2 b VU1 V 11 U1 V1
be open balls, again with U2?V2 6= ?, ?0 ? C?0 (U2?V ? 2d2), ? ? C0 (R \{|?|, |?| < 1}).2
Then there exists ? > 0 such that, for every h ? (0, 1], u0 ? C?0 (U1 ? V1), we can
find a u? ? C?([??, ?]? U2 ? V2), compactly supported, satisfying
ih?su?+ h
2(?x ??y)u? = r, u?(0, x, y) = ?0(x, y)?(hDx, hDy)u0(x, y),
such that we have the estimate
? ?
1 1
?? ? ?xu?(s, x, x) ? ? ?2 2 ?L2([??,?]?L2(U ?V )) . h ? |?x| |?y| u0 ? . (6.22)2 2 L2(U1?V1)
184
Moreover, if u is a solution to (6.2) written in the above local coordinate with initial
data u?(0) then we have u(s, x, y) = u?(s, x, y) +R(s, x, y) where
?R(s, x, x) ? . hpositive powerL2([??,?]?L2(U2?V2)) ?u0 ?L2(U ?V ). (6.23)1 1
Again, the proof of Proposition 6.11 relies on the following proposition
Proposition 6.12. Consider T : L2(R2d)? L2(R1+d) defined by
?
i
(TF )(s, x) = e ?(s,x,?,?)
d?d?
h qh(s, x, ?, ?)F (?, ?) (6.24)
(2?h)2d+1
where ?(s, x, ?, ?) = ?(s, x, ?)??(s, x, ?) and qh vanishes either on the complement
of 1 < |?|2 + |?|2 < 4 or (s, x) 6? [??, ?]?X, X compact, and |??s,xqh| ? C?|? ? ?|.4
If ? is sufficiently small then there exists a constant C, depending on finitely many
derivatives of qh, so that the following holds
? ?
3d+1 1
? ?TF ? ? 2L2(R1+d) ? Ch ? | | | |?? ?? 2 ? F ? . (6.25)
L2(R2d)
Proof of Proposition 6.12. Expanding the L2 norm of TF and making the change
of variables (?, ?) 7? (? ? ?, ? + ?), we get
?
? ?2 ? ? ? K(?, ?, ?
?, ??)F? (?, ?)F? (??, ??)
TF 2 d?d?d? d? | | 1 | d?1 1 d?1? + ? 2 ? ? ?| 2 |?? + ??| 2 |?? ? ??| 2
185
1
where F? (?, ?) := |? + ?| 2 | ? | d?1? ? 2 F (? ? ?, ? + ?) and
?
? ? 1 iK(?, ?, ? , ? ) = e ?(s,x)h q?h(s, x, ?, ?, ?
?, ??) dxds
(2?h)4d+2
with the phase function ?(s, x) = ?(s, x, ?, ?, ??, ??) given by
?(s, x) = ?(s, x, ? ? ?, ? + ?)? ?(s, x, ?? ? ??, ?? + ??)
and
|q?h(s, x, ?, ?, ??, ??)| ? C|?||??|.
Applying the method of non-stationary phase along with Lemma 6.7 and 6.8, we
obtain the estimate
?
1 |?||??|F? (?, ?)F? (??, ???) d?d?d??d???TF ?2 . ? . (6.26)2 h4d+2 |?+?|?|???|?1 (1 + h?1|? ? ??|+ h?1?? ? ? ? ?? ? ???)N
Note that we have again used the fact that g(?, ?) ? ? ? ? since g ? Id. Next, write
? = (p, ??) then consider the integration with respect to ?. Without loss of generality,
take ? = (|?|, 0, . . . , 0), then we have the integral
?
? ? F? (?, p, ??)F? (?
?, p?, ???)
dpd??dp d?? ? ?
|?|?2+p2+| | ? ?1 ??? 2 1 (1 + h |? ? ? |+ h?1?|?|p? |??|p??)N
? G(?, p)G(?
??, p?). dpdp ?
|?|2+p2.1 (1 + h?1|? ? ??|+ h?1?|?|p? |??|p??)N
186
(? ) 1
where 2G(?, p) = d?? |F? (?, p, ??)|2 . Finally, we see that
?
1 |?||??|G(?, p)G(??, p?)
(6.26) . d?dpd??dp? ? ?
h4d+2 ? (1 + h?1|? ? ??|+ h?1?|?|p? |??|p??)N
1 ? ? G(?, ?/|?|)G(??, ? ?/|??|). ? d?d?d? d?h4d+2 (1 + h?1|? ? ??|+ h?1|? ? ? ?|)N
1
Young?s ineq. . ? d?d? |G(?, ?/|?|)|
2
h3d+1
1
. d?dpd?? |F? (?, p, ??)|2.
h3d+1
Hence we arrive at the desired inequality.
6.2.3 Bourgain Refinement of the Collapsing Estimates
In this subsection we study the collapsing estimates where the spectral vari-
ables corresponding to the two spatial variables of M ?M are localized to different
ranges, that is, we choose ? ? C?0 (R\{0}) and
P 1 21P 1 F0 = F0 (6.27)
h m
for any h,m ? (0, 1] where F0 = ?0 or ?0. The proof of the estimates is based on
Hani?s work on the bilinear Strichartz estimates on closed manifold [Han12] .
Proposition 6.13. For every 0 < h < m ? 1, ? = h and ? ? C?0 (R) withm
supp? ? [1 , 1]. Assume P 11P 21 F0 = F0. Suppose F (t) is a solution to either (6.1)2
h m
187
or (6.2), then we have the estimate
?? ?1? diagF ? 2 ?? ?L2([??h,?h],H?1(M)) . ?D1 D2 F0 ? (6.28a)
L2(M?M)
of some ? > 0, or equivalently
? diagF ? ??L2([??,?],H?1(M)) . ?D1D2 F0 ?L2(M?M) . (6.28b)
Let us state the local version of Proposition 6.13.
Proposition 6.14. Let U1 ? V1 ? R2d be a product of open balls U , V ? Rd1 1 ,
en?dowed with ?Riemannian metrics g?, g?, respectively, such that U1 ? V1 6= ? and
g ? ?? = gU ?V ? ? . Let U b U and V b V be open balls, with U ? V 6= ?,1 1 U1 V 2 1 2 1 2 21
? ? C?0 0 (U2 ? V2), ? ? C?(R2d0 \{|?|, |?| < 1}). Then there exists ? > 0 such that,2
for every h,m ? (0, 1] with h < m and ? = h , w ? C?0 0 (U1 ? V1), we can find am
w? ? C?([??, ?]? U2 ? V2), compactly supported, satisfying the problem
ih? 2 2sw? + h ?xw? ? h ?yw? = r,
w?(0, x, y) = ?0(x, y)?(hDx,mDy)w0(x, y)
with estimates
? ?
1 1
?? ? ?xw?(s, x, x) ?L2([??,?]?H1(U ?V )) . h? 2 ? |?x| 2 |? |??w2 2 y 0 ? . (6.29)L2(U1?V1)
Moreover, if w is a solution to either (6.1) or (6.2), satisfying (6.27), written in local
188
coordinates with initial data w?(0) then we have w(s, x, y) = w?(s, x, y) + R(s, x, y)
where
?R(s, x, x) ? positive powerL2([??,?]?L2(U2?V )) . h ?w0 ?L22 (U1?V ).1
Proof of Proposition 6.14. Similar to the proof of Proposition 6.5, we consider the
WKB approximation (6.5) where ah is supported in [??, ?]?K? [1/2, 1]? [?/2, ?].
Making the rescaling ? ?7 ??, it follows
?
i
? w?(s, x, x) ? e {?(s,x,?)??(s,x,??)}j h (?(j?(s, x), ?)? ?j?(s, x, ??))
? ? ? d?d?ah(s, x, ?, ??)w?0 , .
h m hd+1md
for j = 1, . . . , d. Similar to the proof of Proposition 6.5, we first prove the following
proposition.
Proposition 6.15. Consider T : L2(R2d)? L2(R1+d) defined by
?
i
(TF )(s, x) = e ?(s,x,?,?)
d?d?
h qh,m(s, x, ?, ?)F (?, ?) (6.30)
mdhd+k
where ?(s, x, ?, ?) = ?(s, x, ?) ? ?(s, x, ??) and qh,m vanishes either when |?| ?6 1
or |?| 6? 1, or (s, x) 6? [??, ?] ? X, X compact, and |??s,xqh,m| ? C?|? ? ??|k. If ?
is sufficiently small then there exists a constant C, depending only on finitely many
189
derivatives of qh,m, so that the following holds
? ?
d 1 1
?TF ? ? Ch? ?km?d+ ?2 2 2L (R1+d) ? |?|k? |?|? ?2 ?F ? . (6.31)
L2(R2d)
Remark 6.16. The proof of Proposition 6.15 is similar to the proof of Theorem 1.1
given in [Han12]. However, for completeness, we have included a sketch of the proof
of Proposition 6.15 and refer the reader to [Han12] for the details. The key ingredient
in the argument is the uniform transversality condition satisfied by the two surfaces
?t,x?(t, x, ?) and ? ?1t,x[?? ?(t, x, ??)] in T(t,x)Rd+1 whenever |?|g ? |?|g ? 1. More
precisely, let n1 and n2 be unit normal vectors to the two surfaces, respectively, then
for every ? ? (0, 1] there exists ?0 such that for all ? ? ?0 we have
|?n1(?), n2(?)?g| ? 1? ?
whenever |?|g ? 1 and |?|g ? 1.
Sketch of the Proof of Proposition 6.15. Consider the change of variables (?, ?) ?7
(? ? ??, ?) which gives
?
i
(TF )(t, x) = e ?(s,x,????,?)
d?d?
h qh,m(s, x, ? ? ??, ?)F (? ? ??, ?) .
mdhd+1
190
Let us note that for t sufficiently small (6.11) gives
?? ?1 ?2 ???2??Tg?1(x)??(t, x, ??) = ? ?+O(t),? ???(t, x) ?
Id?d
which is clearly maximal rank when t is sufficiently small, and the unit normal vector
to ?(t, x, ?) is given by
? ?
n1(?) = ? 1 ??? 1 ???+O(t).
1 + 4|?|2g 2g?1(x)?
Hence we have
1 ?2 2(? ? ??)Tg?1(x)
n1(? ? ??)T ?(t, x, ??) = ? ? +O(t) =6 0
? ???(t, x) 1 + 4|? ? ??|2g
on the region |?|g ? 1 and |?|g ? 1 whenever t is sufficiently small. In particular, if
we consider the unit vector v1 in the direction of g(x)?1(? ? ??), it is clear that
???? ???g? ?1? ?(x)(? ? ??), v1?g ??? ? ? |? ? ??|g &? 11 + 4|? ? ??|2 ? 2g 1 + 4|? ? ??|g
whenever ? is sufficiently small. Hence the transversality condition holds.
Now let us rewrite ? in terms of a new basis ? = pv1 + ?? or, simply, ? = (p, ??).
191
Then we have that
?TF ???L2??? ?d??dpd? i= e {?(t,x,?? ? ???) ?(t,x,??)}? ??h a?h,m(t, x, ?, ?)F (? ? ??, ?)
?
hd+1md ?2 ?
d?? ??? i ?? dpd? e {?(t,x,????)??(t,x,??)}h a?h,m(t, x, ?, ?)F (? ? ??, ?)? .hd+1md ?2
Let us define the freezing operator S : L2(Rd+1)? L2(Rd+1?? ).
?
i
S (w) := dpd? e {?(t,x,????)??(t,x,??)}?? h a?h,m(t, x, ?, p, ??)F (? ? ??, ?)
where ?? is frozen. Thus, it suffices to prove that
d 1
?S??(w) ? 2 2L2([??,?]?Rd) . h m ?F ?L2(R2d)
since |??| . 1. Using a TT ?-argument, we see that
?
?S??(w) ?2 ?L2 = d?dpd? dp
? K(?, p, ??, p?)G(?, p)G(??, p?)
where G(?, p) = G (?, p) = |? ? ?(p, ??)|a?? |(p, ??)|bF (? ? ?(p, ??), (p, ??)) and
?
K(?, p, ??, p?
i
) = dtdx e {?(t,x,?,p)??(t,x,?
?,p?)}
h c(t, x, ?, p, ??, p?).
for some smooth compactly supported function c. By applying Lemma 2.1 in
192
[Han12], we can estimate the kernel as follow
|K(?, p, ??, p?)| . (1 + h?1|? ? ??|+m?1|p? p?|)?N .
Finally, by Young?s inequality, we have that
?
G(?, p)G(??, p?)
d?dpd??dp? 2
(1 + h?
. ?K ? 1?G ? .
1| L 2? ? ??|+m?1|p? p?|)N L (d?dp)
where
?
? ? d d(?/h)d(p/m)K = h m . hdL1
(1 + h?1|?|+m?
m.
1p)N
Thus we arrive at the desired result.
The remainder of the proof of Proposition 6.14 follows exactly the same line
of arguments as in the proof of Proposition 6.5.
6.2.4 Proof of Theorem 6.1 for (6.1)
In this section we prove Proposition 6.1 for (6.1). The key ingredients involved
in establishing (6.3a) are Proposition 6.3, Proposition 6.13, and the following two
lemmas.
Lemma 6.17. Assume 0 < h < m ? 1 with ? = h and ? ? C?(R) with supp? ?
m
193
[1 , 1]. Assume P 1P 21 1 ?0 = ?0. Then we have the estimate2
h h
??? ??P diag ? ? . ? ?D D??1 1 ?0 ? 2
m 2 ? 21 L (M?M)L ([ ?,?],H? (M))
for some ? > 0.
Proof of Lemma 6.17. It suffices to prove the statement in local coordinates. By
Theorem 2.1 in [BGT04] and Proposition 6.3, we have that
???? ?N?1? ? ??? ?(?1 ? P 1 diag ?)? mj?j(x,mD)??(?2 ? diag ?)m? ? ?
?
j=0
? ? L2([??h,?h],H?1(Rd))
. hN ? 1D 2 ??1 D2 ?0 ?
L2(M?M)
for any N ? 1. Finally, following the proofs of Proposition 6.5 and Proposition 6.6,
we see that
??? ??2??(x,mD)?
?(?2 ? diag[P 11P 21 ?]) ?
h h L2([??h,?h],H?1(M))
K(?, ?, ??, ??? ? )F? (?, ?)F? (?
?, ??)
. d?d?d? d?
| | 1 | ? | d?1 | ? ?| 1 | ? ? ?| d?1? + ? 2 ? ? 2 ? + ? 2 ? ? 2
+ lowe?r order terms
1 |?|2?d?2
. d??d?1d? |h(?, ?)|2
h3d+1 ? d(|?|2 + ?2) 2
?2
. d?d? |?|2|?|d?2|F (?, ?)|2
h3d+1
where the last inequality is a result of the facts that |?| ? ? and ? ? 1, which are
consequences of rescaling and the restriction imposed by ?(x,mD). The remainder
194
of the argument is similar to the proof of Proposition 6.5.
Lemma 6.18. Assume 0 < h < h? < m ? 1. Then we have that
1
?DgP 1 diag[P 1P 21 1 ?], DgP 1 diag[P 1 2 N 2 ?? 21 P 1 ?]? .N h ?D1 D2 ?0 ?? ? L2(M?M)m h h m h h
for any N and where the ??, ?? is the standard inner product on L2([??h, ?h]?M).
Proof of Lemma 6.18. It suffices to prove the statement in local coordinates. Ap-
plying Theorem 2.1 in [BGT04], Cauchy-Schwarz inequality and Proposition 6.3, we
see that
??? ??? ??x?
?(?1 ? P diag[P 1 2 ?1 1P 1 ?]),?x? (?1 ? P 1 diag[P 11 P 21 ?])??
? m h h m h? h?? ?. ?? ?x[?(x,mD)??(??2 diag[P
1 2
1P 1??])],?x[?(x,mD)?
?(?2 diag[P
1 2
1 P 1 ?])]??
h h h? h?
1 2
+ hlarge positive power ?? 2 ?? ?D1 D2 ?0 ? = I + small term
L2(M?M)
where the inner product is defined on L2([??h, ?h] ? Rd). Finally, using WKB
approximation, we have that
? ( ) ( )
d?d?d??d?? ?? ? ? ? ? ?
?
I . ? Kh?,h(?, ?, ? , ? )F? , F? ? ,h2d+1(h )2d+1 h h h h?
where the kernel Kh,h? can be estimated as follows
| ? ? | 1 1Kh,h?(?, ?, ? , ? ) .N .
(1 + h?1|? ? ??|+ h?1D N N) (1 + h?1)N
195
since |?| ? 1 and |??|  1. Hence by the same arguments as in the proof of
Proposition 6.5 and Proposition 6.6, we arrive at the desired estimate.
Proof of Estimate (6.3a). By the almost orthogonality property of the spectral lo-
calization of diag ?, we see that
??
?D diag ? ?2 2g iL2([??,?]?M) . ?DgP2 diag ? ?L2([??,?]?M) .
i=0
Next, employing the standard Littlewood-Paley product decomposition, we see that
for each fixed i we have
( ? ? ? ? )
P diag ? ? + + + P 1 22i 2i diag(P2jP2k?)
2i2j?2k 2i?2j?2k 2i?2j2k 2i?2k2j
=: HHi<? +HHi?? +HLi>? + LHi>?.
Then it follows
?DgP 22i diag ? ?L2([??,?]?M)
. ?DgHHi<? ?2L2([??,?]?M) + ?DgHH ? ?
2
i? L2([??,?]?M)
+ ?DgHL 2 2i>? ?L2([??,?]?M) + ?DgLHi>? ?L2([??,?]?M) .
Next, let us estimate each term.
196
For the first term, observe by Lemma 6.17 and Lemma 6.18 we have that
?D?gHHi?<? ?
2
L2([??,?]?M)
. ?DgP 1 2 1 22i diag(P2jP2j?), DgP2i diag(P j?P j??)?2 2
2i?2j 2?i2j?
? ? ?D 1 2 ?2gP2i diag(P2jP2j?) L2([??,?]?M)
2?i2j?
? ? ?. 22(i j) ?D D??P 1 2 ?21 2 2jP2j?0 2 .L (M?M)
j=i+1
Then taking the summation in i yields
?? ?? ? ?
?D HH ? ?2 ? ?? 1 2 ?2g i< L2([??,?]?M) . D1D2 P2jP2j?0 L2(M?M)
i=1 j=2
. ?D D?? 21 2 ?0 ?L2(M?M) .
We can handle the HHi?? term in a similar manner.
Next, we consider the termHLi>?. By Cauchy-Schwarz inequality and Propo-
sition 6.13, we have that
?D HL ? ?2g? i>?L2([??,?]?M)
? ?DgP2i diag(P 12iP 2 1 22k?), DgP2i diag(P2iP2k??)?(2i?2k 2?i2k
?
? ?? )i?1 2. Dg diag(P 1 22iP2k?) L2([??,?]?M)
?k=1i?1
. ?? ? ? ?D D?P 1P 2 ?2 ? ? 1 ?21 2 2i 2k?0 2 . D D P ? .L (M?M) 1 2 2i 0 L2(M?M)
k=1
197
Finally, by almost orthogonality, we obtain the desired result. The proof is similar
for the LHi>? term.
6.2.5 Proof of Theorem 6.1 for (6.2)
Lemma 6.19. Let 0 < h < m ? 1 with ? = h and ? ? C?(R) with supp? ? [1 , 1].
m 2
Assume P 1 21P 1 ?0 = ?0. Then we have the estimate
h h
??? ?? 1P 1 diag ? ? . ? ?D D?2 ?1 2 ?0 ?L2m (M?M)L2([??,?],H?1(M))
for some ? > 0.
Sketch of Proof of Lemma 6.19. If suffices to consider the modification of the proof
of Proposition 6.12. In the current case, we have |?| ? ? which means
2 ?? F? (?, ?)F? (??, ??) d?d?d??d??
RHS (6.26) . ? ?
h4d+2 ?|?|?1,|??|?1 (1 + h?1|? ? ??|+ h?1?? ? ? ? ?? ? ???)N
?2 G(?, p)G(??, p?) d??dpd??dp?. ? ?h4d+2 (1 + h?1|? ? ??|+ h?1?|?|p? |??|p??)N
?
. d?d? |?|2|?|d?2|F (?, ?)|2.
h3d+1
The rest of the proof is standard.
Let us now complete the proof of Theorem 6.1 for (6.2).
Proof of Estimate (6.3b). Fix i. As in the proof of estimate (6.3a), we immediately
198
have that
?DgP 22i? ?L2([??,?]?M)
. ?DgHHi<? ?2L2([??,?]?M) + ?D HH ? ?
2
g i? L2([??,?]?M)
+ ?DgHL 2 2i>? ?L2([??,?]?M) + ?DgLHi>? ?L2([??,?]?M) .
For the first term, applying Lemma 6.19 and Cauchy-Schwarz inequality yields
?DgH?H
2
i<???L2([??,?]?M)? ?. D 1 2 ?2gP2i diag(P2jP2j?) L2([??,?]?M)
2i2j???2j
+ ?DgP2i diag(P 1 22jP2j?), DgP2i diag(P 1 2j?P j??)?2 2
?2i2j?2j? ? ?
i?j ? ?? 2. 2 D1D2 P 1 P 2 ? ?2j 2j 0 L2(M?M)
j=i+1 ?? ? ? ?
+ 2i?j ?D D??P 1 P 2? ?? ?1 2 2j 2j 0( L
2(M?M)
j?=i+1 ? ?? ? ) 1? j? 2?j? 22 2 D1D??P 1 P 2 ? ?2 2j 2j 0 2 .L (M?M)
j=j?+1
199
Then it follows from Cauchy-Schwarz inequality that
??
?DgHHi<? ?2L2([??,?]?M)
i=1?? ?
. ? ?D D?? 1 21 2 P2jP 2 ?2j?0 L2(M?M)
j=1 ?? ?? ?+ D D?? 1 2 ?1 2 P2j?P2j??0 L2(M?M)
j?=2 ( ?? j??j ?? ? ) 12? 2 D D?2 ? 1 2 ?21 2 P2jP2j?0 L2(M?M)
?? j=j?+1? ? ?2. D1D??P 1 2 ?2 2jP2j?0 L2(M?M)
j=1 (??? ? ) 1? 2+ D D??P 1 2 21 2 j?P j?? ?0
( 2 2 L
2(M?M)
j?=1 ?? ?? ? ?? ? ) 12j ?j? 22 2 D D??1 2 P 12jP 22j? ?0 L2(M?M)
j?=1 j=j?+1
. ?D D??1 2 ? ?
2
0 L2(M?M)
The remainder of the proof is similar to case of (6.1).
200
Chapter 7: Conclusion and Discussion
In many ways, the time-dependent HFB system for bosons offers a remarkable
generalization of cubic NLS and Hartree equation. From a mathematical physics
perspective, the system provides a nontrivial correction to the mean-field equation
which allows for the studies of effective dynamics of the excitation of quantum gas.
In fact, Theorem 4.1 states that the time-dependent HFB system is rich enough to
capture details of the many-body system and provide Fock space estimates for the
true dynamics of quasifree states.
To put the results of Chapter 4 in context, we compare Theorem 4.1 and
Remark 4.2 to Theorem 1.1 in [BCS17]. Reader should note that the theorems are
similar in spirit, but the nature of the results are different. In [BCS17], the authors
imposed a condition on the structure of the pair excitation function k, which only
depends dynamically on the evolution of the condensate, then use it to obtain Fock
space approximation to the true dynamic when 0 < ? < 1. In fact, they were able
to prove a global-in-time Fock space error estimate which is double-exponential in
time.
On the other hand, Theorem 4.1 allows for the dynamical development of
the pair excitation function, which in some sense is more general than the results
201
in [BCS17]. But, the regularity assumption on the initial data in Theorem 4.1 im-
poses restriction on the form of k. In particular, one could show that the regularity
assumption rules out the case of the coherent states, i.e. the case k = 0. However,
the recent result of Grillakis and Machedon in [GM18] suggests that more general
data is permissible. It is conjectured that we could even consider coherent states ini-
tial data and obtain Fock space estimate on the development of correlation structure
in the long time dynamics of the initial state. Nevertheless, even with the restric-
tion, the set of permissible initial conditions in Theorem 4.1 is still comparable to
that of [BCS17].
Lastly, the error estimate of Theorem 4.1 is valid for a much longer period of
time when compare to Theorem 1.1 in [BCS17]. The improvement in time of the
Fock space estimate in Theorem 4.1 is a result of our analysis of ?. More precisely,
the improvement is the result of (4.1a), which ultimately helped us to avoid the
usage of Gr?nwall estimate. In fact, we believe the Fock space estimate could be
further improved if we improve our estimates for ?. One conjecture we expect to
be true is that there exists ? > 0 such that
? exp(?T )?exact(t)? ?approx(t) ?F . 1?? (7.1)
N 2
for all t ? [0, T ] for all N . In fact, to establish (7.1) it suffices to show
?? some powerx?y?(t, x, y) ?L?(dt)L2(dxdy) . N . (7.2)
202
Note that (7.2) is interesting in its own right. Problems of this flavor can be traced
back to the works of Bourgain on the growth of Sobolev norms for linear Schr?dinger
equations (c.f. [Bou99a,Bou99b]).
A close examination of (7.2) shows that the current approach of estimating
?x?y?(t) via energy method will not be sufficient in establishing the estimate.
In fact, to prove (7.2), we will need to employ global-in-time Strichartz estimates.
Unfortunately, obtaining global-in-time Strichartz estimates for the time-dependent
HFB equations is in general a formidable task. In chapter 5, we establish the
interaction Morawetz-type estimate for ? using the Virial interaction potential
?
V a(t) = dxdy ?(t, x, x)a(x? y)?(t, y, y).
The approach adopts the method used in [CPT12] to establish the interaction
Morawetz estimates for the BBGKY (Gross-Pitaevskii) hierarchy. However, a major
difference between the BBGKY (Gross-Pitaevskii) hierarchy and the time-dependent
HFB system is the obvious fact that the time-dependent HFB system is nonlinear
whereas the Gross-Pitaevskii hierarchy is linear. Hence instead of following [CPT12]
and use the second marginal density, i.e. consider the Virial interaction potential
?
V a(t) = dxdy L2,2(t, x, y, x, y)a(x? y),
we replaced L2,2(t, x, y, x, y) by ?(t, x, x)?(t, y, y). In doing so, we were able to prove
203
the interaction Morawetz estimate
??(t, x, x) ?L2(dtdx) . ??0 ?H1/2(dx) .
Of course, the immediate question that follows is whether a similar type of estimate
holds for ?. The calculation using L2,2 in the Virial interaction potential is highly
involved and we have no guarantee whether the idea will bear fruit.
204
Bibliography
[AEM+95] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and
E. A. Cornell, Observation of Bose-Einstein condensation in a dilute
atomic vapor, science 269 (1995), no. 5221, 198?201.
[Arn97] V. I. Arnold, Mathematical methods of classical mechanics, 2 ed., Grad-
uate Texts in Mathematics, vol. 60, Springer, 1997.
[BBC+18] V. Bach, S. Breteaux, T. Chen, J. Fr?hlich, and I. M. Sigal, The time-
dependent Hartree-Fock-Bogoliubov equations for Bosons, arXiv preprint
arXiv:1602.05171v2 (2018), 1?46.
[BCS17] C. Boccato, S. Cenatiempo, and B. Schlein, Quantum many-body fluctu-
ations around nonlinear schr?dinger dynamics, Annales Henri Poincar?
18 (2017), no. 1, 113?191.
[BdOS15] N. Benedikter, G. de Oliveira, and B. Schlein, Quantitative derivation
of the Gross-Pitaevskii equation, Communications on Pure and Applied
Mathematics 68 (2015), no. 8, 1399?1482.
[BDZ08] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold
gases, Reviews of Modern Physics 80 (2008), no. 3, 885?964.
[Ber66] F. A. Berazin, The method of second quantization, Pure and Applied
Physics, vol. 24, Academic Press, 1966.
[BGT04] N. Burq, P. G?rard, and N. Tzvetkov, Strichartz inequalities and the
nonlinear Schr?dinger equation on compact manifolds, American Journal
of Mathematics 126 (2004), no. 3, 569?605.
[Bog47] N. Bogoliubov, On the theory of superfluidity, Journal of Physics 11
(1947), no. 1, 23.
[Bou99a] J. Bourgain, Growth of Sobolev norms in linear Schr?dinger equations
with quasi-periodic potential, Communications in Mathematical Physics
204 (1999), no. 1, 207?247.
205
[Bou99b] , On growth of Sobolev norms in linear Schr?dinger equations
with smooth time dependent potential, Journal d?Analyse Math?matique
77 (1999), no. 1, 315?348.
[BR03] O. Bratteli and D. W. Robinson, Operator algebras and quantum sta-
tistical mechanics 2: Equilibrium states. models in quantum statistical
mechanics, Theoretical and Mathematical Physics, Springer, 2003.
[BSS18] N. Benedikter, J. Sok, and J. P. Solovej, The Dirac?Frenkel principle
for reduced density matrices, and the Bogoliubov?de Gennes equations,
Annales Henri Poincar? 19 (2018), no. 4, 1167?1214.
[BSTH95] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Evidence
of Bose-Einstein condensation in an atomic gas with attractive interac-
tions, Physical Review Letters 75 (1995), no. 9, 1687.
[CH16a] X. Chen and J. Holmer, Focusing quantum many-body dynamics: the rig-
orous derivation of the 1D focusing cubic nonlinear Schr?dinger equa-
tion, Archive for Rational Mechanics and Analysis 221 (2016), no. 2,
631?676.
[CH16b] , The rigorous derivation of the 2D cubic focusing NLS from
quantum many-body evolution, International Mathematics Research No-
tices (2016), rnw113.
[CH17] , Focusing quantum many-body dynamics II: the rigorous deriva-
tion of the 1D focusing cubic nonlinear Schr?dinger equation from 3D,
Analysis & PDE 10 (2017), no. 3, 589?633.
[Che12] X. Chen, Second order corrections to mean field evolution for weakly
interacting bosons in the case of three-body interactions, Archive for Ra-
tional Mechanics and Analysis 203 (2012), no. 2, 455?497.
[Cho16] J.J.W. Chong, Dynamics of large boson systems with attractive interac-
tion and a derivation of the cubic focusing NLS in R3, arXiv preprint
arXiv:1608.01615 (2016), 1?23.
[Cho17] J. J. W. Chong, Dynamical Hartree-Fock-Bogoliubov approximation of
interacting bosons, arXiv preprint arXiv:1711.00610 (2017), pp. 1?26.
[Cho18] , Uniform in n global well-posedness of the time-dependent
Hartree?Fock?Bogoliubov equations in R1+1, Letters in Mathematical
Physics 108 (2018), no. 10, 2255?2283.
[CHP17] T. Chen, Y. Hong, and N. Pavlovi?, Global well-posedness of the NLS
system for infinitely many fermions, Archive for Rational Mechanics and
Analysis 224 (2017), no. 1, 91?123.
206
[CLS11] L. Chen, J.O. Lee, and B. Schlein, Rate of convergence towards hartree
dynamics, Journal of Statistical Physics 144 (2011), no. 4, 872?903.
[CPT12] T. Chen, N. Pavlovi?, and N. Tzirakis, Multilinear Morawetz identi-
ties for the Gross-Pitaevskii hierarchy, Contemporary Mathematics 581
(2012), 39?62.
[DG13] J. Derezi?ski and C. G?rard, Mathematics of quantization and quantum
fields, Cambridge University Press, 2013.
[DMA+95] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. Van Druten, D. S.
Durfee, D. M. Kurn, and W. Ketterle, Bose-Einstein condensation in a
gas of sodium atoms, Physical Review Letters 75 (1995), no. 22, 3969.
[Dys57] F. J. Dyson, Ground-state energy of a hard-sphere gas, Physical Review
106 (1957), no. 1, 20.
[ES09] L. Erd?s and B. Schlein, Quantum dynamics with mean field interac-
tions: a new approach, Journal of Statistical Physics 134 (2009), no. 5,
859?870.
[ESY06] L. Erd?s, B. Schlein, and H.-T. Yau, Derivation of the Gross-Pitaevskii
hierarchy for the dynamics of Bose-Einstein condensate, Communica-
tions on Pure and Applied Mathematics 59 (2006), no. 12, 1659?1741.
[ESY07] , Derivation of the cubic non-linear Schr?dinger equation from
quantum dynamics of many-body systems, Inventiones Mathematicae
167 (2007), no. 3, 515?614.
[ESY09] , Rigorous derivation of the gross-pitaevskii equation with a large
interaction potential, Journal of the American Mathematical Society 22
(2009), no. 4, 1099?1156.
[ESY10] , Derivation of the Gross-Pitaevskii equation for the dynamics
of Bose-Einstein condensate, Annals of Mathematics 172 (2010), no. 1,
291?370.
[EY01] L. Erd?s and H.-T. Yau, Derivation of the nonlinear Schr?dinger equa-
tion from a many-body Coulomb system, Advances in Theoretical and
Mathematical Physics 5 (2001), no. 6, 1169?1205.
[FKS09] J. Fr?hlich, A. Knowles, and S. Schwarz, On the mean-field limit of
bosons with coulomb two-body interaction, Communications in Mathe-
matical Physics 288 (2009), no. 3, 1023?1059.
[Fol89] G. B. Folland, Harmonic analysis in phase space, Annals of Mathematics
Studies, no. 122, Princeton University Press, 1989.
207
[GM13a] M. Grillakis and M. Machedon, Beyond mean field: On the role of pair
excitations in the evolution of condensates, Journal of Fixed Point The-
ory and Applications 14 (2013), no. 1, 91?111.
[GM13b] , Pair excitations and the mean field approximation of interacting
Bosons, I, Communications in Mathematical Physics 324 (2013), no. 2,
601?636.
[GM17] , Pair excitations and the mean field approximation of interacting
Bosons, II, Communications in Partial Differential Equations 42 (2017),
no. 1, 24?67.
[GM18] , Uniform in N estimates for a Bosonic system of Hartree-Fock-
Bogoliubov type, arXiv preprint arXiv:1808.06448 (2018), pp. 1?44.
[GMM10] M. Grillakis, M. Machedon, and D. Margetis, Second-order corrections
to mean field evolution of weakly interacting Bosons. I., Communications
in Mathematical Physics 294 (2010), no. 1, 273?301.
[GMM11] , Second-order corrections to mean field evolution of weakly in-
teracting Bosons. II, Advances in Mathematics 228 (2011), no. 3, 1788?
1815.
[GMM17] , Evolution of the boson gas at zero temperature: Mean-field
limit and second-order correction, Quarterly of Applied Mathematics
75 (2017), no. 1, 69?104.
[Gol16] F. Golse, On the dynamics of large particle systems in the mean field
limit, Macroscopic and Large Scale Phenomena: Coarse Graining, Mean
Field Limits and Ergodicity, Springer, 2016, pp. 1?144.
[Gro61] E. P. Gross, Structure of a quantized vortex in boson systems, Il Nuovo
Cimento (1955-1965) 20 (1961), no. 3, 454?477.
[Gro63] , Hydrodynamics of a superfluid condensate, Journal of Mathe-
matical Physics 4 (1963), no. 2, 195?207.
[GV79a] J. Ginibre and G. Velo, The classical field limit of scattering theory for
non-relativistic many-boson systems. I, Communications in Mathemati-
cal Physics 66 (1979), no. 1, 37?76.
[GV79b] , The classical field limit of scattering theory for non-relativistic
many-boson systems. II, Communications in Mathematical Physics 68
(1979), no. 1, 45?68.
[Han12] Z. Hani, A bilinear oscillatory integral estimate and bilinear refinements
to strichartz estimates on closed manifolds, Analysis & PDE 5 (2012),
no. 2, 339?363.
208
[Hep74] K. Hepp, The classical limit for quantum mechanical correlation func-
tions, Communications in Mathematical Physics 35 (1974), no. 4, 265?
277.
[HLLS10] C. Hainzl, E. Lenzmann, M. Lewin, and B. Schlein, On blowup for time-
dependent generalized Hartree?Fock equations, Annales Henri Poincar?
11 (2010), no. 6, 1023?1052.
[H?r94] L. H?rmander, The analysis of linear partial differential operators
III: Pseudo-differential operators, Classics in Mathematics, vol. 256,
Springer, 1994.
[KP10] Antti Knowles and Peter Pickl, Mean-field dynamics: singular potentials
and rate of convergence, Communications in Mathematical Physics 298
(2010), no. 1, 101?138.
[KT98] M. Keel and T. Tao, Endpoint strichartz estimates, American Journal
of Mathematics 120 (1998), no. 5, 955?980.
[Kuz15a] E. Kuz, Exact evolution versus mean field with second-order correction
for Bosons interacting via short-range two-body potential, arXiv preprint
arXiv:1511.00487 (2015), 1?38.
[Kuz15b] , Rate of convergence to mean field for interacting Bosons, Com-
munications in Partial Differential Equations 40 (2015), no. 10, 1831?
1854.
[LEN+17] R. Lopes, C. Eigen, N. Navon, D. Cl?ment, R. P. Smith, and Z. Hadz-
ibabic, Quantum depletion of a homogeneous Bose-Einstein condensate,
Physical Review Letters 119 (2017), no. 19, 190404.
[Lew15] M. Lewin, Mean-field limit of Bose systems: rigorous results, arXiv
preprint arXiv:1510.04407 (2015), 1?26.
[LHY57] T. D. Lee, K. Huang, and C. N. Yang, Eigenvalues and eigenfunctions
of a Bose system of hard spheres and its low-temperature properties,
Physical Review 106 (1957), no. 6, 1135.
[Lie63] E. H. Lieb, Exact analysis of an interacting Bose gas. II. The excitation
spectrum, Physical Review 130 (1963), 1616?1624.
[LL63] E. H. Lieb and W. Liniger, Exact analysis of an interacting Bose gas. I.
The general solution and the ground state, Physical Review 130 (1963),
1605?1616.
[LS78] J.E. Lin and W. A. Strauss, Decay and scattering of solutions of a non-
linear schr?dinger equation, Journal of Functional Analysis 30 (1978),
no. 2, 245?263.
209
[LS02] E. H. Lieb and R. Seiringer, Proof of Bose-Einstein condensation for
dilute trapped gases, Phys. Rev. Lett. 88 (2002), no. 17, 170409.
[LSSY05] E. H. Lieb, R. Seiringer, J. P. Solovej, and J. Yngvason, The mathemat-
ics of the Bose gas and its condensation, vol. 34, Springer Science &
Business Media, 2005.
[LSY00] E. H. Lieb, R. Seiringer, and J. Yngvason, Bosons in a trap: A rigorous
derivation of the Gross-Pitaevskii energy functional, Phys. Rev. A 61
(2000), no. 4, 043602.
[NN17] P. T. Nam and M. Napi?rkowski, Bogoliubov correction to the mean-field
dynamics of interacting bosons, Advances in Theoretical and Mathemat-
ical Physics 21 (2017), no. 3, 683?738.
[Pic10] P. Pickl, Derivation of the time dependent Gross-Pitaevskii equation
without positivity condition on the interaction, Journal of Statistical
Physics 140 (2010), no. 1, 76?89.
[Pic11] , A simple derivation of mean field limits for quantum systems,
Letters in Mathematical Physics 97 (2011), no. 2, 151?164.
[Pit61] L. P. Pitaevskii, Vortex lines in an imperfect bose gas, Sov. Phys. JETP
13 (1961), no. 2, 451?454.
[PO56] O. Penrose and L. Onsager, Bose-Einstein condensation and liquid he-
lium, Physical Review 104 (1956), no. 3, 576.
[RS80] M. Reed and B. Simon, Functional analysis, Methods of Modern Math-
ematical Physics, vol. 1, Academic Press, 1980.
[RS09] I. Rodnianski and B. Schlein, Quantum fluctuations and rate of conver-
gence towards mean field dynamics, Communications in Mathematical
Physics 291 (2009), no. 1, 31?61.
[Sha62] D. Shale, Linear symmetries of free boson fields, Transactions of the
American Mathematical Society 103 (1962), no. 1, 149?167.
[Sog93a] C. D. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in
Mathematics, vol. 105, Cambridge University Press, 1993.
[Sog93b] , On local existence for nonlinear wave equations satisfying vari-
able coefficient null conditions, Communications in partial differential
equations 18 (1993), no. 11, 1795?1821.
[Sol14] J. P. Solovej, Many body quantum mechanics, Lecture Notes. Summer
(2014), 1?102.
[Spo80] H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian
limits, Reviews of Modern Physics 52 (1980), no. 3, 569.
210
[SW71] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean
spaces, Princeton Mathematical Series, no. 32, Princeton University
Press, 1971.
[Tao06] T. Tao, Nonlinear dispersive equations: local and global analysis, CBMS
Regional Conference Series in Mathematics, no. 106, American Mathe-
matical Society, 2006.
[Wu61] T. T. Wu, Some nonequilibrium properties of a Bose system of hard
spheres at extremely low temperatures, Journal of Mathematical Physics
2 (1961), no. 1, 105?123.
[XLM+06] K. Xu, Y. Liu, D. E. Miller, J. K. Chin, W. Setiawan, and W. Ketterle,
Observation of strong quantum depletion in a gaseous Bose-Einstein con-
densate, Physical Review Letters 96 (2006), no. 18, 180405.
211