arXiv subjects

Shunlin Shen

Publications and source records attributed to Shunlin Shen.

15 recordsLinked to original sources

arXiv2026

Unconditional Uniqueness for the Energy-critical and Energy-supercritical Quadratic NLS

We study the quadratic nonlinear Schr\"{o}dinger equation in energy-critical and energy-supercritical regimes and establish unconditional uniqueness at critical regularity on both $\mathbb{T}^{d}$ and $\mathbb{R}^{d}$. We introduce a new infinite quadratic hierarchy that featuring a linear structure for tensor product forms, instead of marginal densities. Consequently, this newly constructed hierarchy differs from the quantum Gross-Pitaevskii hierarchy, and its structure is in fact closer to that of the classical Boltzmann hierarchy. We prove this quadratic hierarchy admits combinatorial structures that are compatible with the bilinear $U$-$V$ estimates we prove for the quadratic nonlinearity. These tools enable us to establish unconditional uniqueness at the critical regularity via a quadratic hierarchy approach, and thus to provide an affirmative answer to Bourgain's uniqueness concern [3,p.152] for the quadratic nonlinearity in the weak-form bilinear case.

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arXiv2026

Norm inflation for the cubic hyperbolic NLS on $\mathbb T^2$

We prove norm inflation for the cubic hyperbolic nonlinear Schr\"odinger equation in $H^s(\mathbb T^2)$ for every $s\in(-\infty,0)\cup(0,\frac12]$. The scaling-critical point $s=0$ is excluded by conservation of the $L^2$ norm. The strong ill-posedness below and above the scaling-critical point arises from two completely different mechanisms. Particularly in the scaling-subcritical regime, this dynamical instability stems from the hyperbolic nature. Together with the local well-posedness result in \cite{WangHNLS}, this gives a sharp dichotomy away from the mass space $L^2(\mathbb T^2)$: local well-posedness holds for $s>\frac12$, whereas norm inflation occurs for all $s\le \frac12$ with $s\ne0$.

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arXiv2026

Sharp $H_{x}^{s}$ Ill-posedness of the Hard-sphere Boltzmann Equation

We investigate the ill-posedness mechanism of the hard-sphere Boltzmann equation in $H_{x}^{s}$ Sobolev space. Via a direct construction, we prove a strong-weak type ill-posedness result in the low-regularity regime $s<1$, establishing a sharp threshold in connection to the local $s>1$ well-posedness result [11]. Instead of originating from the large-velocity growth of the collision kernel, this illposedness is generated by the loss term and dispersive effects. Consequently, we prove a dispersion-driven nonlinear instability mechanism for the hard-sphere Boltzmann equation, and provide a capstone of the ill-posedness series [18,20].

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arXiv2026

Asymptotic stability threshold of the 2-D monotone shear flow with no-slip boundary condition

In this paper, we investigate the asymptotic stability threshold problem for the 2-D Navier-Stokes equations in a finite channel with no-slip boundary conditions, around monotone shear flow $(U(t,y),0)$. We establish that the flow is asymptotically stable under perturbations satisfying $\|u^{\mathrm{in}}\|_{H^2}\leq c\nu^{\frac12}$. To achieve the stability threshold $\nu^{\frac{1}{2}}$, the key ingredients of the proof include: sharp resolvent estimates for the vorticity based on weak-type resolvent bounds; weighted space-time estimates for the vorticity; pointwise estimates for the velocity. Furthermore, we handle the nonlinear term through a divergence formulation, which facilitates the sharp application of the aforementioned space-time estimates.

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arXiv2025

Asymptotic stability of the symmetric flow via inviscid damping and enhanced dissipation

In this paper, we establish the inviscid damping and enhanced dissipation estimates for the linearized Navier-Stokes system around the symmetric flow in a finite channel with the non-slip boundary condition. As an immediate consequence, we prove the asymptotic stability of the symmetric flow in the high Reynolds number regime. Namely, if the initial velocity perturbation $u^{\mathrm{in}}$ satisfies $\Vert u^{\mathrm{in}}-(U(y),0) \Vert_{H^5}\leq c \nu^{\frac{2}{3}}$, then inviscid damping and enhanced dissipation estimates also hold for the solution to the Navier-Stokes system.

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arXiv2025

Global stability of the inhomogeneous sheared Boltzmann equation in torus

Homo-energetic solutions to the spatially homogeneous Boltzmann equation have been extensively studied, but their global stability in the inhomogeneous setting remains challenging due to unbounded energy growth under self-similar scaling and the intricate interplay between spatial dependence and nonlinear collision dynamics. In this paper, we introduce an approach for periodic spatial domains to construct global-in-time inhomogeneous solutions in a non-conservative perturbation framework, characterizing the global dynamics of growing energy. The growth of energy is shown to be governed by a long-time limit state that exhibits features not captured in either the homogeneous case or the classical Boltzmann theory. The core of our proof is the derivation of new energy estimates specific to the Maxwell molecule model. These estimates combine three key ingredients: a low-high frequency decomposition, a spectral analysis of the matrix associated with the second-order moment equation, and a crucial cancellation property in the zero-frequency mode of the nonlinear collision term. This last property bears a close analogy to the null condition in nonlinear wave equations.

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arXiv2025

$l^{2}$-decoupling and the unconditional uniqueness for the Boltzmann equation

We broaden the application of the $l^{2}$-decoupling theorem to the Boltzmann equation. We prove Strichartz estimates for the linear problem in the $\mathbb{T}^d$ setting. We establish space-time bilinear estimates, and hence the unconditional uniqueness of solutions to the $\mathbb{R}^d$ and $\mathbb{T}^d$ Boltzmann equation for the Maxwellian particle and soft potential with an angular cutoff, adopting a unified hierarchy scheme originally developed for the nonlinear Schr\"{o}dinger equation.

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arXiv2024

Derivation of the compressible Euler equations from the dynamics of interacting Bose gas in the hard-core limit regime

We investigate the dynamics of short-range interacting Bose gases with varying degrees of diluteness and interaction strength. By applying a combined mean-field and semiclassical space-time rescaling to the dynamics in both the Gross--Pitaevskii and hard-core limit regimes, we prove that the local one-particle mass, momentum, and energy densities of the many-body system can be quantitatively approximated by solutions to the compressible Euler system in the strong sense, up to the first blow-up time of the fluid description, as the number of particles tends to infinity. In the hard-core limit regime, two novel results are presented. First, we rigorously prove, for the first time, that the internal energy of the fluid takes the form $4\pi \mathfrak{c}_{0}\rho^{2}$ (equivalently, pressure $P=2\pi \mathfrak{c}_{0}\rho^{2}$), arising solely from the kinetic energy density of the many-body system, rather than the interaction energy density, marking a fundamental difference from the Gross--Pitaevskii and other mean-field regimes. Second, the newly discovered coupling constant $\mathfrak{c}_{0}$ is the electrostatic capacity of the interaction potential, corresponding to the scattering length of the hard-core potential. Furthermore, in other limiting regimes, including those beyond the Gross--Pitaevskii regime, we find that the limiting equation is described by an eikonal system, offering a rigorous first-principle justification for using the ``geometric optics approximation'' to describe the dynamics of ultracold Bose gases.

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arXiv2024

Global derivation of the 1D Vlasov-Poisson equation from quantum many-body dynamics with screened Coulomb potential

We study the 1D quantum many-body dynamics with a screened Coulomb potential in the mean-field setting. Combining the quantum mean-field, semiclassical, and Debye length limits, we prove the global derivation of the 1D Vlasov-Poisson equation. We tackle the difficulties brought by the pure state data, whose Wigner transforms converge to Wigner measures. We find new weighted uniform estimates around which we build the proof. As a result, we obtain, globally, stronger limits, and hence the global existence of solutions to the 1D Vlasov-Poisson equation subject to such Wigner measure data, which satisfy conservation laws of mass, momentum, and energy, despite being measure solutions. This happens to solve the 1D case of an open problem regarding the conservation law of the Vlasov-Poisson equation raised in [18] by Diperna and Lions.

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arXiv2023

Sharp Global Well-posedness and Scattering of the Boltzmann Equation

We consider the 3D Boltzmann equation for the Maxwellian particle and soft potential with an angular cutoff. We prove sharp global well-posedness with initial data small in the scaling-critical space. The solution also remains in $L^{1}$ if the initial datum is in $L^{1}$, even at such low regularity. The key to existence, uniqueness and regularity criteria is the new bilinear spacetime estimates for the gain term, the proof of which is based on novel techniques from nonlinear dispersive PDEs including the atomic $U$-$V$ spaces, multi-linear frequency analysis, dispersive estimates, etc. To our knowledge, this is the first 3D sharp global result for the Boltzmann equation.

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arXiv2023

Well/Ill-posedness of the Boltzmann Equation with Soft Potential

We consider the Boltzmann equation with the soft potential and angular cutoff. Inspired by the methods from dispersive PDEs, we establish its sharp local well-posedness and ill-posedness in $H^{s}$ Sobolev space. We find the well/ill-posedness separation at regularity $s=\frac{d-1}{2}$, strictly $\frac{1}{2}$-derivative higher than the scaling-invariant index $s=\frac{d-2}{2}$, the usually expected separation point.

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arXiv2023

On the mean-field and semiclassical limit from quantum $N$-body dynamics

We study the mean-field and semiclassical limit of the quantum many-body dynamics with a repulsive $\delta$-type potential $N^{3\beta}V(N^{\beta}x)$ and a Coulomb potential, which leads to a macroscopic fluid equation, the Euler-Poisson equation with pressure. We prove quantitative strong convergence of the quantum mass and momentum densities up to the first blow up time of the limiting equation. The main ingredient is a functional inequality on the $\delta$-type potential for the almost optimal case $\beta\in(0,1)$, for which we give an analysis of the singular correlation structure between particles.

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arXiv2022

The unconditional uniqueness for the energy-supercritical NLS

We consider the cubic and quintic nonlinear Schrödinger equations (NLS) under the $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ energy-supercritical setting. Via a newly developed unified scheme, we prove the unconditional uniqueness for solutions to NLS at critical regularity for all dimensions. Thus, together with [18,19], the unconditional uniqueness problems for $H^{1}$-critical and $H^{1}$-supercritical cubic and quintic NLS are completely and uniformly resolved at critical regularity for these domains. One application of our theorem is to prove that defocusing blowup solutions of the type in [54] is the only possible $C([0,T);\dot{H}^{s_{c}})$ solution if exist in these domains.

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arXiv2021

The derivation of the compressible Euler equation from quantum many-body dynamics

We study the three dimensional many-particle quantum dynamics in mean-field setting. We forge together the hierarchy method and the modulated energy method. We prove rigorously that the compressible Euler equation is the limit as the particle number tends to infinity and the Planck's constant tends to zero. We establish strong and quantitative microscopic to macroscopic convergence of mass and momentum densities up to the 1st blow up time of the limiting Euler equation. We justify that the macroscopic pressure emerges from the space-time averages of microscopic interactions, which are in fact, Strichartz-type bounds. We have hence found a physical meaning for Strichartz type bounds which were first raised by Klainerman and Machedon in this context.

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arXiv2020

The rigorous derivation of the $\mathbb{T}^{2}$ focusing cubic NLS from 3D

We derive rigorously the 2D periodic focusing cubic NLS as the mean-field limit of the 3D focusing quantum many-body dynamics describing a dilute Bose gas with periodic boundary condition in the $x$-direction and a well of infinite-depth in the $z$-direction. Physical experiments for these systems are scarce. We find that, to fulfill the empirical requirement for observing NLS dynamics in experiments, namely, the kinetic energy dominates the potential energy, it is necessary to impose an extra restriction on the system parameters. This restriction gives rises to an unusual coupling constant.

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