生成时间: 2026年01月05日 04:09:21

重点关注论文数量: 0

了解领域论文数量: 3

不相关论文数量: 3

了解领域论文（摘要翻译）

1. 具有奇异或退化矩阵值权的 $p(x)$-Laplacian 变分不等式的全局正则性估计

Global regularity estimates for $p(x)$-Laplacian variational inequalities with singular or degenerate matrix-valued weights

作者: Minh-Phuong Tran, Duc-Quang Bui, Thanh-Nhan Nguyen

类别: math.AP

发布日期: 2026-01-02

arXiv ID: 2601.00652v1

链接: http://arxiv.org/abs/2601.00652v1

摘要翻译

中文标题: 具有奇异或退化矩阵值权的 $p(x)$-Laplacian 变分不等式的全局正则性估计

摘要翻译: 我们建立了与 $p(x)$-Laplacian 型算子相关的椭圆变分不等式弱解的全局梯度界，该算子涉及退化或奇异的矩阵权。在矩阵值权、域的适当几何平坦性以及给定数据的最优正则性假设下，我们旨在研究问题结构对解的可积性水平的影响。为此，我们在两个方面发展了正则性理论：带权 Calderón-Zygmund 型估计和带权 Orlicz 型估计。我们结果的一个显著特点是，通过构造性的水平集（level-set）方法，可以在最小程度依赖结构常数的标度参数下导出这些估计。这些正则性结果是尖锐的，因为它们能够在允许的参数集内，利用近乎最优的标度参数来构造水平集估计。

2. 带阻尼双调和波方程反问题的 Lipschitz 稳定性

Lipschitz Stability for an Inverse Problem of Biharmonic Wave Equations with Damping

作者: Minghui Bi, Yixian Gao

类别: math.AP

发布日期: 2026-01-02

arXiv ID: 2601.00648v1

链接: http://arxiv.org/abs/2601.00648v1

摘要翻译

好的，请看以下翻译：

中文标题: 带阻尼双调和波方程反问题的 Lipschitz 稳定性

摘要翻译: 本文针对带阻尼双调和波方程，建立了同时恢复变量密度系数和初始位移的 Lipschitz 稳定性。数据由解的 Laplacian 的边界柯西数据构成，即 $Δu |_{\partial Ω}$ 和 $ \partial_{n}(Δu)|_{\partial Ω}$。我们首先证明了相关的系统算子生成一个压缩半群，从而确保了正问题的适定性。随后，通过乘子技术推导出一个关键的可观测性不等式。在此基础上，得到了反问题的显式稳定性估计。这些估计表明，双调和结构本身增强了参数辨识的稳定性，并且稳定性常数通过因子 $ (1 + γ)^{1/2} $ 明确地依赖于阻尼系数。这项工作为无损检测和动态反演中的应用提供了严格的理论基础。

3. 基于负耦合的多尺度反应-扩散系统中零模奇点的全局动力学与稳定性

Global Dynamics and Stabilization of Zero-Mode Singularities in Multi-Scale Reaction-Diffusion Systems via Negative Coupling

作者: Pengyue Hou

类别: math.AP, math.DS, nlin.CD

发布日期: 2026-01-02

arXiv ID: 2601.00638v1

链接: http://arxiv.org/abs/2601.00638v1

摘要翻译

中文标题: 基于负耦合的多尺度反应-扩散系统中零模奇点的全局动力学与稳定性

摘要翻译: 本文为多尺度负耦合系统（MNCS）建立了一个严格的数学框架，该动力学模型描述了具有定向、符号结构相互作用的分层状态空间。我们研究了在有界区域 $Ω\subset \mathbb{R}^d$ ($d \le 3$) 上满足齐次 Neumann 边界条件的反应-扩散系统的稳定性问题。该设定的一个关键特征是“零模奇点”，即拉普拉斯算子具有平凡的零特征值 ($λ_0=0$)，从而为空间平均值不提供线性耗散。我们严格证明了系统的全局适定性，以及在相空间 $\mathbb{H}=(L^2(Ω))^N$ 中存在紧致全局吸引子 $\mathcal{A}$。利用 Moser-Alikakos 迭代技术，我们建立了均匀的 $L^\infty(Ω)$ 界，克服了在三维情形下从 $H^1$ 到 $L^\infty$ 的 Sobolev 嵌入缺失问题。这些界限使得我们能够通过 Kaplan-Yorke 迹公式推导出吸引子分形维数的显式上界估计。我们证明了维数的标度律为 $d_F(\mathcal{A}) \sim \max\{0, \mathcal{K}_{\mathcal{A}}-γ\}^{d/2}$，这证实了负耦合强度 $γ$ 起到了压缩相空间的全局正规化作用。理论结果通过刚性稳定的二阶指数时间差分（ETD2）格式结合离散余弦变换（DCT）进行了验证，该方法严格强制执行了无通量边界条件。

不相关论文（基本信息）

1. 局部Lipschitz Lévy噪声驱动的非自治随机可逆Selkov格点系统的极限行为

Limiting Behavior of Non-Autonomous Stochastic Reversible Selkov Lattice Systems Driven by Locally Lipschitz Lévy Noises

作者: Guofu Li, Jianxin Wu, Yunshun Wu

类别: math.AP

发布日期: 2026-01-02

arXiv ID: 2601.00600v1

链接: http://arxiv.org/abs/2601.00600v1

摘要: This work investigates the long-term distributional behavior of the reversible Selkov lattice systems defined on the set $\mathbb{Z}$ and driven by locally Lipschitz \emph{Lévy noises}, which possess two pairs of oppositely signed nonlinear terms and whose nonlinear couplings can grow polynomially with any order $p \geq 1$. Firstly, based on the global-in-time well-posedness in $L^{2}(Ω, \ell^2 \times \ell^2)$, we define a \emph{continuous} non-autonomous dynamical system (NDS) on the metric space $(\mathcal{P}_{2}(\ell^2 \times \ell^2), d_{\mathcal{P}(\ell^2 \times \ell^2)})$, where $d_{\mathcal{P}(\ell^2 \times \ell^2)}$ is the dual-Lipschitz distance on $\mathcal{P}(\ell^2 \times \ell^2)$, the space of probability measures on $\ell^2 \times \ell^2$. Specifically, we establish that this non-autonomous dynamical system admits a unique pullback measure attractor, characterized via measure-valued complete solutions and orbits in the sense of Wang (DOI.org/10.1016/j.jde.2012.05.015). Moreover, when the deterministic external forcing terms are periodic in time, we demonstrate that the pullback measure attractors are also periodic. We also study the upper semicontinuity of pullback measure attractors as $(ε_1, ε_2, γ_1, γ_2) \rightarrow (0, 0, 0, 0)$. The main difficulty in proving the pullback asymptotic compactness of the NDS in $(\mathcal{P}_{2}(\ell^2 \times \ell^2), d_{\mathcal{P}(\ell^2 \times \ell^2)})$ is caused by the lack of compactness in infinite-dimensional lattice systems, which is overcome by using uniform tail-ends estimates. And the inherent structure of the Selkov system precludes the possibility of any unidirectional dissipative influence arising from the interaction between the two coupled equations, thereby obstructing the emergence of a dominant energy-dissipation mechanism along a single directional pathway.

2. 有界区域上分数阶 $p$-拉普拉斯哈代索伯列夫算子的全局紧性结果

Global compactness results for fractional $p$-Laplace Hardy Sobolev operator on a bounded domain

作者: Nirjan Biswas

类别: math.AP

发布日期: 2026-01-02

arXiv ID: 2601.00589v1

链接: http://arxiv.org/abs/2601.00589v1

摘要: In this paper, we establish a Struwe type global compactness result for a class of nonlinear critical Hardy-Sobolev exponent problems driven by the fractional $p$-Laplace Hardy-Sobolev operator.

3. 基于深度学习增强的傅里叶方法求解稀疏远场数据的多频反源问题

A Deep Learning-Enhanced Fourier Method for the Multi-Frequency Inverse Source Problem with Sparse Far-Field Data

作者: Hao Chen, Yan Chang, Yukun Guo, Yuliang Wang

类别: math.AP, math.NA

发布日期: 2026-01-01

arXiv ID: 2601.00427v1

链接: http://arxiv.org/abs/2601.00427v1

摘要: This paper introduces a hybrid computational framework for the multi-frequency inverse source problem governed by the Helmholtz equation. By integrating a classical Fourier method with a deep convolutional neural network, we address the challenges inherent in sparse and noisy far-field data. The Fourier method provides a physics-informed, low-frequency approximation of the source, which serves as the input to a U-Net. The network is trained to map this coarse approximation to a high-fidelity source reconstruction, effectively suppressing truncation artifacts and recovering fine-scale geometric details. To enhance computational efficiency and robustness, we propose a high-to-low noise transfer learning strategy: a model pre-trained on high-noise regimes captures global topological features, offering a robust initialization for fine-tuning on lower-noise data. Numerical experiments demonstrate that the framework achieves accurate reconstructions with noise levels up to 100%, significantly outperforms traditional spectral methods under sparse measurement constraints, and generalizes well to unseen source geometries.