arXiv subjects

Zhimeng Ouyang

Publications and source records attributed to Zhimeng Ouyang.

At least 19 recordsLinked to original sources

arXiv2026

Kinetic Wedge Layers and Diffusive Limit of Neutron Transport in Polygonal Domains

We establish the diffusive limit of the stationary neutron transport equation with velocity-dependent inflow data in polygonal domains. On a bounded convex polygon we assemble a composite approximation from an interior harmonic field, flat side layers, and kinetic wedge layers, and we prove that the solution converges to this composite in $L^{\infty}$ at an explicit algebraic rate, and, uniformly on compact subsets of the interior, to the harmonic field itself. We also give a complete formulation and well-posedness theory for the kinetic wedge layer, including its algebraic decay. The proof combines a characteristic stability estimate, a weighted Mellin mapping theorem, a two-depth construction and matching scheme, and a shifted superharmonic barrier for the wedge corrector. As a secondary result, we prove convergence at the square-root rate in $L^{2}$ on any bounded simple polygon, including those with reentrant vertices. That argument needs only an endpoint-truncated side layer and a two-test cancellation, and no kinetic wedge layer at all.

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arXiv2026

Dynamics of Chemotactic Gliding-Aggregation in Myxobacteria on Bounded Domains: Stochastic Modeling, Analysis, and Deep Neural Network Simulations

Bacterial chemotactic movement and collective aggregation have long attracted substantial interest in mathematical biology and applied modeling. Classical Keller--Segel-type systems, however, are typically formulated under idealized laboratory assumptions, such as smooth agar substrates, and thus cannot adequately capture the gliding dynamics of myxobacteria in naturally rough environments like soil. In this paper, we propose a unified framework that integrates stochastic modeling, rigorous analysis, and deep neural network-based simulation of chemotactic gliding--diffusion and aggregation processes on bounded domains. Starting from a lattice-based discrete agent description and a subordinated Langevin equation driven by an inverse stable subordinator at the microscopic level, we characterize anomalous gliding dynamics on rough surfaces and derive a macroscopic time-nonlocal Keller--Segel-type chemotaxis model with logarithmic sensitivity. We then establish a comprehensive solution theory for the resulting model, covering mass conservation, novel regularity results, local well-posedness in any spatial dimension, and global well-posedness in two and three. The analysis relies on several newly developed ingredients, including a fractional Lyapunov functional, a variational inequality adapted to the time-nonlocal structure, logarithmic Sobolev-type estimates, Bregman distance techniques, and a weighted bootstrap mechanism adapted to the singular sensitivity and time-nonlocal memory. Finally, we design a mesh-free, positivity-preserving, multi-objective, time-marching physics-informed neural network method with separate architectures and tailored variable transformations. Numerical experiments on complex geometries, including a butterfly-shaped domain, demonstrate the robustness, accuracy, and flexibility of the proposed computational framework across a range of Keller--Segel-type systems.

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arXiv2024

Lattice Approximations to NLS

In this paper, we prove that solutions of the discrete NLS lattice model for $L^2$ initial data with double frequency components converge to solutions of a coupled system of cubic NLS.

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arXiv2024

The Local-well-posedness of the relativistic Vlasov-Maxwell-Landau system with the specular reflection boundary condition

We prove the local-in-time well-posedness of the relativistic Vlasov-Maxwell-Landau system in a bounded domain $Ω$ with the specular reflection condition. Our result covers the case when $Ω$ is a non-convex domain, e.g., solid torus. To the best of our knowledge, this is the first local well-posedness result for a nonlinear kinetic model with a self-consistent magnetic effect in a three-dimensional $\textbf{bounded}$ domain.

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arXiv2024

The modified Korteweg--de Vries limit of the Ablowitz--Ladik system

For slowly-varying initial data, solutions to the Ablowitz-Ladik system have been proven to converge to solutions of the cubic Schrödinger equation. In this paper we show that in the continuum limit, solutions to the Ablowitz-Ladik system with $H^1$ initial data may also converge to solutions of the modified Korteweg--de Vries equation. To exhibit this new limiting behavior, it suffices that the initial data is supported near the inflection points of the dispersion relation associated with the Ablowitz-Ladik system. Our arguments employ harmonic analysis tools, Strichartz estimates, and the conservation of mass and energy. Correspondingly, they are applicable beyond the completely integrable models of greatest interest to us.

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arXiv2023

Hilbert Expansion for Coulomb Collisional Kinetic Models

The relativistic Vlasov-Maxwell-Landau (r-VML) system and the relativistic Landau equation (r-LAN) are fundamental models that describe the dynamics of an electron gas. In this paper, we introduce a novel weighted energy method and establish the validity of the Hilbert expansion for the r-VML system and r-LAN equation. As the Knudsen number shrinks to zero, we rigorously demonstrate the relativistic Euler-Maxwell limit and relativistic Euler limit, respectively. This successfully resolves the long-standing open problem regarding the hydrodynamic limits of Landau-type equations.

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arXiv2023

Conditional $L^{\infty}$ estimates for the non-cutoff Boltzmann equation in a bounded domain

We consider weak solutions of the inhomogeneous non-cutoff Boltzmann equation in a bounded domain with any of the usual physical boundary conditions: in-flow, bounce-back, specular-reflection and diffuse-reflection. When the mass, energy and entropy densities are bounded above, and the mass density is bounded away from vacuum, we obtain an estimate of the $L^\infty$ norm of the solution depending on the macroscopic bounds on these hydrodynamic quantities only. It is a regularization effect in the sense that the initial data is not required to be bounded.

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arXiv2023

Modified Wave Operators for the Wave-Klein-Gordon System

We consider a coupled Wave-Klein-Gordon system in 3D, which is a simplified model for the global nonlinear stability of the Minkowski space-time for self-gravitating massive fields. In this paper we study the large-time asymptotic behavior of solutions to such systems, and prove modified wave operators for small and smooth data with mild decay at infinity. The key novelty comes from a crucial observation that the asymptotic dynamics are dictated by the resonant interactions. As a consequence, our main results include the derivation of a resonant system with good error bounds, and a detailed description of the asymptotic dynamics of such quasilinear evolution system of hyperbolic and dispersive type.

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arXiv2023

Diffusive Limit of the Boltzmann Equation in Bounded Domains

The rigorous justification of the hydrodynamic limits of kinetic equations in bounded domains has been actively investigated in recent years. In spite of the progress for the diffuse-reflection boundary case, the more challenging in-flow boundary case, in which the leading-order boundary layer effect is non-negligible, still remains open. In this work, we consider the stationary and evolutionary Boltzmann equation with the in-flow boundary in general (convex or non-convex) bounded domains, and demonstrate their incompressible Navier-Stokes-Fourier (INSF) limits in $L^2$. Our method relies on a novel and surprising gain of $\varepsilon^{\frac{1}{2}}$ in the kernel estimate, which is rooted from a key cancellation of delicately chosen test functions and conservation laws. We also introduce the boundary layer with grazing-set cutoff and investigate its BV regularity estimates to control the source terms of the remainder equation with the help of Hardy's inequality.

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arXiv2023

Diffusive Limit of the Unsteady Neutron Transport Equation in Bounded Domains

The justification of hydrodynamic limits in non-convex domains has long been an open problem due to the singularity at the grazing set. In this paper, we investigate the unsteady neutron transport equation in a general bounded domain with the in-flow, diffuse-reflection, or specular-reflection boundary condition. Using a novel kernel estimate, we demonstrate the optimal $L^2$ diffusive limit in the presence of both initial and boundary layers. Previously, this result was only proved for convex domains when the time variable is involved. Our approach is highly robust, making it applicable to all basic types of physical boundary conditions.

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arXiv2022

Continuum limit for the Ablowitz--Ladik system

We show that solutions to the Ablowitz--Ladik system converge to solutions of the cubic nonlinear Schrödinger equation for merely $L^2$ initial data. Furthermore, we consider initial data for this lattice model that excites Fourier modes near both critical points of the discrete dispersion relation and demonstrate convergence to a decoupled system of nonlinear Schrödinger equations.

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arXiv2022

Determining the collision kernel in the Boltzmann equation near the equilibrium

We consider an inverse problem for the nonlinear Boltzmann equation near the equilibrium. Our goal is to determine the collision kernel in the Boltzmann equation from the knowledge of the Albedo operator. Our approach relies on a linearization technique as well as the injectivity of the Gauss-Weierstrass transform.

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arXiv2022

Hilbert Expansion for the Relativistic Landau Equation

In this paper, we study the local-in-time validity of the Hilbert expansion for the relativistic Landau equation. We justify that solutions of the relativistic Landau equation converge to small classical solutions of the limiting relativistic Euler equations as the Knudsen number shrinks to zero in a weighted Sobolev space. The key difficulty comes from the temporal and spatial derivatives of the local Maxwellian, which produce momentum growth terms and are uncontrollable by the standard $L^2$-based energy and dissipation. We introduce novel time-dependent weight functions to generate additional dissipation terms to suppress the large momentum. The argument relies on a hierarchy of energy-dissipation structures with or without weights. As far as the authors are aware of, this is the first result of the Hilbert expansion for the Landau-type equation.

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arXiv2022

The Vlasov-Poisson-Landau System with the Specular-Reflection Boundary Condition

We consider the Vlasov-Poisson-Landau system, a classical model for a dilute collisional plasma interacting through Coulombic collisions and with its self-consistent electrostatic field. We establish global stability and well-posedness near the Maxwellian equilibrium state with decay in time and some regularity results for small initial perturbations, in any general bounded domain (including a torus as in a tokamak device), in the presence of specular reflection boundary condition. We provide a new improved $L^{2}\rightarrow L^{\infty }$ framework: $L^{2}$ energy estimate combines only with $S^{p}$ estimate for the ultra-parabolic equation.

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arXiv2021

On the Quantum Boltzmann Equation near Maxwellian and Vacuum

We consider the non-relativistic quantum Boltzmann equation for fermions and bosons. Using the nonlinear energy method and mild formulation, we justify the global well-posedness when the density function is near the global Maxwellian and vacuum. This work is a generalization and adaptation of the classical Boltzmann theory. Our main contribution is a detailed analysis of the nonlinear operator $Q$ in the quantum context. This is the first piece of a long-term project on the quantum kinetic equations.

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arXiv2021

Scattering map for the Vlasov-Poisson system

We construct (modified) scattering operators for the Vlasov-Poisson system in three dimensions, mapping small asymptotic dynamics as $t\to -\infty$ to asymptotic dynamics as $t\to +\infty$. The main novelty is the construction of modified wave operators, but we also obtain a new simple proof of modified scattering. Our analysis is guided by the Hamiltonian structure of the Vlasov-Poisson system. Via a pseudo-conformal inversion we recast the question of asymptotic behavior in terms of local in time dynamics of a new equation with singular coefficients which is approximately integrated using a generating function.

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arXiv2020

Asymptotic Analysis of Boltzmann Equation in Bounded Domains

Consider 3D Boltzmann equation in convex domains with diffusive-reflection boundary condition. We study the hydrodynamic limits as the Knudsen number and Strouhal number $ε\rightarrow 0^+$. Using the Hilbert expansion, we rigorously justify that the solution of stationary/evolutionary problem converges to that of the steady/unsteady Navier-Stokes-Fourier system. This is the first paper to justify the hydrodynamic limits of nonlinear Boltzmann equations with hard-sphere collision kernel in bounded domain in the $L^{\infty}$ sense. The proof relies on a novel analysis on the boundary layer effect with geometric correction. The difficulty mainly comes from three sources: 3D domain, boundary layer regularity, and time dependence. To fully solve this problem, we introduce several techniques: (1) boundary layer with geometric correction; (2) remainder estimates with $L^2-L^{6}-L^{\infty}$ framework. Keywords: boundary layer; Milne problem; geometric correction; remainder estimates.

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arXiv2020

$L^2$ decay for the linearized Landau equation with the specular boundary condition

In this paper, we develop an alternative approach to establish the $L^2$ decay estimate for the linearized Landau equation in a bounded domain with specular boundary condition. The proof is based on the methodology of proof by contradiction motivated by [Guo, Comm. Pure Appl. Math., 55(9):1104-1135, 2002] and [Guo, Arch. Ration. Mech. Anal., 197(3):713-809, 2010].

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