arXiv subjects

Zhongwei Shen

Publications and source records attributed to Zhongwei Shen.

At least 19 recordsLinked to original sources

arXiv2026

Resolvent Estimates for the Stokes Operator in a Three-Dimensional Lipschitz Domain

By refining the approach developed in \cite{AGH-2015, GS2026a}, we establish $L^p$ resolvent estimates for the Stokes operator in a bounded Lipschitz domain $Ω$ in $\R^3$ for any $(3/2)-\e< p\le \infty$, where $\e>0$ depends on $Ω$. As a consequence, the Stokes operator generates a uniformly bounded analytic semigroup in $L^p_σ(Ω)$. The results are particularly surprising, as it is long believed that the $L^p$ resolvent estimates in three-dimensional Lipschitz domains may fail for large $p$. In the Appendix we prove a theorem on interpolation between $L^2_σ(Ω)$ and $L^\infty_σ(Ω)$.

math.AP↗

arXiv2026

Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups

Koopman spectral analysis turns nonlinear stochastic dynamics into a linear evolution of observables and gives access to decay rates, oscillatory modes, and metastable behavior. In practice, however, EDMD, SDMD, and related estimators depend strongly on where the trajectory data are collected. If most trajectories start in regions that carry little spectral information, the leading eigenvalues and eigenfunctions can be poorly estimated even with a rich dictionary. We propose \emph{Reinforced SDMD}, a data-acquisition method that couples Stochastic Dynamic Mode Decomposition with reinforcement learning. The RL agent chooses trajectory-initialization regions, SDMD updates the Koopman approximation, and a spectral-consistency reward evaluates the estimated eigenpairs on the newly generated data. An exploration bonus is added to avoid repeatedly sampling only a small part of the state space. We test multi-armed bandits, DQN, and PPO on stochastic double-well, Duffing, and FitzHugh--Nagumo systems. The learned policies place more samples in regions that are useful for estimating the leading Koopman eigenpairs. We also give an error-propagation analysis showing how SDMD operator error enters the corresponding bandit, approximate value-iteration, and approximate policy-iteration bounds.

math.DS↗

arXiv2026

Quantitative homogenization of elliptic equations with infinitely many scales

In this paper, we develop a general homogenization theory for elliptic equations with coefficients that oscillate periodically at infinitely many scales $\varepsilon = (\varepsilon_1, \varepsilon_2, \cdots) \in (0,1)^\infty$, with $\varepsilon_1>\varepsilon_2>\cdots$ and $\varepsilon_n \to 0$ as $n \to \infty$. Such problems arise naturally in the study of fractal materials and diffusion in fluids. Under suitable scale-separation assumptions, we prove a qualitative homogenization theorem and obtain optimal $L^2$ convergence rates. We also establish interior and boundary Lipschitz estimates that are uniform in $\varepsilon$.

math.AP↗

arXiv2026

Large-scale harmonic measures and nontangential maximal functions in periodic homogenization

In this paper, we consider the elliptic operators $\mathcal{L}_\varepsilon = -\nabla\cdot (A(X/\varepsilon) \nabla )$ with periodic coefficients in a bounded domain $Ω$ without any local smoothness assumption on $A = A(Y)$, where $\varepsilon \ll \text{diam}(Ω)$ is a microscopic scale. Due to the irregularity of the coefficients at $\varepsilon$ scale, we introduce the correct forms of the large-scale nontangential maximal functions for the Dirichlet, Neumann and regularity problems that measure the behaviors of solutions at an $\varepsilon$ distance away from the boundary. The $L^p$ estimates uniform in $\varepsilon$ are established for these nontangential maximal functions for the same and optimal ranges of $p$ as the Laplace operator in the Lipschitz or $C^1$ domains. With some additional regularity assumption on the coefficients, the large-scale estimates combined with the small-scale estimates recover the classical full-scale estimates of the nontangential maximal functions. Our proofs are based on the notion of large-scale $\mathcal{L}_\varepsilon$-harmonic measures, the periodic structure of operators in the transversal direction to the boundaries, and the homogenization tools, including convergence rates and large-scale regularity.

math.AP↗

arXiv2026

FusionRegister: Every Infrared and Visible Image Fusion Deserves Registration

Spatial registration across different visual modalities is a critical but formidable step in multi-modality image fusion for real-world perception. Although several methods are proposed to address this issue, the existing registration-based fusion methods typically require extensive pre-registration operations, limiting their efficiency. To overcome these limitations, a general cross-modality registration method guided by visual priors is proposed for infrared and visible image fusion task, termed FusionRegister. Firstly, FusionRegister achieves robustness by learning cross-modality misregistration representations rather than forcing alignment of all differences, ensuring stable outputs even under challenging input conditions. Moreover, FusionRegister demonstrates strong generality by operating directly on fused results, where misregistration is explicitly represented and effectively handled, enabling seamless integration with diverse fusion methods while preserving their intrinsic properties. In addition, its efficiency is further enhanced by serving the backbone fusion method as a natural visual prior provider, which guides the registration process to focus only on mismatch regions, thereby avoiding redundant operations. Extensive experiments on three datasets demonstrate that FusionRegister not only inherits the fusion quality of state-of-the-art methods, but also delivers superior detail alignment and robustness, making it highly suitable for infrared and visible image fusion method. The code will be available at https://github.com/bociic/FusionRegister.

cs.CV↗

arXiv2025

Exponential Decays of Steklov Eigenfunctions for the Magnetic Laplacian

Consider the Dirichlet-to-Neumann map $Λ_β$ associated with the Schrödinger operator $(D+β\A)^2$ with a magnetic potential in a bounded Lipschitz domain $Ω$, where $β>1$ is the field strength parameter. Assume that the magnetic field $\B=\nabla \times \A$ is of finite type. We show that if $β>β_0$, the ground state for $Λ_β$ decays exponentially away from a neighborhood of the subset of $\partialΩ$, on which $\B$ vanishes to the maximal order.

math.AP↗

arXiv2025

Resolvent Estimates in $L^\infty$ for the Stokes Operator in Nonsmooth Domains

We establish resolvent estimates in spaces of bounded solenoidal functions for the Stokes operator in a bounded domain $Ω$ in $R^d$ under the assumptions that $Ω$ is $C^1$ for $d\ge 3$ and Lipschitz for $d=2$. As a corollary, it follows that the Stokes operator generates a uniformly bounded analytic semigroup in the spaces of bounded solenoidal functions in $Ω$. The smoothness conditions on $Ω$ are sharp. The case of exterior domains with nonsmooth boundaries is also studied.The key step in the proof involves new estimates which connect the pressure to the velocity in the $L^q$ average, but only on scales above certain level.

math.AP↗

arXiv2025

Boundary Value Problems for the Magnetic Laplacian in Semiclassical Analysis

This paper is concerned with the magnetic Laplacian $P^h (\A)=(h D+\A)^2$ in semiclassical analysis, where $h$ is a semiclassical parameter. We study the $L^2$ Neumann and Dirichlet problems for the equation $P^h(\A)u=0$ in a bounded Lipschitz domain $Ω$. Under the assumption that the magnetic field $\nabla \times \A$ is of finite type on $\overlineΩ$, we establish the nontangential maximal function estimates for $(h D+\A)u$, which are uniform for $0< h< h_0$. This extends a well-known result due to D. Jerison and C. Kenig for the Laplacian in Lipschitz domains to the magnetic Laplacian in the semiclassical setting. Our results are new even for smooth domains.

math.AP↗

arXiv2025

A Data-Driven Framework for Koopman Semigroup Estimation in Stochastic Dynamical Systems

We present Stochastic Dynamic Mode Decomposition (SDMD), a novel data-driven framework for approximating the Koopman semigroup in stochastic dynamical systems. Unlike existing methods, SDMD explicitly incorporates sampling time into its approximation, ensuring numerical stability and precision. By directly approximating the Koopman semigroup instead of the generator, SDMD avoids computationally expensive matrix exponential computations, which offers a more efficient and practical pathway for analyzing stochastic dynamics. The framework further integrates neural networks to automate basis selection, which reduces the reliance on manual intervention while maintaining computational efficiency. Rigorous theoretical guarantees, including convergence in the large data limit, zero-limit of sampling time, and large dictionary size, establish the method's reliability. Numerical experiments on canonical stochastic systems validate SDMD's effectiveness in approximating eigenvalues and eigenfunctions of the stochastic Koopman operator.

math.DS↗

arXiv2025

Quasi-potential of stationary and quasi-stationary densities: existence, regularity, and applications

The present paper is devoted to the large deviation principle (LDP), with particular emphasis on the regularity of the quasi-potential for densities of stationary and quasi-stationary distributions of randomly perturbed dynamical systems. Our framework is set up within a positively invariant set contained in the basin of attraction of a maximal attractor of the unperturbed system. Such a setting with a general maximal attractor is anticipated in many applications.

math.DS↗

arXiv2025

Stochastic stability of physical measures in conservative systems

Given the significance of physical measures in understanding the complexity of dynamical systems as well as the noisy nature of real-world systems, investigating the stability of physical measures under noise perturbations is undoubtedly a fundamental issue in both theory and practice. The present paper is devoted to the stochastic stability of physical measures for conservative systems on a smooth, connected, and closed Riemannian manifold. It is assumed that a conservative system admits an invariant measure with a positive and mildly regular density. Our findings affirm, in particular, that such an invariant measure has strong stochastic stability whenever it is physical, that is, for a large class of small random perturbations, the density of this invariant measure is the zero-noise limit in $L^{1}$ of the densities of unique stationary measures of corresponding randomly perturbed systems. Stochastic stability in a stronger sense is obtained under additional assumptions. Examples are constructed to demonstrate that stochastic stability could occur even if the invariant measure is non-physical. The high non-triviality of constructing such examples asserts the sharpness of the stochastic stability conclusion. Similar results are established for conservative systems on bounded domains.

math.DS↗

arXiv2025

ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals

Analyzing the long-term behavior of high-dimensional nonlinear dynamical systems remains a significant challenge. While the Koopman operator framework provides a powerful global linearization tool, current methods for approximating its spectral components often face theoretical limitations and depend on predefined dictionaries. Residual Dynamic Mode Decomposition (ResDMD) advanced the field by introducing the \emph{spectral residual} to assess Koopman operator approximation accuracy; however, its approach of only filtering precomputed spectra prevents the discovery of the operator's complete spectral information, a limitation known as the `spectral inclusion' problem. We introduce ResKoopNet (Residual-based Koopman-learning Network), a novel method that directly addresses this by explicitly minimizing the \emph{spectral residual} to compute Koopman eigenpairs. This enables the identification of a more precise and complete Koopman operator spectrum. Using neural networks, our approach provides theoretical guarantees while maintaining computational adaptability. Experiments on a variety of physical and biological systems show that ResKoopNet achieves more accurate spectral approximations than existing methods, particularly for high-dimensional systems and those with continuous spectra, which demonstrates its effectiveness as a tool for analyzing complex dynamical systems.

cs.LG↗

arXiv2025

The Magnetic Laplacian with a Higher-order Vanishing Magnetic Field in a Bounded Domain

This paper is concerned with spectrum properties of the magnetic Laplacian with a higher-order vanishing magnetic field in a bounded domain. We study the asymptotic behaviors of ground state energies for the Dirichlet Laplacian, the Neumann Laplacian, and the Dirichlet-to-Neumann operator, as the field strength parameter $β$ goes to infinite. Assume that the magnetic field does not vanish to infinite order, we establish the leading orders of $β$. We also obtain the first terms in the asymptotic expansions with remainder estimates under additional assumptions on an invariant subspace for a Taylor polynomial of the magnetic field. Our aim is to provide a unified approach to all three cases.

math.AP↗

arXiv2025

Mesoscopic and Macroscopic Entropy Balance Equations in a Stochastic Dynamics and Its Deterministic Limit

Entropy, its production, and its change in a dynamical system can be understood from either a fully stochastic dynamic description or from a deterministic dynamics exhibiting chaotic behavior. By taking the former approach based on the general diffusion process with diffusion $\tfrac{1}α{\bf D}(\bf x)$ and drift $\bf b(\bf x)$, where $α$ represents the ``size parameter'' of a system, we show that there are two distinctly different entropy balance equations. One reads ${\rm d} S^{(α)}/{\rm d} t = e^{(α)}_p + Q^{(α)}_{ex}$ for all $α$. However, the leading $α$-order, ``extensive'', terms of the entropy production rate $e^{(α)}_p$ and heat exchange rate $Q^{(α)}_{ex}$ are exactly cancelled. Therefore, in the asymptotic limit of $α\to\infty$, there is a second, local ${\rm d} S/{\rm d} t = \nabla\cdot{\bf b}({\bf x}(t))+\left({\bf D}:{\bf Σ}^{-1}\right)({\bf x}(t))$ on the order of $O(1)$, where $\tfrac{1}α{\bf D}(\bf x(t))$ represents the randomness generated in the dynamics usually represented by metric entropy, and $\tfrac{1}α{\bf Σ}({\bf x}(t))$ is the covariance matrix of the local Gaussian description at ${\bf x}(t)$, which is a solution to the ordinary differential equation $\dot{\bf x}={\bf b}(\bf x)$ at time $t$. This latter equation is akin to the notions of volume-preserving conservative dynamics and entropy production in the deterministic dynamic approach to nonequilibrium thermodynamics {\it à la} D. Ruelle. As a continuation of [17], mathematical details with sufficient care are given in four Appendices.

math-ph↗

arXiv2024

Conti-Fuse: A Novel Continuous Decomposition-based Fusion Framework for Infrared and Visible Images

For better explore the relations of inter-modal and inner-modal, even in deep learning fusion framework, the concept of decomposition plays a crucial role. However, the previous decomposition strategies (base \& detail or low-frequency \& high-frequency) are too rough to present the common features and the unique features of source modalities, which leads to a decline in the quality of the fused images. The existing strategies treat these relations as a binary system, which may not be suitable for the complex generation task (e.g. image fusion). To address this issue, a continuous decomposition-based fusion framework (Conti-Fuse) is proposed. Conti-Fuse treats the decomposition results as few samples along the feature variation trajectory of the source images, extending this concept to a more general state to achieve continuous decomposition. This novel continuous decomposition strategy enhances the representation of complementary information of inter-modal by increasing the number of decomposition samples, thus reducing the loss of critical information. To facilitate this process, the continuous decomposition module (CDM) is introduced to decompose the input into a series continuous components. The core module of CDM, State Transformer (ST), is utilized to efficiently capture the complementary information from source modalities. Furthermore, a novel decomposition loss function is also designed which ensures the smooth progression of the decomposition process while maintaining linear growth in time complexity with respect to the number of decomposition samples. Extensive experiments demonstrate that our proposed Conti-Fuse achieves superior performance compared to the state-of-the-art fusion methods.

cs.CV↗

arXiv2024

Neumann Problems for the Stokes Equations in Convex Domains

This paper studies the Neumann boundary value problems for the Stokes equations in a convex domain in $\mathbb{R}^d$. We obtain nontangential-maximal-function estimates in $L^p$ and $W^{1, p}$ estimates for $p$ in certain ranges depending on $d$. These ranges are larger than the known ranges for Lipschitz domains. The proof relies on a $W^{2, 2}$ estimate for the Stokes equations in convex domains.

math.AP↗

arXiv2024

From habitat decline to collapse: a spatially explicit approach connecting habitat degradation to destruction

Habitat loss, driven primarily by anthropogenic activity, significantly threatens ecosystem sustainability. While it is well understood that habitat loss is the leading contributor to declines in biodiversity worldwide, the connection between habitat degradation, destruction, and different locomotion strategies remains unclear. We use a reaction-diffusion framework to analyze the effects of habitat loss on population persistence and abundance. We establish necessary and sufficient conditions for the existence of an extinction threshold, beyond which further degradation of the environment predicts deterministic extirpation. Our results offer a robust analytical connection between habitat degradation and destruction, providing a mechanistic understanding of species persistence under varying environmental conditions and differing locomotion strategies.

math.AP↗

arXiv2024

Convergence rates of eigenvalue problems in perforated domains: the case of small volume

This paper is concerned with the Dirichlet eigenvalue problem for Laplace operator in a bounded domain with periodic perforation in the case of small volume. We obtain the optimal quantitative error estimates independent of the spectral gaps for an asymptotic expansion, with two leading terms, of Dirichlet eigenvalues. We also establish the convergence rates for the corresponding eigenfunctions. Our approach uses a known reduction to a degenerate elliptic eigenvalue problem for which a quantitative analysis is carried out.

math.AP↗