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Zehua Zhao

Publications and source records attributed to Zehua Zhao.

At least 37 records · Page 2Linked to original sources
arXiv2023

Singular Levy processes and dispersive effects of generalized Schrödinger equations

We introduce new models for Schrödinger-type equations, which generalize standard NLS and for which different dispersion occurs depending on the directions. Our purpose is to understand dispersive properties depending on the directions of propagation, in the spirit of waveguide manifolds, but where the diffusion is of different types. We mainly consider the standard Euclidean space and the waveguide case but our arguments extend easily to other types of manifolds (like product spaces). Our approach unifies in a natural way several previous results. Those models are also generalizations of some appearing in seminal works in mathematical physics, such as relativistic strings. In particular, we prove the large data scattering on waveguide manifolds $\mathbb{R}^d \times \mathbb{T}$, $d \geq 3$. This result can be regarded as the analogue of \cite{TV2, YYZ2} in our setting and the waveguide analogue investigated in \cite{GSWZ}. A key ingredient of the proof is a Morawetz-type estimate for the setting of this model.

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arXiv2023

On well-posedness results for the cubic-quintic NLS on $\mathbb{T}^3$

We consider the periodic cubic-quintic nonlinear Schrödinger equation \begin{align}\label{cqnls_abstract} (i\partial_t +Δ)u=μ_1 |u|^2 u+μ_2 |u|^4 u\tag{CQNLS} \end{align} on the three-dimensional torus $\mathbb{T}^3$ with $μ_1,μ_2\in \mathbb{R} \setminus\{0\}$. As a first result, we establish the small data well-posedness of \eqref{cqnls_abstract} for arbitrarily given $μ_1$ and $μ_2$. By adapting the crucial perturbation arguments in \cite{zhang2006cauchy} to the periodic setting, we also prove that \eqref{cqnls_abstract} is always globally well-posed in $H^1(\mathbb{T}^3)$ in the case $μ_2>0$.

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arXiv2022

On decaying properties of nonlinear Schrödinger equations

In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing Nonlinear Schrödinger equation with various initial data, deterministic and random. We show that nonlinear solutions enjoy the same decay rate as the linear ones. The regularity assumption on the initial data is much lower than in previous results (see \cite{fan2021decay} and the references therein) and moreover we quantify the decay, which is another novelty of this work. Furthermore, we show that the (physical) randomization of the initial data can be used to replace the $L^1$-data assumption (see \cite{fan2022note} for the necessity of the $L^1$-data assumption). At last, we note that this method can be also applied to derive decay estimates for other nonlinear dispersive equations.

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arXiv2022

On scattering asymptotics for the 2D cubic resonant system

In this paper, we prove scattering asymptotics for the 2D (discrete dimension) cubic resonant system. This scattering result was used in Zhao \cite{Z1} as an assumption to obtain the scattering for cubic NLS on $\mathbb{R}^2\times \mathbb{T}^2$ in $H^1$ space. Moreover, the 1D analogue is proved in Yang-Zhao \cite{YZ}. Though the scheme is also tightly based on Dodson \cite{D}, the 2D case is more complicated which causes some new difficulties. One obstacle is the failure of `$l^2$-estimate' for the cubic resonances in 2D (we also discuss it in this paper, which may have its own interests). To fix this problem, we establish weaker estimates and exploit the symmetries of the resonant system to modify the proof of \cite{YZ}. At last, we make a few remarks on the research line of `long time dynamics for NLS on waveguides'.

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arXiv2022

A note on decay property of nonlinear Schrödinger equations

In this note, we show the existence of a special solution $u$ to defocusing cubic NLS in $3d$, which lives in $H^{s}$ for all $s>0$, but scatters to a linear solution in a very slow way. We prove for this $u$, for all $ε>0$, one has $\sup_{t>0}t^ε\|u(t)-e^{itΔ}u^{+}\|_{\dot{H}^{1/2}}=\infty$. Note that such a slow asymptotic convergence is impossible if one further pose the initial data of $u(0)$ be in $L^{1}$. We expect that similar construction hold the for other NLS models. It can been seen the slow convergence is caused by the fact that there are delayed backward scattering profile in the initial data, we also illustrate why $L^{1}$ condition of initial data will get rid of this phenomena.

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arXiv2022

Global uniform in $N$ estimates for solutions of a system of Hartree-Fock-Bogoliubov type in the case $β<1$

We extend the results of the 2019 paper by the third and fourth author globally in time. More precisely, we prove uniform in $N$ estimates for the solutions $ϕ$, $Λ$ and $Γ$ of a coupled system of Hartree-Fock-Bogoliubov type with interaction potential $V_N(x-y)=N^{3 β}v(N^β(x-y))$ with $β<1$. The potential satisfies some technical conditions, but is not small. The initial conditions have finite energy and the "pair correlation" part satisfies a smallness condition, but are otherwise general functions in suitable Sobolev spaces, and the expected correlations in $Λ$ develop dynamically in time. The estimates are expected to improve the Fock space bounds from the 2021 paper of the first and fifth author. This will be addressed in a different paper.

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arXiv2022

On the decay property of the cubic fourth-order Schrödinger equation

In this short paper, we prove that the solution of the cubic fourth-order Schrödinger equation (4NLS) on $\mathbb{R}^d$ ($5 \leq d \leq 8$) enjoys the same (pointwise) decay property as its linear solution does. This result is proved via a bootstrap argument based on the corresponding global result Pausader \cite{Pau1}. This result can be extended to more general dispersive equations (including some more 4NLS models) with scattering asymptotics.

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arXiv2021

Long time behavior of stochastic NLS with a small multiplicative noise

We prove the global space-time bound for the mass critical nonlinear Schrödinger equation perturbed by a small multiplicative noise in dimension three. The associated scattering behavior are also obtained. We also prove a global Strichartz space-time bound for the linear stochastic model, which is new itself and serves a prototype model for the nonlinear case. The proof combines techniques from \cite{fan2018global}, \cite{fan2020long} as well as local smoothing estimates for linear Schrödinger operators.

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arXiv2021

Global Well-posedness and scattering for fourth-order Schrödinger equations on waveguide manifolds

In this paper, we study the well-posedness theory and the scattering asymptotics for fourth-order Schrödinger equations (4NLS) on waveguide manifolds (semiperiodic spaces) $\mathbb{R}^d\times \mathbb{T}^n$, $d \geq 5$, $n=1,2,3$. The tori component $\mathbb{T}^n$ can be generalized to $n$-dimensional compact manifolds $\mathcal{M}^n$. First, we modify Strichartz estimates for 4NLS on waveguide manifolds, with which we establish the well-posedness theory in proper function spaces via the standard contraction mapping method. Moreover, we prove the scattering asymptotics based on an interaction Morawetz-type estimate established for 4NLS on waveguides. At last, we discuss the higher dimensional analogue, the focusing scenario and give some further remarks on this research line. This result can be regarded as the waveguide analogue of Pausader \cite{Pau2,Pau1,Pau3} and the 4NLS analogue of Tzvetkov-Visciglia \cite{TV2}.

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arXiv2020

Dynamical Hartree-Fock-Bogoliubov Approximation of Interacting Bosons

We consider a many-body Boson system with pairwise particle interaction given by $N^{3β-1}v(N^βx)$ for $0<β<1$ and $v$ a non-negative spherically-symmetric function. Our main result is the extension of the local-in-time Fock space (norm) approximation of the exact dynamics of squeezed states proved in Grillakis and Machedon, Comm. PDEs, (2017) for $0<β<\frac{2}{3}$ to a global-in-time norm approximation for $0<β<1$. Our work can also be viewed as a generalization of the results of Boccato, Cenatiempo, and Schlein, Ann. Henri Poincaré, (2017) to a more general set of initial data that includes coherent states along with an improved error estimate. The key ingredients in establishing the norm approximation are the work of Grillakis and Machedon on the local wellposedness theory, Grillakis and Machedon, Comm. PDEs, (2019), some recent established global estimates in Chong, Grillakis, Machedon, and Zhao, arXiv:2008.01753 (2020), and our quantitative result on the uniform in $N$ global wellposedness of the time-dependent Hartree-Fock-Bogoliubov (TBHFB) system.

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arXiv2020

Global Well-posedness for the focusing cubic NLS on the product space $\mathbb{R} \times \mathbb{T}^3$

In this paper, we prove the global well-posedness for the focusing, cubic nonlinear Schrödinger equation on the product space $\mathbb{R} \times \mathbb{T}^3$ with initial data below the threshold that arises from the the ground state in the Euclidean setting. The defocusing analogue was discussed and proved in Ionescu-Pausader \cite{IPRT3} (Comm. Math. Phys. 312 (2012), no. 3, 781-831).

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arXiv2020

Global estimates for the Hartree-Fock-Bogoliubov equations

We prove that certain Sobolev-type norms, slightly stronger than those given by energy conservation, stay bounded uniformly in time and $N$. This allows one to extend the local existence results of the second and third author globally in time. The proof is based on interaction Morawetz-type estimates and Strichartz estimates (including some new end-point results) for the equation $\{ \frac{1}{i}\partial_t-Δ_{x}-Δ_{y}+\frac{1}{N}V_N(x-y) \}Λ(t, x, y) =F$ in mixed coordinates such as $L^p(dt) L^q(dx) L^2(dy)$, $L^p(dt) L^q(dy) L^2(dx)$, $L^p(dt) L^q(d(x-y)) L^2(d(x+y))$. The main new technical ingredient is a dispersive estimate in mixed coordinates, which may be of interest in its own right.

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arXiv2020

Long time dynamics for defocusing cubic NLS on three dimensional product space

In this article, we study long time dynamics for defocusing cubic NLS on three dimensional product space. First, we apply the decoupling method in Bourgain-Demeter \cite{BD} to establish a bilinear Strichartz estimate. Moreover, we prove global well-posedness for defocusing, cubic NLS on three dimensional product space with rough initial data ($H^s$, $s>\frac{5}{6}$) based on I-method and the bilinear estimate. At last, we discuss the growth of higher Sobolev norm problem which is tightly linked to the weak turbulence phenomenon.

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arXiv2019

Well-posedness for energy-critical nonlinear Schrödinger equation on waveguide manifold

In this article, we utilize the scale-invariant Strichartz estimate on waveguide which is developed recently by Barron \cite{Barron} based on Bourgain-Demeter $l^2$ decoupling method \cite{BD} to give a unified and simpler treatment of well-posedness results for energy critical nonlinear Schrödinger equation on waveguide when the whole dimension is three and four.

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arXiv2018

Dynamics of subcritical threshold solutions for energy-critical NLS

In this paper, we study the dynamics of subcritical threshold solutions for focusing energy critical NLS on $\mathbb{R}^d$ ($d\geq 5$) with nonradial data. This problem with radial assumption was studied by T. Duyckaerts and F. Merle in \cite{DM} for $d=3,4,5$ and later by D. Li and X. Zhang in \cite{LZ} for $d \geq 6$. We generalize the conclusion for the subcritical threshold solutions by removing the radial assumption for $d\geq 5$. A key step is to show exponential convergence to the ground state $W(x)$ up to symmetries if the scattering phenomenon does not occur. Remarkably, an interaction Morawetz-type estimate are applied.

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arXiv2018

Global well-posedness and scattering for the defocusing cubic Schrödinger equation on waveguide $\mathbb{R}^2$ $\times$ $\mathbb{T}^2$

We consider the problem of large data scattering for the defocusing cubic nonlinear Schrödinger equation on $\mathbb{R}^2$ $\times$ $\mathbb{T}^2$. This equation is critical both at the level of energy and mass. The key ingredients contain a global-in-time Stricharz estimate, resonant system approximation and profile decomposition. Assuming the large data scattering for the 2d cubic resonant system, we can prove the large data scattering for this problem.

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