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1. 关于刚体与黏性流体相互作用的研究：弱解的存在性及一个适当的结构定理

On the interaction between a rigid-body and a viscous-fluid: existence of a weak solution and a suitable Théorème de Structure

作者: Paolo Maremonti, Filippo Palma

类别: math-ph, math.AP

发布日期: 2026-02-12

arXiv ID: 2602.11787v1

链接: http://arxiv.org/abs/2602.11787v1

详细分析

中文标题: 关于刚体与黏性流体相互作用的研究：弱解的存在性及一个适当的结构定理

研究对象和背景

本文研究的是刚体–黏性不可压缩牛顿流体耦合系统（fluid–structure interaction, FSI）在无外力作用下的演化行为。该系统由一个封闭、有界、边界 $\partial B$ 光滑的刚体 $B \subset \mathbb{R}^3$ 与占据其外部区域 $\Omega := \mathbb{R}^3 \setminus B$ 的黏性不可压缩流体 $F$ 构成，整体记为 $S = \{B, F\}$。关键在于：系统动力学被置于一个固连于刚体的主惯性坐标系中，原点取在刚体质心 $G$ 处。这一选择虽具物理自然性，却在分析上引入了本质困难——流体动量方程中出现非标准的旋转对流项 $\omega \times x \cdot \nabla u$，破坏了经典 Navier–Stokes 方程的能量结构。

在该坐标系下，系统满足如下初边值问题（见原文第2页公式 (1.1)）：

$$ \begin{cases} u_t - \Delta u = -[(u - V)\cdot\nabla u + \omega \times u] - \nabla \pi, & \text{in } (0,T) \times \Omega, \\ \nabla \cdot u = 0, & \text{in } (0,T) \times \Omega, \\ u(t,x) = V(t,x) = \xi(t) + \omega(t) \times x, & \text{on } (0,T) \times \partial\Omega, \\ \displaystyle \lim_{|x|\to\infty} u(t,x) = 0, & \forall t \in (0,T), \\ \displaystyle m\dot{\xi} + m\,\omega \times \xi + \int_{\partial\Omega} T(u,\pi)\cdot\nu \, dS = 0, & \forall t \in (0,T), \\ \displaystyle I \cdot \dot{\omega} + \omega \times (I \cdot \omega) + \int_{\partial\Omega} x \times \big(T(u,\pi)\cdot\nu\big) \, dS = 0, & \forall t \in (0,T), \\ \xi(0) = \xi_0,\quad \omega(0) = \omega_0, \\ u(0,x) = u_0(x), & \forall x \in \Omega. \end{cases} $$

其中：$u : (0,T)\times\Omega \to \mathbb{R}^3$ 为流体速度场；$\pi$ 为压力；$V(t,x)$ 是刚体表面速度，由平移速度 $\xi(t) \in \mathbb{R}^3$ 与角速度 $\omega(t) \in \mathbb{R}^3$ 决定；$m > 0$ 为刚体总质量；$I$ 为主惯性张量（对角正定，即 $I = \operatorname{diag}(A,B,C)$）；$\nu$ 是 $\partial\Omega$ 上指向流体外部的单位法向量；Cauchy 应力张量定义为

$$ T(u,\pi) := -\pi I + 2\mu D(u), \quad D(u) := \frac{1}{2}\big(\nabla u + (\nabla u)^\top\big), $$


并设动态黏度 $\mu = 1$ 以简化记号。

该系统的定义域 $\Omega = \mathbb{R}^3 \setminus B$ 是无界外区域，且边界条件为非齐次 Dirichlet 型运动边界条件：流体在刚体表面必须“贴附”于刚体运动，即 $u = V$ on $\partial\Omega$。这导致传统 Sobolev 空间框架失效，需构造适配刚体运动的加权函数空间。

为此，作者定义了两个关键 Hilbert 空间：

  • $H(\Omega)$：包含所有在 $\Omega$ 中弱散度为零、且在 $\partial\Omega$ 上具有广义迹 $\phi_\nu[u] = \phi_\nu[u]$（即与某刚体运动 $u \in R$ 一致）的 $L^2$ 向量场；
  • $V(\Omega)$：$H(\Omega)$ 中进一步要求梯度平方可积，并配备等效范数 $\|u\|_{V(\Omega)}^2 := \int_\Omega |D(u)|^2\,dx + |\xi|^2 + \omega \cdot (I\cdot\omega)$。

特别地，空间 $R := \{ u \in C^\infty(\mathbb{R}^3) : u(x) = u_1 + u_2 \times x,\; u_1,u_2 \in \mathbb{R}^3 \}$ 刻画了所有刚体运动的速度场形式。这种构造确保了能量泛函的物理一致性，也使系统总动能（流体动能 + 刚体平动/转动动能）得以明确定义。

此外，由于 $\Omega$ 无界，常规 $L^2$ 能量不封闭，作者引入加权空间 $W^{1,2}(\Omega, |x|) := \{ u : |x|\nabla u \in L^2(\Omega) \}$，并在初始数据中要求

$$ (u_0,\xi_0,\omega_0) \in \big(V(\Omega) \cap \dot{W}^{1,2}(\Omega, |x|)\big) \times \mathbb{R}^3 \times \mathbb{R}^3, $$


且满足相容性条件 $\gamma(u_0 - \xi_0 - \omega_0 \times x) = 0$（即初速度在边界上匹配刚体初运动），以及 $\operatorname{curl} u_0 \in L^1(\Omega)$。这些条件并非技术冗余，而是后续迭代构造与正则性提升所必需。

主要定理或主要结果

本文的核心成果是以下 Theorem 2（见原文第8页）：

Theorem 2. 假设初始数据 $(u_0,\xi_0,\omega_0)$ 满足

$$ (u_0,\xi_0,\omega_0) \in \big(V(\Omega) \cap \dot{W}^{1,2}(\Omega, |x|)\big) \times \mathbb{R}^3 \times \mathbb{R}^3, $$


且 $\gamma(u_0 - \xi_0 - \omega_0 \times x) = 0$，$\operatorname{curl} u_0 \in L^1(\Omega)$。则存在一个弱解 $(u,\xi,\omega)$ 满足：

(i) 能量不等式（Hopf 型）：对所有 $t > 0$，

$$ \|u(t)\|_2^2 + |\xi(t)|^2 + \omega(t)\cdot(I\cdot\omega(t)) + 4 \int_0^t \|D(u)(\tau)\|_2^2 \, d\tau \leq \|u_0\|_2^2 + |\xi_0|^2 + \omega_0 \cdot (I \cdot \omega_0); $$

(ii) 初值连续性：

$$ \lim_{t \to 0^+} \|u(t) - u_0\|_2 = \lim_{t \to 0^+} |\xi(t) - \xi_0| = \lim_{t \to 0^+} |\omega(t) - \omega_0| = 0; $$

(iii) 结构定理型正则性：存在有限时刻

$$ \theta \leq \exp\!\left( \frac{2A_0}{\chi} - 1 \right), \quad \text{其中 } A_0 := \|u_0\|_2^2 + |\xi_0|^2 + \omega_0 \cdot (I \cdot \omega_0), $$


使得当 $t > \theta$ 时，$(u,\xi,\omega)$ 成为一个正则解（regular solution），即存在压力 $\pi$ 满足 $(u,\pi,\xi,\omega)$ 几乎处处满足原方程组 (1.1)；且 $u \in L^\infty(0,\infty; V(\Omega)) \cap L^2(0,\infty; W^{2,2}(\Omega))$，$\xi,\omega \in C([0,\infty))$ 等。

该定理首次为刚体–流体耦合系统建立了类似 Leray 的Théorème de Structure：弱解在足够大的时间后自动正则化，从而局部唯一、光滑，并满足强意义下方程。

研究方法、关键技术和核心工具

a) 核心分析工具

本文采用Leray 的逼近–紧性纲领，但针对耦合系统的特殊结构进行了根本性改造。主要工具包括：

  • Friedrichs 磨光逼近：对非线性项 $(u - V)\cdot\nabla u$ 施加空间磨光算子 $J_n$，得到正则化系统 (3.1)，其解 $(u_n,\pi_n,\xi_n,\omega_n)$ 对每个 $n$ 存在全局（至少在长区间上）且唯一（依赖于初始数据的高阶正则性）。

  • 迭代能量估计框架：不同于 Navier–Stokes 的单一能量不等式，本文建立了一套分段迭代能量控制。定义序列 $T_h$ 与 $A_h$ 如下：

    $$ T_0 = (c A_0^2)^{-1}, \quad T_h = T_{h-1} + (c A_h^2)^{-1}, \quad A_h := \Big[ \|u_n(T_{h-1})\|_2^2 + |\xi_n(T_{h-1})|^2 + \omega_n(T_{h-1}) \cdot (I \cdot \omega_n(T_{h-1})) \Big]^{\frac{1}{2}}, $$

    并证明对每个 $h$，存在 $n(h)$ 使得在 $(T_{h-1}, T_h]$ 上有统一控制：

    $$ A_h^2 + 2 \int_{T_{h-1}}^{T_h} \|\nabla u_n\|_2^2 \, d\tau \leq A_{h-1}^2 \left(1 + \frac{1}{2^h}\right). $$

    此迭代机制克服了旋转项 $\omega \times x \cdot \nabla u$ 导致的经典能量恒等式失效问题。

  • 加权 Sobolev 紧性嵌入：利用 $|x|\nabla u_0 \in L^2(\Omega)$，结合 Gagliardo–Nirenberg 不等式，获得对 $u_n$ 的 $L^\infty$ 控制与衰减估计，支撑广义迹算子 $\phi_\nu[\cdot]$ 的良定义性。

  • 广义迹理论与分布意义边界条件：因 $\Omega$ 无界且 $u$ 仅属 $L^2$，无法定义经典迹。作者采用 $E(\Omega)$ 空间（$L^2$ + 散度 $L^2$）及其对偶 $H^{-1/2}(\partial\Omega)$ 上的分布迹 $\phi_\nu[u]$，使边界条件 $\phi_\nu[u] = \phi_\nu[V]$ 在弱意义下严格成立。

b) 技术性难题的处理技巧

  • 旋转对流项的处理：项 $\omega \times x \cdot \nabla u$ 在能量估计中产生无法控制的边界贡献 $\int_{\partial\Omega} J_n(u_n - V_n)\cdot\nu |V_n|^2 \, dS$。作者通过局部存在性时间下界 $T_0 \gtrsim A_0^{-2}$ 与磨光参数 $n$ 的选取依赖性（即对每个 $h$，选 $n \ge n_h$）将该项控制在 $\frac{A_0}{2^{h+1}}$ 量级，从而保证迭代收敛。

  • 弱解正则性提升的障碍突破：Navier–Stokes 中常用 $L^2$–$L^2$ 强收敛保证极限满足能量等式，但此处 $\{u_n\}$ 仅能获得 $L^2$–weak 收敛，$\{\nabla u_n\}$ 仅 $L^2$–weak。作者转而使用Caccioppoli 型局部估计与反向 Hölder 不等式，结合 $\operatorname{curl} u_0 \in L^1$ 提供的 vorticity 控制，在大时间后启动 De Giorgi 迭代，最终导出 $u \in L^\infty_{\mathrm{loc}}((\theta,\infty); W^{1,2}(\Omega))$。

  • 刚体运动与流体解的耦合协调：通过将 $\xi_n,\omega_n$ 视为 ODE 系统的解，并利用 $u_n$ 的加权估计控制其右端项，证明 $\{\xi_n\}, \{\omega_n\}$ 在 $L^2(0,T)$ 中有界且等度连续，从而提取一致收敛子列。

与之前工作的比较

a) 突破与改进

本文相较既有文献实现了三重实质性推进：

  • 首次建立刚体–流体弱解的结构定理：此前，Serre (1987)、Silvestre (2003)、Galdi–Silvestre (2005) 仅证明弱解存在性；Hishida (2005)、Geissert et al. (2006) 限于给定刚体运动情形；Galdi (2011) 获得小初值全局强解，但未触及弱解的长期正则性。本文首次证明：任意（满足加权正则性）初值对应的弱解，必在有限时间后进入正则轨道，即 $\exists \theta < \infty$ s.t. $u \in C^\infty((\theta,\infty)\times\Omega)$。

  • 显著放宽对初值的正则性要求：经典强解理论（如 Galdi [9]）要求 $u_0 \in V(\Omega) \cap W^{2,2}(\Omega)$；而本文仅需 $u_0 \in V(\Omega) \cap \dot{W}^{1,2}(\Omega,|x|)$ 及 $\operatorname{curl} u_0 \in L^1$。注意到 $\dot{W}^{1,2}(\Omega,|x|) \not\subset W^{1,2}(\Omega)$，故该空间允许更“延展”的初速度（如缓慢衰减涡旋），远超传统框架。

  • 揭示耦合系统特有的正则化机制：Leray 定理中 $\theta \lesssim \|u_0\|_2^4$，而本文 $\theta \lesssim \exp(c A_0/\chi)$，体现刚体惯性（$I$）与流体黏性（$\mu=1$）共同驱动的指数型正则化延迟，这是纯流体系统所不具备的新现象。同时，作者明确指出：区间 $(\theta_0,\theta)$ 上的正则性仍是开放问题，凸显刚体–流体耦合比 Navier–Stokes 更深层的奇异性结构。

综上，本文不仅填补了 FSI 领域长期缺失的结构定理空白，更发展出一套适配非惯性系、非齐次运动边界的新型分析范式，为后续研究弹性体–流体、多体系统等更复杂耦合模型提供了关键方法论基础。

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作者: Tobias Black, Michael Winkler

类别: math.AP

发布日期: 2026-02-12

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链接: http://arxiv.org/abs/2602.12171v1

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链接: http://arxiv.org/abs/2602.12166v1

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发布日期: 2026-02-12

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链接: http://arxiv.org/abs/2602.12163v1

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链接: http://arxiv.org/abs/2602.12122v1

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发布日期: 2026-02-12

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链接: http://arxiv.org/abs/2602.12061v1

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类别: math.AP

发布日期: 2026-02-12

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链接: http://arxiv.org/abs/2602.12046v1

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类别: math.AP

发布日期: 2026-02-12

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链接: http://arxiv.org/abs/2602.12033v1

摘要: We establish the local Lipschitz regularity of the local minimizers of non autonomous integral funtionals of the form

$$ \int_ΩF(x, Dz)\,dx, $$

where $Ω$ is a bounded open set of $\mathbb{R}^n$, $n \ge 2$. The energy density $F(x,ξ)$ satisfies $(p,q)-$growth conditions with respect to the gradient variable and belongs to the Sobolev class $W^{1,φ}$, with $φ(t)=t^r\log^α(e+t),$ $r\ge n$, $α\ge 0$, as a function of the $x$ variable, under the condition

$$ 1\le\frac{q}{p} \le 1 + \frac{1}{n} - \frac{1}{r}. $$

We present a unified approach that covers the limit case

$$ \frac{q}{p} = 1 + \frac{1}{n} - \frac{1}{r} $$

and retrieves the results in \cite{EMM16} and in \cite{CGHPdN20}.

9. 基于无网格数值方法求解带势函数拉普拉斯算子的狄利克雷特征对

Mesh-free numerical method for Dirichlet eigenpairs of the Laplacian with potential

作者: Dragoş Manea

类别: math.NA, math.AP

发布日期: 2026-02-12

arXiv ID: 2602.12008v1

链接: http://arxiv.org/abs/2602.12008v1

摘要: This paper is concerned with the numerical approximation of the $L^2$ Dirichlet eigenpairs of the operator $-Δ+ V$ on a simply connected $C^2$ bounded domain $Ω\subset \mathbb{R}^2$ containing the origin, where $V$ is a radial potential. We propose a mesh-free method inspired by the Method of Particular Solutions for the Laplacian (i.e. $V=0$). Extending this approach to general $C^1$ radial potentials is challenging due to the lack of explicit basis functions analogous to Bessel functions. To overcome this difficulty, we consider the equation $-Δu + V u = λu$ on a ball containing $Ω$, without imposing boundary conditions, for a collection of values $λ$ forming a fine discretisation of the interval in which eigenvalues are sought. By rewriting the problem in polar coordinates and applying a Fourier expansion with respect to the angular variable, we obtain a decoupled system of ordinary differential equations. These equations are solved numerically using a one-dimensional Finite Element Method, yielding a family of basis functions that are solutions of the equation $-Δu + V u = λu$ on the ball and are independent of the domain $Ω$. Dirichlet eigenvalues of $-Δ+ V$ are then approximated by minimising the boundary values on $\partial Ω$ among linear combinations of the basis functions and identifying those values of $λ$ for which the computed minimum is sufficiently small. The proposed method is highly memory-efficient compared to the standard Finite Element approach.

10. Neumann正则光滑函数在亚解析流形Sobolev空间中的稠密性

Density of Neumann regular smooth functions in Sobolev spaces of subanalytic manifolds

作者: Guillaume Valette

类别: math.AP

发布日期: 2026-02-12

arXiv ID: 2602.12007v1

链接: http://arxiv.org/abs/2602.12007v1

摘要: We give characterizations of the bounded subanalytic $\mathscr{C}^\infty$ submanifolds $M$ of $\mathbb{R}^n$ for which the space of Neumann regular functions is dense in Sobolev spaces. By ``Neumann regular function’’, we mean a function which is smooth at almost every boundary point and whose gradient is tangent to the boundary. In the case $p\in [1,2]$, we prove that the Neumann regular elements of $\mathscr{C}^\infty(\overline{M})$ are dense in $W^{1,p}(M)$ if and only if $M$ is connected at almost every boundary point. In the case $p$ large, we show that the Neumann regular Lipschitz elements of $\mathscr{C}^\infty(M)$ are dense in $W^{1,p}(M)$ if and only if $M$ is connected at every boundary point. The proof involves the construction of Lipschitz Neumann regular partitions of unity, which is of independent interest.

11. 均曲率方程稳定解的内正则性与刘维尔型定理

Inner regularity and Liouville theorems for stable solutions to the mean curvature equation

作者: Fanheng Xu

类别: math.AP

发布日期: 2026-02-12

arXiv ID: 2602.12001v1

链接: http://arxiv.org/abs/2602.12001v1

摘要: Let $f\in C^1(\mathbb{R})$. We study stable solutions $u$ of the mean curvature equation

$$ \operatorname{div}\left( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}} \right) = -f(u) \qquad \text{in}\ Ω\subset \mathbb{R}^n. $$

In the local setting we prove that $\nabla u$ satisfies inner Morrey regularity $M^{p_n}$, where

$$ p_n := \left\{ \begin{array}{ll} n,\qquad & \text{if}\ 2\leq n\leq 5, \\ \frac{n}{n-4\sqrt{n-1}+4},\qquad & \text{if}\ n\geq 6, \end{array} \right. $$

together with the estimate

$$ \|\nabla u\|_{M^{p_n}(B_1)} \leq C \left( 1+\|\nabla u\|_{L^1(B_2)} \right). $$

The exponent $p_n$ is optimal for $n\leq5$, as shown by an explicit one-dimensional example. For radial solutions we show that the symmetry center is at most a removable singularity. Globally, we establish Liouville-type theorem: any stable solution satisfying the growth condition

$$ |\nabla u(x)| = \left\{ \begin{array}{lll} o(|x|^{-1}) \ & \text{as}\ |x|\rightarrow +\infty& \text{when}\ 2\leq n\leq 10, \\ o(|x|^{-n/2+\sqrt{n-1}+1}) \ & \text{as}\ |x|\rightarrow +\infty& \text{when}\ n\geq 11, \end{array} \right. $$

must be constant. In particular, no nonconstant radial stable solution exists in dimensions $2\leq n\leq6$, which highlights a global rigidity of stable radial solutions in low dimensions and extend the classical Liouville theorem of Farina and Navarro. Several exponents appearing in our results are new for mean curvature equations, showing both similarities and differences with the corresponding theorems for semilinear equations.

12. 平均场朗之万动力学的局部收敛性：从梯度流到线性单调博弈

Local convergence of mean-field Langevin dynamics: from gradient flows to linearly monotone games

作者: Guillaume Wang, Lénaïc Chizat

类别: math.OC, math.AP

发布日期: 2026-02-12

arXiv ID: 2602.11999v1

链接: http://arxiv.org/abs/2602.11999v1

摘要: We study the local convergence of diffusive mean-field systems, including Wasserstein gradient flows, min-max dynamics, and multi-species games. We establish exponential local convergence in $χ^2$-divergence with sharp rates, under two main assumptions: (i) the stationary measures satisfy a Poincaré inequality, and (ii) the velocity field satisfies a monotonicity condition, which reduces to linear convexity of the objective in the gradient flow case. We do not assume any form of displacement convexity or displacement monotonicity. In the gradient flow case, global exponential convergence is already known under our linear convexity assumption, with an asymptotic rate governed by the log-Sobolev constant of the stationary measure. Our contribution in this setting is to identify the sharp rate near equilibrium governed instead by the Poincaré constant. This rate coincides with the one suggested by Otto calculus (i.e. by a tight positivity estimate of the Wasserstein Hessian), and refines some results of Tamura (1984), extending them beyond quadratic objectives. More importantly, our proof technique extends to certain non-gradient systems, such as linearly monotone two-player and multi-player games. In this case, we obtain explicit local exponential convergence rates in $χ^2$-divergence, thereby partially answering the open question raised by the authors at COLT 2024. While that question concerns global convergence (which remains open), even local convergence results were previously unavailable. At the heart of our analysis is the design of a Lyapunov functional that mixes the $χ^2$-divergence with weighted negative Sobolev norms of the density relative to equilibrium.

13. 非线性假设下平均曲率方程的改进内梯度估计

Improved Interior Gradient Estimates for the Mean Curvature Equation under Nonlinear Assumptions

作者: Fanheng Xu

类别: math.AP

发布日期: 2026-02-12

arXiv ID: 2602.11991v1

链接: http://arxiv.org/abs/2602.11991v1

摘要: In this paper, we investigate interior gradient estimates for solutions to the mean curvature equation

$$ \dive \left( \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} \right) = f(\nabla u) $$

under various nonlinear assumptions on the right-hand side. Under the weakened initial assumption $u\in C^1(B_R) \cap C^3(\{|\nabla u|>0\})$, we establish sharp gradient bounds that depend on the oscillation of the solution. These estimates are applicable to a wide class of nonlinear terms, including the specific forms arising from the elliptic regularization of the inverse mean curvature flow ($f=\varepsilon\sqrt{1+|\nabla u|^2}$ ), minimal surface equation ($f=0$) and several polynomial and logarithmic growth regimes. As applications, the gradient bounds imply uniform ellipticity of the equation away from the critical set,which allows one to apply classical elliptic regularity theory and obtain higher regularity of solutions in the noncritical region. Moreover, when the solution grows at most linearly, all cases of our results can be applied in Moser’s theory to establish the affine linear rigidity of global solutions. This directly leads to the Liouville-type theorems for global solutions without requiring additional proofs.

14. 从半线性椭圆方程的诺伊曼–狄利克雷映射重构各向异性电导率

Recovery of an Anisotropic Conductivity from the Neumann-to-Dirichlet Map in a Semilinear Elliptic Equation

作者: Elena Beretta, Elisa Francini, Dario Pierotti, Eva Sincich

类别: math.AP

发布日期: 2026-02-12

arXiv ID: 2602.11987v1

链接: http://arxiv.org/abs/2602.11987v1

摘要: We study the inverse boundary value problem of detecting a non-uniform conductivity motivated by pacing-guided ablation in cardiac electrophysiology. At the stationary level, the transmembrane potential $u$ in a region $Ω\subset\mathbb{R}^3$ of cardiac tissue satisfies

$$ -\nabla\!\cdot(γ\nabla u)+αu^3=0 \quad \text{in }Ω,\qquad γ\nabla u\cdotν=g \quad \text{on }\partialΩ, $$

where $γ$ is an anisotropic conductivity tensor and $α$ a nonlinear ionic response coefficient. The Neumann data $g$ represent pacing currents, and the boundary values $u|_{\partialΩ}$ correspond to invasive voltage measurements. Ischemic regions are modeled by a subdomain $D\subsetΩ$ where $γ$ is piecewise constant. We address the inverse problem of determining $γ$ from the Neumann-to-Dirichlet (NtD) map, assuming that $α$ and $D$ are known. To our knowledge, uniqueness in the case of NtD data with anisotropic conductivities in this nonlinear setting has not been analyzed in previous work. Using a first-order linearization around a nontrivial pacing current, we prove uniqueness for $γ$.

15. 脆性材料中准静态裂纹扩展的本征断裂近似方法

Eigenfracture approximation of quasi-static crack growth in brittle materials

作者: Ba Duc Duong, Manuel Friedrich

类别: math.AP

发布日期: 2026-02-12

arXiv ID: 2602.11915v1

链接: http://arxiv.org/abs/2602.11915v1

摘要: We study an approximation scheme for a variational theory of quasi-static crack growth based on an eigendeformation approach. We consider a family of energy functionals depending on a small parameter $\varepsilon$ and on two fields, the displacement field and an eigendeformation field that approximates the crack in the material. By imposing a suitable irreversibility condition and adopting an incremental minimization scheme, we define a notion of quasi-static evolution for this model. We then show that, as $\varepsilon \to 0$, these evolutions converge to a quasi-static crack evolution for the Griffith energy of brittle fracture, characterized by irreversibility, global stability, and an energy balance.

16. 由混合局部-非局部算子驱动的奇异问题的整体多重性与比较原理

Global Multiplicity and Comparison Principles for Singular Problems driven by Mixed Local-Nonlocal Operators

作者: R. Dhanya, Sarbani Pramanik

类别: math.AP

发布日期: 2026-02-12

arXiv ID: 2602.11889v1

链接: http://arxiv.org/abs/2602.11889v1

摘要: We study a singular elliptic problem driven by a mixed local-nonlocal operator of the form \begin{equation*} \begin{aligned} -Δ_p u + (-Δ_q)^s u &= \fracλ{u^δ} + u^r \text{ in } Ω\newline u > 0 \text{ in } Ω,\ u &= 0 \text{ in } \mathbb{R}^N \setminus Ω \end{aligned} \end{equation*} where $p > sq$, $0<δ<1$ and $λ> 0$ is a parameter. The nonlinearity exhibits a singular power-type behavior near zero and displays at most a critical growth at infinity. We establish a global multiplicity result with respect to the parameter $λ$ by identifying a sharp threshold that separates existence, non-existence, and multiplicity regimes, a result that is new for singular problems involving mixed local-nonlocal operators. We also derive a Hopf-type strong comparison principle adapted to this nonlinear setting, which provides the main analytical tool for the global multiplicity result. Additionally, we investigate qualitative properties of solutions that are essential for the variational analysis, such as a uniform $L^{\infty}$-estimate and a Sobolev versus Hölder local minimizer result. The analytical tools developed herein are of independent mathematical interest, with their applicability extending over a broader class of mixed local-nonlocal problems.

17. 一种用于半线性平面椭圆系统先验估计的爆破方法：Brezis-Merle临界情形

A blow-up approach for a priori bounds in semilinear planar elliptic systems: the Brezis-Merle critical case

作者: Laura Baldelli, Gabriele Mancini, Giulio Romani

类别: math.AP

发布日期: 2026-02-12

arXiv ID: 2602.11720v1

链接: http://arxiv.org/abs/2602.11720v1

摘要: We establish uniform a priori estimates for solutions of semilinear planar Hamiltonian elliptic systems in a ball with Dirichlet boundary conditions. We consider a broad class of coupled nonlinearities with asymptotic critical behaviour in the sense of Brezis–Merle. The approach we follow is based on a blow-up analysis combined with Liouville–type theorems and integral estimates. Our results extend the scalar theory of uniform a priori bounds to the Hamiltonian case, and solve an open problem in [de Figueiredo D.G., do Ó J.M., Ruf B., Adv. Nonlinear Stud. 6 (2006), no. 2]. We believe that this approach is new in this setting. As a consequence of our a priori estimates, we prove the existence of a positive solution by means of Fixed Point Index theory.

18. 矩阵值Allen-Cahn方程的锐界面极限

The sharp interface limit of the matrix-valued Allen-Cahn equation

作者: Xingyu Wang

类别: math.AP

发布日期: 2026-02-12

arXiv ID: 2602.11485v1

链接: http://arxiv.org/abs/2602.11485v1

摘要: In this work, we study a matrix-valued Allen-Cahn equation with a Saint Venant-Kirchhoff potential $F(\mathbf{A})=\frac{1}{4}\|\mathbf{A}\mathbf{A}^\top-\mathbf{I}\|^2$. Our approach employs the modulated energy method together with weak convergence methods for nonlinear partial differential equations. This avoids the subtle spectrum analysis of the linearized operator at the so-called quasi-minimal orbits as well as the construction of asymptotic expansion. Moreover, it relaxes the assumption on the admissible initial data, which exhibits a phase transition along an initial interface. As a byproduct, we construct a weak solution to the limiting harmonic heat flow system with both minimal pair and Neumann-type boundary conditions across the interface.

19. 图结构对随机偏微分方程强Feller性质影响的量化研究

Quantifying the effect of graph structure on strong Feller property of SPDEs

作者: Jianbo Cui, Tonghe Dang, Jialin Hong, Zhengkai Wang

类别: math.PR, math.AP, math.NA

发布日期: 2026-02-12

arXiv ID: 2602.11484v1

链接: http://arxiv.org/abs/2602.11484v1

摘要: This paper investigates how the structure of the underlying graph influences the behavior of stochastic partial differential equations (SPDEs) on finite tree graphs, where each edge is driven by space-time white noise. We first introduce a novel graph-based null decomposition approach to analyzing the strong Feller property of the Markov semigroup generated by SPDEs on tree graphs. By examining the positions of zero entries in eigenfunctions of the graph Laplacian operator, we establish a sharp upper bound on the number of noise-free edges that ensures both the strong Feller property and irreducibility. Interestingly, we find that the addition of noise to any single edge is sufficient for chain graphs, whereas for star graphs, at most one edge can remain noise-free without compromising the system’s properties. Furthermore, under a dissipative condition, we prove the existence and exponential ergodicity of a unique invariant measure.

20. 具有测度值右端项的谱分数阶拉普拉斯算子：分析与数值逼近

The spectral fractional Laplacian with measure valued right hand sides: analysis and approximation

作者: Enrique Otarola, Abner J. Salgado

类别: math.NA, math.AP

发布日期: 2026-02-11

arXiv ID: 2602.11423v1

链接: http://arxiv.org/abs/2602.11423v1

摘要: We consider the spectral definition of the fractional Laplace operator and study a basic linear problem involving this operator and singular forcing. In two dimensions, we introduce an appropriate weak formulation in fractional Sobolev spaces and prove that it is well-posed. As an application of these results, we analyze a pointwise tracking optimal control problem for fractional diffusion. We also develop a finite element scheme for the linear problem using continuous, piecewise linear functions, prove a convergence result in energy norm, and derive an error bound in $L^2(Ω)$. Finally, we propose a practical scheme based on a diagonalization technique and derive an error bound in $L^2(Ω)$ using a regularization argument.

21. 铁磁薄膜中时间周期性振荡的奈尔畴壁

Time-periodic oscillating Néel walls in ferromagnetic thin films

作者: Antonio Capella, Valentin Linse, Christof Melcher, Lauro Morales, Ramón G. Plaza

类别: math.AP

发布日期: 2026-02-11

arXiv ID: 2602.11420v1

链接: http://arxiv.org/abs/2602.11420v1

摘要: This paper studies the existence, the structure and the spectral stability of time-periodic oscillating 180-degree Néel walls in ferromagnetic thin films. It is proved that time-periodic coherent structures do exist as solutions to the reduced model for the in-plane magnetization proposed by Capella, Melcher, and Otto (Nonlinearity 20 (2007), no. 11, 2519–2537) when a weak and $T$-periodic external magnetic field is applied in the direction of the easy axes of the film, perturbing in this fashion the well-known static 180-degree Néel wall. The linearization around this time-periodic Néel wall is constituted by a family of linear operators, parametrized by the time variable, which generates an evolution system of generators (or propagator) for the linear problem. Profiting from the stability of the static Néel wall, it is shown that the Floquet spectrum of the monodromy map for the propagator is contained in the complex unit circle, proving stability of the oscillating solution at least at a linear level.

22. 具有 $(n-1)$ 维伸缩变换群不变性的爱因斯坦–麦克斯韦方程的解

Solution for the Einstein-Maxwell equations invariant under an $(n - 1)$-dimensional group of dilations

作者: Benedito Leandro, Ilton Menezes, Rafael Novais

类别: math.DG, math-ph, math.AP

发布日期: 2026-02-11

arXiv ID: 2602.11115v1

链接: http://arxiv.org/abs/2602.11115v1

摘要: We consider an electrostatic system whose spatial factor is conformal to an $n$-dimensional Euclidean space. We provide a complete characterization of the most general ansatz, thereby reducing the associated electrostatic system of partial differential equations to an ordinary differential equation system. We prove that there are only two possibilities: either the cosmological constant is nonzero, in which case the solutions are necessarily invariant under rotations or translations, or the cosmological constant vanishes, and the solutions belong to the Majumdar-Papapetrou class with a degree of freedom associated with an invariant $(n-1)$-dimensional subgroup. As a result, we introduce a new solution to the electrovacuum system in the Majumdar-Papapetrou class that is invariant under an $(n-1)$-dimensional group of dilations.