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RESEARCH STATEMENT
ZHIHUI XIE
My current research interests are roughly analysis and partial differential equations (PDE). One
theme in my research has been applying ideas and techniques from harmonic analysis to study
dispersive equations and problems that arise in mathematical physics. I obtained my PhD from
UT-Austin under the supervision of Nataša Pavlović. Currently I am a postdoc working with
Christof Sparber at UIC. I have been mostly working on questions related to dispersive equations.
Some specific examples of my current and future projects include:
• derivation of the nonlinear Schrödinger equation (NLS) with general power nonlinearities
from a quantum many-body system,
• analysis of the Cauchy problem for the Gross-Pitaevskii (GP) hierarchy,
• low regularity uniqueness of solution to GP hierarchy in subcritical and critical spaces,
• the continuum limit of discrete NLS with fractional Laplacian,
• ground state conjecture for the cubic GP hierarchy.
The sections below will describe my research interests in details.
1. From quantum many-body systems to nonlinear dispersive PDE
One of the central topics in mathematical physics is the analysis of an interacting quantum
mechanical system of Bosonic particles. Since 1970s, many people have worked on developing
rigorous mathematical models for this system. To name a few, early papers on the derivation of
Hartree type equation can be found in Hepp [22], also in Spohn [38] via the BBGKY hierarchy, and a
more recent paper by Fröhlich, Tsai and Yau [19]. In [35], Rodnianski and Schlein provide estimates
on the convergence rate of the evolution under mean field limit, and second-order corrections in
the two-body interaction setting by Grillakis, Machedon and Margetis [20, 21] and three-body
interaction setting by X. Chen [7]. These works are through the Fock space approach. There is
also work due to Fröhlich, Graffi and Schwarz [16] that combined the semi-classical limit and the
mean field limit. During the last few years, Erdös, Schlein and Yau further developed the BBGKY
approach in their celebrated works [12, 14, 15, 13], which have generated further research interests
surrounding this topic. Before we present our results we give a very brief overview of the derivation
of the cubic NLS following the approach of Erdös-Schlein-Yau [13].
In an N -particle system, the time evolution of the wave function ψN follows the Schrödinger
equation
(1.1)
i∂t ψN = HN ψN ,
where the Hamiltonian is given by:
HN := −
N
X
∆j +
j=1
X
VN (xi − xj ).
i<j
Here, VN (x) := N 3β V (N β x) is the rescaled potential with parameter β > 0. Since the number
of particles N is usually large (up to ∼ 1030 ) and particles interact with each other, it is almost
unrealistic to find an explicit solution to (1.1). The numerical methods cannot help either because
of the size of N . However, physicists are usually interested in the macroscopic properties of the
system consisting of large amount of single molecules. This is in the spirit of statistical mechanics:
1
the observable properties are reflected by averaging over individual particles. Though N is finite,
one expects that the limit N → ∞ is a good approximation of the system in the macroscopic
regime.
For further investigation, one defines the density matrix γN ∈ C(RdN × RdN ) as the orthogonal
projection of ψN onto L2 (RdN , dx) with kernel given by
γN (t, x, x0 ) := ψN (t, x)ψN (t, x0 ),
(1.2)
(k)
and the k-particle density matrix γN (by tracing out the last N − k space variables) as
Z
(k)
0
(1.3)
γN (t, xk , xk ) :=
ψN (t, xk , xN −k )ψN (t, x0k , xN −k )dxN −k ,
Rd(N −k)
x0k
where xk = (x1 , x2 , · · · , xk ) and
= (x01 , x02 , · · · , x0k ) are both in Rdk .
The strategy of Erdös-Schlein-Yau’s proof roughly follows three steps:
(k)
(k)
(1) show that each solution γN (t) admits at least one limit point γ∞ (t) as N → ∞;
(2) prove that any limit point satisfies the GP hierarchy (which is an infinite hierarchy of
coupled linear non-homogeneous PDE that admits the so-called factorized solutions where
each factor satisfies the cubic NLS);
(3) establish uniqueness of solutions to the GP hierarchy.
The most difficult part of this approach is step (3). To handle step (3) Erdös-Schlein-Yau [13]
discovered the powerful combinatorial argument that resolves the issue of the factorial growth
of number of terms coming from iterated Duhamel expansions. Later in [31], Klainerman and
Machedon gave a short proof of the uniqueness of solutions to the GP hierarchy by reorganizing
the combinatorial argument from [13] and using techniques from nonlinear dispersive equations.
However, the result of [31] is conditional (a certain space-time bound on the solutions of the GP
is assumed in their argument). Kirkpatrick, Schlein and Staffilani [28] were the first to use the
approach of Klainerman and Machedon for the derivation of the NLS. Such a line of work was
continued in e.g. [4, 5, 8, 9, 28, 6].
2. Derivation of a Schrödinger Equation with General Power-type Nonlinearity
2.1. Problem setting and main results. In [4] T. Chen and Pavlović considered Hamiltonians
with three-body interactions in 1D and 2D and proved that factorized solutions of the corresponding
GP hierarchy are determined by solutions to a quintic NLS. In such a way, they derived the quintic
NLS from quantum many-body systems. In the same paper [4], T. Chen-Pavlović predicted that,
if both two-body and three-body interactions are present in the model, then this would produce
an NLS with linear combination of cubic and quintic nonlinearities. In [41], we gave a proof of
this claim. We also generalize the prediction in [4] and derive an NLS with a linear combination of
power-type nonlinearities (such equations were studied in e.g. [40]).
In particular, we consider the quantum many-body system with the following Hamiltonian:
(2.1)
HN :=
N
X
i=1
p0
X
1
(−∆xi ) +
Np
p=1
X
(p)
VN
xi1 − xi2 , · · · , xi1 − xip+1 ,
1≤i1 <···<ip+1 ≤N
(p)
where VN (a1 , · · · , ap ) = N pdβ V (p) (N β a1 , · · · , N β ap ) is the rescaled (p + 1)-body interaction po(k)
tential and β is a positive parameter. Rewriting (1.1) with respect to γN,t gives
(2.2)
(k)
(k)
i∂t γN (t) = [HN , γN (t)].
Here, [A, B] := AB−BA denotes the commutator. After expanding (2.2), we obtain a coupled chain
(k)
(k+i)
of equations: the equation for γN (t) involves γN (t), i = 1, 2, · · · , p. We call this set of equations
2
the BBGKY hierarchy, and its limiting hierarchy (N → ∞) the Gross-Pitaevskii hierarchy. As
part of my PhD thesis, we proved the following theorem in [41].
Theorem 2.1. Let p0 ≥ 1 be a fixed integer. Suppose that for all 1 ≤ p ≤ p0 the potential
V (p) ∈ W p,∞ (Rdp ) and V (p) ≥ 0 is translation-invariant. Let d ∈ {1, 2} and 0 < β < 2dp10 +2 .
{ψN }N ≥1 is a family of functions that satisfy the uniform bound:
(2.3)
sup
N
1
hψN , HN ψN i < ∞.
N
And assume {ψN }N ≥1 exhibits asymptotic factorization:
(1)
∃φ ∈ L2 (Rd ) such that Tr γN − |φi hφ| → 0
as
N → 0,
(1)
where γN is the 1-particle marginal density associated with ψN . Then we have the k-particle
marginal density will stay factorized:
(k)
(2.4)
Tr γN (t) − |φ(t)i hφ(t)|⊗k → 0 as N → ∞.
(k)
Here γN (t) is the k-particle
density associated to ψN (t) = e−iHN t ψN , and φt solves the
Pmarginal
p0
NLS: i∂t φ(t) = (−∆)φ(t)+ p=1 bp |φ(t)|2p φ(t), with initial condition φ0 = φ and potential constant
R
bp = Rpd V (p) (x)dx < ∞.
The proof of this theorem follows the “three step” strategy. However, to establish the step (1)
we have to obtain a new energy estimate for the many-body system with Hamiltonian (2.1). This
requires careful analysis of various interaction terms. The step (3) is obtained via generalizing the
combinatorial organization of Klainerman-Machedon. In 1D and 2D, we are able to prove that
limits from step (2) satisfy the analogue of the Klainerman and Machedon condition.
2.2. Future research projects. The main result in [41] holds only for d = 1, 2 due to the failure
of a crucial Strichartz type estimates in 3D. One direction of my future research is to extend our
theorem to include d = 3 since this is the most realistic dimension.
In the 3D case, T. Chen and Pavlović [5] derived the defocusing cubic GP via proving convergence
of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces motivated by the solution
spaces based on space-time norms. This was done for potentials VN with scaling parameter β ∈
(0, 14 ). The case β ∈ ( 31 , 1) is important since it justifies the Gross-Pitaevskii theory that the manybody effect should be modeled by a strong self-interaction. An improvement in this directions
include [9] in which the upper bound for β is 32 .
3. Uniqueness of solutions to the Gross-Pitaevskii hierarchy à la de Finetti
3.1. Problem background. In a recent work [3] T. Chen, Hainzl, Pavlović and Seiringer proposed
1 ∗ for the
a new method to prove the unconditional uniqueness of solutions in the space L∞
t∈[0,T ) H
cubic GP hierarchy. By unconditional uniqueness, we mean uniqueness of solutions in the Sobolev
space H s itself, while uniqueness in the intersection of the Sobolev space and auxiliary spaces is
called conditional. The proof in [3] is based on the quantum de Finetti theorems and is much
simpler compared to the previous proofs. The de Finetti theorem is a quantum analogue of the
Hewitt-Savage theorem in probability theory. A strong version of it [39, 26] states that if a sequence
(k,α) (k)
For α ≥ 0, the space Hα := {γ (k) }∞
γ |) < M 2k , M > 0 is a constant , where the differential
k=1 Tr (|S
Q
Q
α
α
k
k
(k)
0
(k)
2
0 2
operator S (k,α) :=
(xk , x0k ) =
∈
j=1 (1 − ∆xj ) (1 − ∆xj ) . In factorized case, γ
j=1 φ(xj )φ(xj ), so γ
∗
Hα if and only if kφkH α < M . The homogeneous space Ḣα is defined the same way as Hα , but with the S (k,α)
Q
α
α
replace by homogeneous operator R(k,α) := kj=1 (∆xj ) 2 (∆x0j ) 2 .
3
of bosonic density matrices is admissible †, then there exists a unique Borel probability measure µ,
supported on the unit sphere and invariant under unit complex multiplication of φ, such that
Z
(k)
(3.1)
γ = dµ(φ)|φi hφ|⊗k .
Here the measure µ depends on φ. There is also a weak version of the de Finetti theorem in which
(3.1) still holds but with µ supported on a unit ball. The weak de Finetti theorem applies to the
limiting bosonic density matrices γ (k) that are obtained as the weak-∗ limit on marginals, such γ (k)
originates naturally in derivation of NLS.
The authors in [3] proved that if {γ (k) (t)}k∈N is a mild solution to the 3D cubic GP hierarchy
1
that lives in L∞
t∈[0,T ) H , then it is unique for the given initial data. Furthermore, if the initial data
(k)
γ0 satisfies (3.1) with φ0 , then the time evolved sequence γ (k) (t) also satisfies (3.1) with φ(t), so
long as φ(t) solves the cubic NLS with initial data φ0 .
For the factorized case, the H1 norm on the γ (k) level is corresponding to H 1 Sobolev norm on
the φ level. Since cubic NLS is ill-posed below H sc for sc := d2 − 1, in 3D case we can expect to
1
have the unconditional uniqueness in lower regularity space L∞
t H2 .
3.2. Unconditional uniqueness for cubic GP hierarchy with low regularity. Together with
Hong (postdoc at UT-Austin) and Taliaferro (PhD student at UT-Austin), we revisit the problem
in [3]. We are able to prove the unconditional uniqueness for the cubic GP hierarchy at lower
regularity spaces in all space dimensions.
Theorem 3.1. [Unconditional uniqueness for cubic GP hierarchy [24]] Let {γ (k) (t)}∞
k=1 be a mild
(k) ∞
∞
α
solution to the cubic GP hierarchy with initial data {γ0 }k=1 ∈ Lt∈[0,T ) H , which is either admissible or obtained at each t from a weak-∗ limit (so that the weak de Finetti theorem holds). Then
{γ (k) (t)}∞
k=1 is the unique solution for the given initial data when α satisfies the following:
(3.2)
α
≥ d6 , d = 1, 2;
> sc , d ≥ 3.
1
+
For d = 3, we have the unconditional uniqueness at regularity L∞
t H 2 , which is almost critical.
We remark that condition (3.2) coincides with the currently known regularity requirement for
unconditional uniqueness of solutions at the NLS level.
3.3. Unconditional uniqueness of quintic GP hierarchy with critical regularity. (joint
with Y.Hong and K.Taliaferro) Another project that we have been worked on is the uniqueness of
mild solutions to the 3D quintic GP hierarchy in critical space. In our previous work [24], the use
of dispersive estimates is crucial to obtain optimal subcritical low regularity uniqueness. We obtain
the desired decay on the solution due to time integrability. However, the same approach cannot be
applied for uniqueness in a scaling-critical space. The space H 1 for 3D quintic NLS is exactly the
endpoint case, under which the time integrability fails. We overcome this difficulty by combining
the convolution inequalities with the Strichartz estimates. Compared to previous works, we design
new multilinear estimates only by Strichartz estimates and negative Sobolev norm bound. We also
remark that we make use of Beckner’s convolution estimates [1] to prove the multilinear estimates.
Moreover, the method we used can also be applied to the Hartree hierarchy.
Our main results [25] are the following:
†
γ
(k)
Admissible means each density matrix can be obtained from taking the partial trace of the adjacent one, i.e.,
= Trk+1 γ (k+1) , where Trk+1 denotes partial trace over the (k + 1)-th factor.
4
Theorem 3.2 (Uniqueness of small solutions to the quintic GP hierarchy). There exists a small
1
M > 0 such that the following holds. Suppose that {γ (k) (t)}k∈N is a mild solution in L∞
t∈[0,T ) Ḣ
to the quintic GP hierarchy with initial data {γ (k) (0)}k∈N , which is either admissible or a limiting
hierarchy for each t. If Tr (|R(k,1) γ (k) |) < M 2k for all t ∈ [0, T ), then {γ (k) (t)}k∈N is the only
solution for given initial data.
Theorem 3.3 (Unconditional uniqueness for the quintic Hartree hierarchy). Suppose that V (·, ·) ∈
Lrx,y (R3 × R3 ) for some r > 1. Let {γ (k) (t)}k∈N ∈ Ḣ1 be a mild solution to the quintic Hartree
hierarchy with initial data {γ (k) (0)}k∈N , which is either admissible or a limiting hierarchy for each
t. If there exists M > 0 such that Tr (|R(k,1) γ (k) |) < M 2k for all t ∈ [0, T ), then {γ (k) (t)}k∈N is the
only solution for given initial data.
3.4. Further extensions. The de Finetti theorems provides an alternative formula for the density
matrices in factorized form. By taking advantage of this tool people are able to give new proofs on
unconditional uniqueness for hierarchies. In [37], Sohinger adapted the method in [4] for cubic GP
hierarchy on a periodic setting and X.Chen-Smith considered a nonlinear nonlinear radial hierarchy
in [10].
For cubic GP, although we are able to obtain unconditional uniqueness at the critical regularity
spaces in 1D and 2D. The endpoint regularity cases for d ≥ 3 are open both at the NLS and GP
level.
In [4], the authors derived the quintic NLS as the mean field limit of a non-relativistic Bose gas
with 3-particle interactions. As part of their analysis, the uniqueness of solutions to the quintic GP
hierarchy is proved. However, the substantial space-time estimates used in [4] hold for 1D and 2D
only. The derivation for the quintic NLS in three and higher dimensions is still open. The above
theorems 3.2 and 3.3 provide an alternative proof on the uniqueness of mild solutions to quintic
NLS without the derivation part. We plan to continue this line of work.
4. Continuum Limit of Discrete NLS with Fractional Laplacian
4.1. Problem setting and motivation. I have worked with Nataša Pavlović on the discrete
Hartree type equations on a lattice and its behavior in the continuum limit (as mesh size goes to
zero).
The work by Kirkpatrick, Lenzmann and Staffilani [27] looked at a cubic discrete NLS with
long-range interactions on lattice hZ. They established that in the continuum limit as h → 0+ ,
the limiting dynamics is given by an NLS on R with fractional Laplacian (−∆)σ . One of the tools
they used is the Sobolev embedding H s (R) ⊂ L∞ (R), which imposes restriction on s: s > 12 . This
work motivated us to consider a derivation of the Boson star equation (containing convolution type
nonlinearities) as the continuum limit of certain discrete problems in multi-dimensions. Topics
about the Boson stars have attracted much attention in the past few years. For recent works on
the Boson star equation, we refer the reader to [17, 18, 32, 33, 23].
4.2. Current research projects. Let s > 0, J = {Jn }n∈Zd with Jn ∼ |n|−d−2s be a sequence of
nonnegative numbers, λ = ±1. Suppose the lattice function uh (t, xn ) : [0, T ) × hZd → C satisfies
the initial value problem:
(
d
2
P
m )| uh (t,xn )
n )−uh (xm ))
d
+ λh |uh (t,x
,
i dt
uh (t, xn ) = m6=n Jn−m (uh (x
β(h)
|x
−x
|
n
m
(4.1)
uh (0, xn ) = vh0 (xn ), xn = nh with n ∈ Zd ,
where β(h) is some positive scaling factor. We are interested in what happens when the mesh size
h goes to 0. It is expected that uh tends to a solution of the following initial value problem
(
λ
i∂t u = c(−∆)σ u + ( |x|
∗ |u|2 )u,
(4.2)
u(0, x) = v 0 (x),
5
where u(t, x) : [0, T ) × Rd → C, c and σ are constants depend on s and J.
In this project we hope to have:
(A) local and global (for small v 0 in the focusing case) well-posedness of the continuous problem
(4.2) in Sobolev spaces (we note that the 3D case was discussed in [32] and very recently
in [23]);
(B) the well-posedness of the discrete problem (4.1) in discrete Sobolev spaces;
(C) the convergence under continuum limit, i.e., for every T ∈ (0, ∞), d ≥ 2 and s > 12 , we have
ph uh * u weakly- ∗ in L∞ ([0, T ]; H σ (Rd ))
as h → 0+ ,
for certain σ depending on s and J. Here, ph denotes the piecewise linear interpolation, uh
and u are the unique solutions of (4.1) and (4.2) respectively.
In order to obtain results in the energy space H σ with σ small, we cannot rely on the Sobolev
embedding. This issue has already been addressed in the continuous case in 3D in the work of
Lenzmann [32], where it is proved that the solution map is Lipschitz using fractional Leibniz rule
and Hardy’s inequality. However, no such well-posedness results are available in the discrete setting.
In order to obtain local well-posedness in the discrete setting, we have to develop discrete analogues
of various results from harmonic analysis including fractional Leibniz rule, Hardy’s inequality and
Littlewood-Paley’s estimates. We use such estimates to carry out details of step (C) above.
4.3. Future research projects. As mentioned above, we try to prove local well-posedness in
steps (A) and (B) via proving that the corresponding solution map is Lipschitz. However, in the
1
continuous setting, recently Herr and Lenzmann [23] have proved local well-posedness in H 4 (R3 )
using Strichartz estimates. I plan to investigate whether such techniques can be developed and
used in the discrete setting as well.
Generally speaking, mathematicians and engineers who are working on numerical solutions of
PDE need to justify the effectiveness of discrete models before they can conduct analysis. It
is necessary to have well-posedness of the discrete problems and continuum convergence before
running numerical approach. I am interested in studying if the same regularity of solutions in the
continuous settings can also be established in the discrete setting for other physically relevant PDE.
5. Other research projects
Besides the future extensions on the current projects, there are two other ongoing projects that
I am working on. One is the ground state problem for GP and the other is on the nonlocal fluids
equation.
5.1. Ground state conjecture for GP. For a sequence Γ = (γ (k) )∞
k=1 of bosonic density matrices
2
3k
on L (R ), we define the energy functional E by
Z
1
1
(−∆x γ (1) )(x, x) − γ (2) (x, x, x, x)dx.
(5.1)
E(Γ) =
4
R3 2
By a ground state for the focusing cubic GP hierarchy, we mean an energy minimizer among all
admissible hierarchies or all limiting hierarchies satisfying Tr(γ (1) ) = λ.
We classify ground states for the GP hierarchy using those for NLS. To this end, we recall their
variational properties. For the NLS it is known that
Theorem 5.1 (Ground state for NLS). There exists a unique minimizer Qλ (up to translation)
for the variational problem
n
o
1
1
inf E(u) = k∇uk2L2 − kuk4L4 : kuk2L2 = λ .
2
4
2
Therefore, if kukL2 = λ and u 6= Qλ (· − a) for any a ∈ R3 , then E(u) > E(Q).
6
With Hong and Taliaferro, we are considering the similar version of theorem 5.1 at the GP level.
That means we have that the ground states for the GP hierarchy are given by Qλ :
Proposition 5.1 (Ground state for GP). The sequence of the form
∞
Z
,
|Qλ (· − a)ihQλ (· − a)|⊗k dµR3 (a)
(5.2)
Γgs,λ :=
k=1
R3
where µR3 is a probability measure on R3 , is a unique kind of minimizers for the cubic GP hierarchy.
We are currently writing a article to report the proof of proposition 5.1.
5.2. Nonlocal nonlinear PDE. The global regularity of 3D Navier-Stokes equation is a well
known open problem. Even for its toy model, the surface quasi-geostrophic equation: θt + u · ∇θt +
(−∆)s θt = 0,‡ the global regularity is still open for the supercritical case s < 12 .§ With Jain (PhD
student at UT-Austin), we are studying the 1D case of the problem in the interval ( 14 , 12 ), and trying
to obtain some regularity results by extending the modulus of continuity method in [30] and the
De Giorgi method in [2] for the critical case.
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‡
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7
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Department of Mathematics, Statistics and Computer Science
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