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1. Inertial instability of Couette flow with Coriolis force

科里奥利力作用下库埃特流的惯性不稳定性

作者: Yanlong Fan, Daozhi Han, Quan Wang

类别: math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.01602v1

链接: http://arxiv.org/abs/2510.01602v1

详细分析

中文标题: 科里奥利力作用下库埃特流的惯性不稳定性

1. 研究对象和背景

本文研究的是在三维全空间 $\mathbb{R}^3$ 中，受科里奥利力作用的不可压缩纳维-斯托克斯方程所描述的库埃特流（Couette flow）的非线性惯性不稳定性。系统的基本方程为：

$$ \begin{cases} \partial_t v + v \cdot \nabla v + f \times v = \nu \Delta v - \nabla p, \\ \nabla \cdot v = 0, \end{cases} $$

其中 $v = (v_1, v_2, v_3)$ 是速度场，$p$ 是压力，$f = (0,0,f)$ 是科里奥利向量，$f > 0$ 称为科里奥利系数，$\nu > 0$ 为粘性系数。

考虑的稳态解是库埃特流：

$$ v = (y, 0, 0), \quad p = p(y), $$

满足地转平衡关系：$f y = -\partial_y p$。

引入扰动 $u = v - (y,0,0)$ 和 $q = p - p(y)$，得到扰动方程组：

$$ \begin{cases} \partial_t u + u \cdot \nabla u + \begin{pmatrix} -u_2 \\ 0 \\ 0 \end{pmatrix} + y \partial_x u + f \begin{pmatrix} -u_2 \\ u_1 \\ 0 \end{pmatrix} = \nu \Delta u - \nabla q, \\ \nabla \cdot u = 0, \end{cases} $$

其中压力项分解为线性部分 $q_L$ 和非线性部分 $q_{NL}$：

$$ q_L = -\Delta^{-1}(2\partial_x u_2 + f(\partial_y u_1 - \partial_x u_2)), \quad q_{NL} = -\Delta^{-1}\mathrm{div}(u \cdot \nabla u). $$

初始条件为 $u(0,x) = u_{in}(x) \in [S(\mathbb{R}^3)]^3$，远场条件为 $\lim_{|x|\to\infty}|u(t,x)| = 0$ 对所有 $t \geq 0$ 成立。函数空间设置在 Sobolev 空间 $H^k(\mathbb{R}^3)$ 上，特别是 $L^2$ 和 $H^1$ 范数被用于衡量不稳定性的增长。

值得注意的是，问题定义在整个 $\mathbb{R}^3$ 空间上，无边界条件，仅要求解在无穷远处趋于零。这与周期域或有界区域中的研究形成对比，增加了谱分析的技术难度。

2. 主要定理或主要结果

定理 1.1（线性不稳定性）：设 $0 < f < 1$，则对任意给定时间 $T > 0$ 和 $\epsilon > 0$，线性化系统 (1.10) 在初边值条件 (1.8)-(1.9) 下存在实值解 $u_{T,\epsilon}(t) \in C([0,T];[H^k(\mathbb{R}^3)]^3)$ 且 $\nabla q_{T,\epsilon}(t) \in C([0,T];[H^k(\mathbb{R}^3)]^3)$，满足以下两个性质：
（i）对于任意 $t \in [0,T]$，有

$$ (e^{t\Lambda'} - \epsilon)\|u_{T,\epsilon}(0)\|_{L^2} \leq \|u_{T,\epsilon}(t)\|_{L^2} \leq (e^{t\Lambda'} + \epsilon)\|u_{T,\epsilon}(0)\|_{L^2}; $$


（ii）对任意正整数 $k$，存在常数 $C_k > 0$（独立于 $T$ 和 $\epsilon$），使得

$$ \|u_{T,\epsilon}(0)\|_{H^k} \leq C_k \|u_{T,\epsilon}(0)\|_{L^2}. $$


其中 $\Lambda' = \sqrt{f(1-f)}$ 表示伪特征函数的增长率上界。

定理 1.2（非线性不稳定性）：当科里奥利系数 $f$ 满足

$$ \frac{2}{17}\left(5 - 2\sqrt{2}\right) < f < \frac{2}{17}\left(5 + 2\sqrt{2}\right) $$


且 $\nu > 0$ 时，库埃特流在哈达玛意义下是非线性不稳定的。即存在常数 $\delta_0 \leq 1$、$\varepsilon_0 > 0$ 及光滑初值 $u_{in} \in [S(\mathbb{R}^3)]^3$，使得对任意小的 $\delta \in (0, \delta_0)$，扰动初值 $u_\delta(0) = \delta u_{in}$ 所对应的强解 $u_\delta \in C([0,T_{\max}); H^1(\mathbb{R}^3)) \cap L^2(0,T_{\max}; H^2(\mathbb{R}^3))$ 存在一个逃逸时间 $T_\delta = \frac{1}{\Lambda}\ln\frac{2\varepsilon_0}{\delta} < T_{\max}$，满足

$$ \|u_\delta(T_\delta)\|_{L^2} > \varepsilon_0, $$


其中 $\Lambda = \frac{2-f}{2}$ 是线性算子半群的增长率上界。

3. 研究方法、关键技术和核心工具

a) 核心分析学工具

  • 半群理论：作者利用 Hille-Yosida 定理从预解估计出发，证明线性算子 $L$ 在 $[L^2(\mathbb{R}^3)]^3$ 上生成一个 $C_0$-半群，并给出其增长上界 $\|e^{tL}\|_{L^2} \leq e^{t\Lambda}$，其中 $\Lambda = \frac{2-f}{2}$。
  • 谱分析与伪谱理论：由于线性算子 $L$ 是非正规的（non-normal），传统的特征值方法失效。作者采用伪谱（pseudo-spectrum）分析，借助定义 2.3 中的 $\varepsilon$-伪特征值和伪特征函数概念，构造出具有近似指数增长行为的“伪本征解”。
  • 能量估计与引导论证（bootstrap argument）：为了克服非线性项的影响，作者通过精细的能量估计控制非线性发展，结合引导法将线性不稳定性提升至非线性层面。附录中提供了非线性能量估计的具体细节。
  • Fourier 分析与反演技术：文中虽未显式使用 Fourier 变换求解，但在处理 $\Delta^{-1}$ 这类非局部算子时，隐含依赖于调和分析工具，如 Riesz 变换和椭圆正则性。

b) 解决技术性难题的主要技巧

  • 伪特征函数构造法：针对线性算子无点谱（point spectrum）的问题（即不存在真正的指数增长特征函数），作者构造了所谓的“伪特征函数”，这些函数在有限时间区间内近似满足指数增长，但不具备理想的渐进行为。这是应对非正规算子导致连续谱主导的关键策略。
  • 时间依赖的误差控制：由于伪特征函数随时间积累误差，不能长期保持指数增长形态，因此作者必须精确控制近似解在固定时间 $T$ 内的行为，并建立 $T$ 与初始扰动大小之间的定量关系，从而确保逃逸时间内的增长足以压倒非线性效应。
  • 分裂算子结构分析：将线性算子 $L$ 分解为 $L = L_0 + L_1 + L_2$，分别对应于基本项、有界项和破坏正规性的 $\partial_x$ 高阶项。这种分解有助于识别非正规来源并逐项估计。

4. 与之前工作的比较

a) 突破与改进

本文的主要突破在于首次在 $\mathbb{R}^3$ 全空间中严格证明了科里奥利力引发的库埃特流的非线性惯性不稳定性，填补了现有文献中的空白。

  • 解决了 $R(f) < 0$ 情形下的非线性不稳定性问题：尽管已有工作指出当 Rayleigh 判据 $R(f) = f(f-1) < 0$（即 $0 < f < 1$）时可能发生惯性不稳定性，但在数学上尚未对此情形建立严格的非线性不稳定性结果。本文成功在此条件下构造了解的增长机制。
  • 推广到非周期、无界区域：以往多数关于旋转流稳定性的研究集中在周期域 $T \times R \times T$ 或有界区域（如 [38,40,44,45,47]），而本文在 $\mathbb{R}^3$ 上进行分析，避免了周期性假设带来的简化，更具物理普适性。
  • 无需变分结构或可分离变量解：传统方法常依赖于系统的变分结构或可分离变量解来寻找不稳定模态。然而，该系统既无自然变分结构，也不支持 $L^2$ 中的可分离解。本文绕开此限制，改用伪谱方法和扰动构造法，拓展了处理复杂流动不稳定性的工具箱。
  • 明确揭示非正规性影响：文章清晰指出线性算子的非正规性导致仅有连续谱、无点谱，从而迫使采用伪特征函数方法。这一洞察深化了对剪切流中瞬态增长与非线性失稳之间关系的理解。

综上所述，本文不仅在结果上推进了对惯性不稳定的数学理解，更在方法论上展示了如何处理缺乏经典谱结构的非正规算子系统的非线性不稳定性问题，为后续研究提供了范式。

2. A viscosity solution approach to the large deviation principle for

stochastic convective Brinkman-Forchheimer equations

基于粘性解方法的随机对流Brinkman-Forchheimer方程大偏差原理研究

作者: Sagar Gautam, Manil T. Mohan

类别: math.PR, math.AP

发布日期: 2025-10-01

ArXiv ID: 2510.01166v1

链接: http://arxiv.org/abs/2510.01166v1

详细分析

中文标题: 基于粘性解方法的随机对流Brinkman-Forchheimer方程大偏差原理研究

1. 研究对象和背景

本文研究的对象是定义在环面 $\mathbb{T}^d$（其中 $d \in \{2,3\}$）上的二维和三维随机对流Brinkman-Forchheimer方程（Stochastic Convective Brinkman-Forchheimer, SCBF），该方程描述了多孔介质中不可压缩流体在高流速下的运动。其数学形式如下：

$$ \begin{cases} \mathrm{d}\boldsymbol{u}_n + \left[ -\mu\Delta\boldsymbol{u}_n + (\boldsymbol{u}_n \cdot \nabla)\boldsymbol{u}_n + \alpha\boldsymbol{u}_n + \beta|\boldsymbol{u}_n|^{r-1}\boldsymbol{u}_n + \nabla p_n \right] \mathrm{d}t = \boldsymbol{f} \mathrm{d}t + \frac{1}{\sqrt{n}} \mathrm{Q}^{\frac{1}{2}} \mathrm{d}\mathrm{W}, \\ \nabla \cdot \boldsymbol{u}_n = 0, \end{cases} $$

其中：

  • $\mu > 0$ 是有效粘性系数（Brinkman系数），
  • $\alpha > 0$, $\beta > 0$ 分别为Darcy和Forchheimer阻力系数，
  • $r \in [1,\infty)$ 为吸收指数，当 $r=3$ 时称为临界情形；
  • $\mathrm{Q}$ 是迹类算子，$\mathrm{W}$ 是Hilbert空间取值的柱Wiener过程；
  • 小噪声强度由因子 $\frac{1}{\sqrt{n}}$ 控制，对应大偏差框架中的小扰动极限 $n \to \infty$。

该系统可视为带阻尼项 $\alpha \boldsymbol{u}_n + \beta |\boldsymbol{u}_n|^{r-1}\boldsymbol{u}_n$ 的Navier-Stokes方程的推广。特别地，当 $\alpha = \beta = 0$ 时退化为经典的随机Navier-Stokes方程。

文中采用抽象投影形式进行分析：设 $Y_n = P \boldsymbol{u}_n$，其中 $P$ 为Helmholtz-Hodge投影，则问题转化为Hilbert空间 $H$ 中的随机演化方程：

$$ \begin{cases} \mathrm{d}Y_n(t) = \left[ -\mu A Y_n(t) - B(Y_n(t)) - \alpha Y_n(t) - \beta C(Y_n(t)) + f(t) \right] \mathrm{d}t + \frac{1}{\sqrt{n}} Q^{1/2} \mathrm{d}W(t), \\ Y_n(0) = y \in H, \end{cases} $$

其中：

  • $A = -\Delta$ 为Stokes算子，
  • $B(u) = P[(u \cdot \nabla)u]$ 为非线性对流项，
  • $C(u) = P(|u|^{r-1}u)$ 为吸收非线性项。

函数空间与区域特性

研究区域为周期环面 $\mathbb{T}^d$（无边界），这使得以下关键恒等式成立：

$$ \int_{\mathbb{T}^d} (-\Delta u) \cdot |u|^{r-1} u \, d\xi = \int_{\mathbb{T}^d} |\nabla u|^2 |u|^{r-1} \, d\xi + \frac{r-1}{4} \int_{\mathbb{T}^d} |u|^{r-3} |\nabla |u|^2|^2 \, d\xi, $$

此恒等式在处理高维情形（尤其是 $d=3$）时至关重要，因为它允许利用吸收项控制对流项的能量增长。此外，由于吸收项不保持零均值性质，作者未假设速度场具有零平均，因此不能使用Poincaré不等式，必须使用完整的 $H^1$ 范数。

主要函数空间包括：

  • $H$: $L^2(\mathbb{T}^d)$ 中的无散向量场闭包，配备标准内积 $(\cdot,\cdot)$ 和范数 $\|\cdot\|_H$；
  • $V$: $H^1(\mathbb{T}^d)$ 中的无散向量场闭包，配备范数 $\|u\|_V^2 = \|u\|_H^2 + \|\nabla u\|_H^2$；
  • $L^{\tilde{p}}$: $L^p(\mathbb{T}^d)$ 中的无散向量场闭包。

嵌入关系 $V \hookrightarrow H$ 是紧的，这对建立紧性和收敛性至关重要。

2. 主要定理或主要结果

定理 1.2（连续路径空间中的大偏差原理）
假设 $\mathrm{Tr}(Q) < \infty$，且外力项 $f : [0,T] \to V$ 有界连续。则对于 $d \in \{2,3\}$ 维情形，当 $r > 3$ 或 $r = 3$ 且满足 $2\beta\mu \geq 1$ 时，随机过程族 $\{Y_n(\cdot)\}_{n \geq 1}$（其中 $Y_n(\cdot) = Y_n(\cdot; 0, y)$ 是SCBF系统 (1.4) 以初值 $Y_n(0) = y$ 的解）在连续路径空间 $C([0,T]; H)$ 上满足大偏差原理，速率函数为 $I(\cdot)$，其表达式见文中的 (3.2)。具体而言：

  • 对任意闭集 $F \subset C([0,T]; H)$，有

    $$ \limsup_{n \to \infty} \frac{1}{n} \log \mathbb{P}\{ \omega \in \Omega : Y_n(\cdot, \omega) \in F \} \leq -\inf_{\xi \in F} I(\xi); $$

  • 对任意开集 $G \subset C([0,T]; H)$，有

    $$ \liminf_{n \to \infty} \frac{1}{n} \log \mathbb{P}\{ \omega \in \Omega : Y_n(\cdot, \omega) \in G \} \geq -\inf_{\xi \in G} I(\xi). $$

该定理表明，在小噪声极限下，系统偏离确定性行为的概率呈指数衰减，且衰减速率由泛函 $I(\cdot)$ 刻画。这是全文的核心结论。

3. 研究方法、关键技术和核心工具

a) 核心分析学工具

  • 粘性解理论（Viscosity Solution Theory）：文章将大偏差问题转化为一个无穷维Hamilton-Jacobi-Bellman（HJB）方程的粘性解收敛问题。通过证明扰动后的二阶HJB方程的粘性解 $U_n$ 在 $n \to \infty$ 下收敛到一阶HJB方程的粘性解 $U$，从而识别出Laplace极限。

  • 比较原理（Comparison Principle）：用于证明HJB方程粘性解的唯一性，并作为收敛性的基础。文中建立了适用于SCBF系统的局部Lipschitz连续哈密顿结构下的比较定理（Theorem 5.5 和 5.7）。

  • Feng-Kurtz 大偏差框架：基于文献 [Feng & Kurtz, 2006] 提出的路径空间大偏差系统方法，结合Varadhan-Bryc准则，将LDP的证明分解为两个部分：Laplace极限的存在性和指数紧性。

  • 半群与生成元方法：通过Itô公式形式推导出 $U_n(t,y)$ 满足的HJB方程：

    $$ (U_n)_t + \frac{1}{2n} \mathrm{Tr}(Q D^2 U_n) - \frac{1}{2} \|Q^{1/2} D U_n\|_H^2 + \langle -\mu A y - B(y) - \alpha y - \beta C(y) + f(t), D U_n \rangle = 0, $$

    并研究其 $n \to \infty$ 的奇摄动极限。

  • 指数矩估计与指数紧性：通过能量估计获得解的一致指数矩界：

    $$ \sup_{n \in \mathbb{N}} \mathbb{E} \left[ \exp\left( \delta \int_0^T \|Y_n(s)\|_V^2 \, ds \right) \right] < \infty, $$

    这对证明“C-指数紧性”至关重要，进而实现从Skorohod空间 $D([0,T]; H)$ 到连续空间 $C([0,T]; H)$ 的LDP提升。

b) 解决技术难题的关键技巧

  • 避免正交性条件 $(B(u), Au) = 0$：在三维情形 $\mathbb{T}^3$ 中，经典Navier-Stokes分析常用的恒等式 $(B(u), Au) = 0$ 不再成立，导致难以控制非线性项。本文借助吸收项 $C(u)$ 的强耗散性质，在 $r > 3$ 或 $r=3$ 且 $2\beta\mu \geq 1$ 条件下，利用恒等式 (2.11) 和Gagliardo-Nirenberg不等式，直接控制 $(B(u), Au)$ 项，无需额外假设。

  • 无需截断非线性项 $B(\cdot)$：以往工作（如 [73]）需对对流项 $B(u)$ 进行截断以保证局部单调性；而本文因吸收项 $C(u)$ 具备天然的单调性（见Lemma 2.5），故无需任何截断操作即可建立比较原理。

  • 修改测试函数空间：为克服指数型积分项带来的困难（如Proposition 6.2中的估计），作者要求测试函数 $\phi$ 及其时间导数 $\phi_t$、梯度 $D\phi$、Hessian $D^2\phi$ 均有界（见Definition 5.1），从而自然引入“有界粘性解”概念，简化验证过程。

  • C-指数紧性过渡：通过验证解族满足 $C$-指数紧性（Definition 3.8），即路径连续模的一致指数可积性，成功将LDP从右连左极函数空间 $D([0,T]; H)$ 提升至连续函数空间 $C([0,T]; H)$。

4. 与之前工作的比较

a) 突破与改进

本文相较于已有工作做出了多项重要推进：

  • 维度扩展至三维：此前基于粘性解方法的大偏差研究（如 [73]）仅限于二维Navier-Stokes方程，受限于三维情形缺乏全局适定性。本文借助Brinkman-Forchheimer模型中吸收项的强耗散效应，首次在 $d=3$ 情形下建立了粘性解框架下的LDP结果。

  • 放宽对非线性结构的依赖：传统分析常依赖 $(B(u), Au) = 0$ 这一特殊代数恒等式来闭合能量估计。本文在三维情形下明确放弃这一条件，转而利用吸收项主导对流项的机制（见Remark 2.3），实现了更普适的估计策略。

  • 无需弱收敛或变分表示法：相比文献 [61] 使用Budhiraja-Dupuis弱收敛方法并依赖Varadhan-Bryc定理，本文完全基于PDE视角，通过粘性解方法直接建立Laplace原理，避免了构造测度变换或紧性论证的技术复杂性。

  • 首次应用于阻尼Navier-Stokes类系统：据作者所知，这是首次将粘性解方法应用于随机CBF方程或带阻尼的Navier-Stokes系统的大偏差研究，填补了该方向的空白。

  • 更强的结果结构：所得LDP不仅适用于Skorohod空间，还进一步升级至连续路径空间 $C([0,T]; H)$，增强了结果的应用价值（例如在路径泛函统计推断中的适用性）。

综上所述，本文通过融合无穷维HJB方程的粘性解理论与流体力学的能量估计技术，发展了一套稳健的大偏差分析新框架，显著拓展了现有方法的适用范围和深度。

3. Lie module analysis of hydrodynamic-type systems

流体动力型系统的李模分析

作者: Łukasz Chomienia, Alfred Michel Grundland

类别: math-ph, math.AP, math.MP, 35Q31, 35A30, 17B80, 53A05

发布日期: 2025-04-20

ArXiv ID: 2504.14756v3

链接: http://arxiv.org/abs/2504.14756v3

详细分析

中文标题: 流体动力型系统的李模分析

1. 研究对象和背景

本文研究的对象是一类拟线性双曲型一阶偏微分方程组，特别是所谓的流体动力型系统（hydrodynamic-type systems）。这类系统的一般形式如下：

$$ \sum_{i=1}^{p} A_i(u) \frac{\partial u}{\partial x_i} = 0, \quad u = (u^1(x), \dots, u^q(x)) \in \mathbb{R}^q, \quad x = (x_1, \dots, x_p) \in \mathbb{R}^p, $$

其中 $A_1, \dots, A_p$ 是依赖于未知函数 $u$ 的 $p \times q$ 矩阵函数。该系统被假定为双曲型，并附加了称为**微分约束（differential constraints, DCs）**的结构条件：

$$ \frac{\partial u^\alpha}{\partial x_i} = \sum_{s=1}^{k} \xi_s(x) \gamma_s^\alpha(u) \lambda_s^i(u), \quad \alpha = 1,\dots,q, \; i = 1,\dots,p, $$

其中 $\xi_s(x) \neq 0$ 是标量函数，$\gamma_s: \mathbb{R}^q \to \mathbb{R}^q$ 和 $\lambda_s: \mathbb{R}^q \to \mathbb{R}^p$ 是仅依赖于 $u$ 的向量场与协向量场，且满足代数特征值关系：

$$ \sum_{i=1}^{p} \sum_{\alpha=1}^{q} A_i^\beta(u) \lambda_s^i(u) \gamma_s^\alpha(u) = 0, \quad \det(A_i^\beta \lambda_s^i) = 0. $$

这些系统允许存在黎曼波解（Riemann wave solutions），其基本构造基于隐式关系：

$$ u^\alpha = f^\alpha(r), \quad r = \phi(\lambda_i(r) x^i), $$

其中 $f$ 是向量场 $\gamma^\alpha \partial / \partial u^\alpha$ 的积分曲线。

特别地，文章以**(1+1)维可压缩欧拉系统（compressible Euler system）**作为主要示例进行深入分析。该系统描述了一维无粘可压缩流体的运动，包含质量、动量和能量守恒律，在适当变量下可写成上述拟线性双曲系统的形式。

研究区域通常取为时空域 $(t,x) \in \mathbb{R}_+ \times \mathbb{R}$，关注的是光滑解在有限时间内是否发生梯度灾变（gradient catastrophe），以及多个非线性波相互作用时的几何结构。文中未明确设定边值条件，而是聚焦于局部存在性和波叠加的内在几何性质。

关键概念是“准直化（quasi-rectifiability）”，即一组向量场能否通过坐标变换局部变为仅依赖单个坐标的伸缩形式（见后文定义）。这推广了经典Frobenius定理中对合分布的概念，用于刻画波之间是否可以独立传播而不产生新类型的波——即所谓“弹性叠加”与“非弹性叠加”的区分。

2. 主要定理或主要结果

定理 5.1（李模到李代数的角度保持变换）
设 $M$ 是一个由流体动力型系统诱导的李模（Lie module），其基底为光滑向量场 $\{\gamma_1, \dots, \gamma_k\}$，定义在流形 $N$ 上，且满足 $ \gamma_1 \wedge \cdots \wedge \gamma_k \neq 0 $ 处处成立。若该李模属于某一特定类（满足可积性条件及角度结构兼容性），则存在唯一的（至多差一个同构）实李代数 $\mathfrak{g}$，以及一个角度保持的变换（angle-preserving transformation），将 $M$ 映射为 $\mathfrak{g}$ 中的向量场集合。此外，变换后的李代数具有如下结构：它是某个有限维李代数与一个无限维阿贝尔李代数的半直积。

命题 4.3（欧拉系统的无限维李代数结构）
对于(1+1)维可压缩欧拉系统，与其相关联的波所生成的最小实李代数具有结构：

$$ \mathfrak{g} = \mathfrak{a} \rtimes \mathfrak{b}, $$

其中 $\mathfrak{a}$ 是无限维阿贝尔李代数，$\mathfrak{b}$ 是某个有限维李代数。此分解有两种自然选择方式，取决于如何选取阿贝尔理想 $\mathfrak{a}$。由于 $\mathfrak{a}$ 是阿贝尔的，这一结构可解释为一系列来自 $\mathfrak{b}$ 的波的迭代作用。

定理 3.5（准直化族的曲率判据）
定义在三维空间 $\mathbb{R}^3$ 上的一组线性无关光滑向量场 $\{X_1, X_2, X_3\}$ 是准直化的，当且仅当它们所诱导的联络的曲率为零，且存在一组正交标架使得每个向量场只沿对应坐标方向伸缩。更一般地，对于任意维数，准直化等价于存在一个平坦仿射联络，使得这些向量场成为其测地向量场的方向。

推论 2.10（准直化向量场的坐标实现）
若 $\{X_1, \dots, X_r\}$ 是流形 $N$ 上的准直化向量场族，$D = \mathrm{span}\{X_1, \dots, X_r\}$ 是其张成的分布，$\eta_1, \dots, \eta_r$ 是对偶1-形式，且存在非零函数 $f_1, \dots, f_r$ 使得限制在 $D$ 的叶上时，$d(f_i \eta_i)|_D = 0$，并且 $f_i \eta_i|_D = dx_i|_D$，则局部存在坐标系 $\{x_1, \dots, x_n\}$，使得每个 $X_i$ 具有形式：

$$ X_i = g_i(x_1, \dots, x_n) \frac{\partial}{\partial x_i}, \quad g_i \neq 0, $$

即为准直化形式。

3. 研究方法、关键技术和核心工具

a) 核心分析学工具

  • 李群与李代数理论：全文的核心工具是李群和李代数的结构分析，尤其是无限维李代数（如Witt代数、Virasoro代数）的应用。作者利用李括号封闭性来刻画波之间的相互作用结构。

  • 微分几何与联络理论：引入了仿射联络（affine connection）的概念，证明在连通李群上存在一种自然的仿射联络，使得所有单参数子群成为测地线。这种几何结构被用来描述波叠加流形上的平行移动（parallel transport）。

  • Frobenius定理及其修正版本：使用经典的Frobenius定理判断分布的可积性，并结合经重标度修正的Frobenius定理（Modified Frobenius Theorem by Rescaling） 来处理非对合但可通过乘子变为对合的向量场族。

  • 外微分与积分因子方法：通过构造对偶1-形式并寻找合适的积分因子 $f_i$，使 $f_i \eta_i$ 在分布限制下闭合，从而实现向量场的局部直化。

  • 曲率与几何不变量分析：提出用曲率张量作为判断准直化性的新标准，将代数条件转化为几何条件，便于应用微分几何工具研究波叠加流形的变形。

b) 技术性难题的解决技巧

  • 重标度变换技术（Rescaling technique）：针对不能直接对合的向量场族 $\{\gamma_1, \dots, \gamma_k\}$，引入光滑函数 $h_i$ 进行重标度 $ \tilde{\gamma}_i = h_i \gamma_i $，使得 $[\tilde{\gamma}_i, \tilde{\gamma}_j] = 0$。这是构造多波解的关键步骤，避免了强可积性假设。

  • 李模到李代数的转换机制：由于原始系统导出的是李模（系数为函数的线性组合），而非常系数李代数，作者设计了一种角度保持的变换，将李模“提升”为有限维实李代数，从而能借用李群几何的丰富工具。

  • 无限维代数的有限维逼近：虽然波空间本身具有无限维结构（如Witt代数），但通过识别其半直积结构 $\mathfrak{a} \rtimes \mathfrak{b}$，将问题归结为有限维李代数 $\mathfrak{b}$ 的重复作用，实现了从无限维到有限维的有效降维。

  • 波叠加区域的拓扑刻画：通过支持集交集的变化定义“叠加初刻”$t_{\min}$ 与“终刻”$t_{\max}$，并利用流映射 $\phi_t$ 分析前后存在的波类型，进而定义叠加指数 $\Gamma_M = \mathrm{card}\,\Gamma_- - \mathrm{card}\,\Gamma_+$，用以量化非弹性效应。

4. 与之前工作的比较

a) 突破与改进

本文相较于已有工作（如[10], [9], [7]等）做出了以下显著推进：

  • 突破了弹性叠加的限制：以往多数研究集中于“弹性波叠加”（elastic superpositions），即波交互不产生新类型波，对应于可积分布或叶状结构。本文首次系统地处理非弹性叠加（non-elastic superpositions），即无法分解为两两独立作用的情形，填补了理论空白。

  • 建立了李模与李代数之间的桥梁：此前关于准直化李代数的研究无法直接应用于李模（因其结构依赖函数系数）。本文提出角度保持的重标度变换，将一类广泛的李模唯一地转化为真实李代数，从而将抽象代数工具引入物理波系统的分析。

  • 提供了新的几何视角：将波叠加流形视为由李子代数对应的子流形之变形构成，并引入平行运输概念来描述其演化。这种微分几何框架比传统的Riemann不变量方法更具普适性和结构性。

  • 提出了基于曲率的准直化判据：传统判据依赖于李括号的线性展开（如 $[X_i,X_j] \in \mathrm{span}\{X_i,X_j\}$），而本文给出一个全新的内蕴几何判据——曲率为零，适用于更高维和更复杂情形，尤其便于数值验证。

  • 获得更强的存在性与结构结果：相比仅构造近似解或数值模拟的工作（如[6]），本文给出了精确的多黎曼波解存在的充分条件，并通过李代数结构揭示了解的整体对称性和演化规律，提升了对非线性叠加机制的理解深度。

综上所述，本文不仅发展了新的数学工具（李模重标度、曲率判据），还将李理论、微分几何与流体力学深度融合，为高维非线性波相互作用提供了首个系统的解析框架。

4. On the vanishing viscosity limit for incompressible flows with

inflow/outflow boundary conditions

关于具有流入/流出边界条件的不可压缩流体的粘性消失极限

作者: Anna L. Mazzucato, Dehua Wang, Wei Wei

类别: math.AP, 35Q30, 35B25, 76N10, 76D10, 35B40

发布日期: 2025-10-01

ArXiv ID: 2510.01101v1

链接: http://arxiv.org/abs/2510.01101v1

详细分析

中文标题: 关于具有流入/流出边界条件的不可压缩流体的粘性消失极限

1. 研究对象和背景

本文研究的是在一般有界区域中，带有流入（inflow）与流出（outflow）边界条件的不可压缩纳维-斯托克斯方程（Navier-Stokes equations, NSE）在粘性系数趋于零时的极限行为，即所谓的粘性消失极限（vanishing viscosity limit）。目标是证明当粘性 $\epsilon \to 0$ 时，NSE 的解收敛到欧拉方程（Euler equations, EE）的解，并建立精确的收敛速率估计。

所考虑的物理模型为三维空间中的不可压缩粘性流体，其运动由如下 Navier-Stokes 方程组 描述：

$$ \begin{cases} \partial_t u^\epsilon - \epsilon \Delta u^\epsilon + (u^\epsilon \cdot \nabla) u^\epsilon + \nabla p^\epsilon = f, & \text{in } \Omega \times [0,T), \\ \mathrm{div}\, u^\epsilon = 0, & \text{in } \Omega \times [0,T), \\ \gamma_- u^\epsilon = \alpha, & \text{on } \Gamma_- \times [0,T), \\ \gamma_+ u^\epsilon = \beta, & \text{on } \Gamma_+ \times [0,T), \\ u^\epsilon|_{t=0} = u_0, & \text{in } \Omega, \end{cases} $$

其中：

  • $\Omega \subset \mathbb{R}^3$ 是一个光滑有界区域；
  • 边界 $\Gamma = \partial\Omega$ 被划分为两个互不相交的闭曲面：$\Gamma_-$（出流边界）和 $\Gamma_+$（入流边界）；
  • $u^\epsilon$ 是速度场，$p^\epsilon$ 是压力，$\epsilon > 0$ 是粘性系数；
  • $f$ 是外力项，$u_0$ 是初始速度；
  • $\gamma_\pm$ 表示在 $\Gamma_\pm$ 上的迹算子；
  • 入流/出流条件要求：$\alpha \cdot n > 0$ 在 $\Gamma_-$ 上（出流），$\beta \cdot n < 0$ 在 $\Gamma_+$ 上（入流），且满足质量守恒：
    $$ \int_{\Gamma_-} \alpha \cdot n \, dS + \int_{\Gamma_+} \beta \cdot n \, dS = 0. $$

对应的欧拉方程（无粘极限情形 $\epsilon = 0$）为：

$$ \begin{cases} \partial_t u^0 + (u^0 \cdot \nabla) u^0 + \nabla p^0 = f, & \text{in } \Omega \times [0,T), \\ \mathrm{div}\, u^0 = 0, & \text{in } \Omega \times [0,T), \\ \gamma_- u^0 \cdot n = \alpha \cdot n, & \text{on } \Gamma_- \times [0,T), \\ \gamma_+ u^0 = \beta, & \text{on } \Gamma_+ \times [0,T), \\ u^0|_{t=0} = u_0, & \text{in } \Omega. \end{cases} $$

值得注意的是，在出流边界 $\Gamma_-$ 上，仅指定了法向速度，而切向速度未被指定，这导致了 NSE 与 EE 解之间存在边界层不匹配问题。然而，由于允许非零法向流动（注入/抽吸），该边界不再是特征型边界，从而有望稳定边界层、改善收敛性质。

文章的关键创新之一是允许斜向流入/流出方向（oblique inflow/outflow），即边界速度方向不必与边界的法向对齐，只要注入/抽吸速率远离零即可（即 $\alpha \cdot n \geq \alpha_0 > 0$）。这一设定推广了以往要求正交注入的工作。

函数空间方面，假设所有数据足够光滑（$C^\infty$ 类），并满足适当的兼容性条件（如初始值与边界值在交点处一致）。文中使用标准 Sobolev 空间 $H^s(\Omega)$ 和 Lebesgue 空间 $L^p$，时间依赖函数空间记作 $L^p_X([0,T))$，例如 $L^\infty_t L^2_x$ 或 $L^2_t H^1_x$。

2. 主要定理或主要结果

定理 2.1（粘性消失极限下的收敛性）
假设数据满足正则性条件 $(2.1)$ 及初始时刻的切向速度兼容性条件 $(2.5)$：$\gamma_-(v_0|_{t=0}) \cdot \tau = 0$。则存在足够小的时间 $T > 0$ 和与 $\epsilon$ 无关的常数 $C > 0$，使得以下估计成立：

$$ \begin{cases} \|v^\epsilon - (v^0 + \varphi_0)\|_{L^\infty_t L^2_x} \leq C \epsilon, \\ \|v^\epsilon - (v^0 + \varphi_0)\|_{L^2_t H^1_x} \leq C \epsilon^{1/2}, \end{cases} $$

此外，若进一步满足时间导数的兼容性条件 $(2.6)$：$\gamma_-((\partial_t v^0)|_{t=0}) \cdot \tau = 0$，则可得到更高阶收敛：

$$ \begin{cases} \|v^\epsilon - (v^0 + \varphi_0) - \epsilon(v^1 + \varphi_1)\|_{L^\infty_t L^2_x} \leq C \epsilon^2, \\ \|v^\epsilon - (v^0 + \varphi_0) - \epsilon(v^1 + \varphi_1)\|_{L^2_t H^1_x} \leq C \epsilon^{3/2}. \end{cases} $$

其中 $v^\epsilon = u^\epsilon - U$，$v^0 = u^0 - U$ 是相对于背景 Stokes 流 $U$ 的扰动；$\varphi_0, \varphi_1$ 是构造的边界层修正项（boundary layer correctors），分别定义于公式 $(3.4)$、$(5.7)$ 和 $(5.8)$ 中。

该定理表明，在适当构造边界层修正后，Navier-Stokes 解以明确的代数速率收敛到 Euler 解，且内域中的收敛率与正交注入情况相同，尽管几何更一般（允许斜向流动）。

3. 研究方法、关键技术和核心工具

a) 核心分析学工具

  • 渐近展开法（asymptotic expansion）：采用多尺度分析，将粘性解 $u^\epsilon$ 展开为外部解（outer expansion）与边界层修正项之和，形式为：

    $$ u^\epsilon \approx u^0 + \varphi_0 + \epsilon(u^1 + \varphi_1) + \cdots $$

    其中 $u^0, u^1$ 满足欧拉类方程，$\varphi_0, \varphi_1$ 是局部定义在边界附近的快速衰减的边界层项。

  • 能量估计（energy estimates）：通过对误差函数进行高阶能量估计，控制 $L^2$ 和 $H^1$ 范数下的收敛性。特别地，利用边界层项的指数衰减性质来吸收边界附近产生的奇性。

  • 曲线坐标系下的微分算子表达式：在靠近出流边界 $\Gamma_-$ 的邻域内引入边界法向坐标（boundary normal coordinates），将 Laplace 算子、梯度、散度等写成协变形式，便于处理非平坦边界和任意注入角度的情形。

  • Prandtl 型方程分析：通过形式渐近分析导出主导边界层行为的一阶 Prandtl 型方程：

    $$ -\epsilon \partial_{\xi_3}^2 \varphi_0 - \alpha_3 \partial_{\xi_3} \varphi_0 \approx 0, $$

    其中 $\xi_3$ 是沿法向的距离变量。此方程的解具有 $e^{-\alpha_3 \xi_3 / \epsilon}$ 形式的指数衰减，说明边界层厚度为 $O(\epsilon)$，远薄于无滑移边界下的 $O(\sqrt{\epsilon})$。

  • 单位分解技术（partition of unity）：由于三维边界可能存在脐点（umbilical points），无法全局使用主曲率坐标系，故采用局部图卡覆盖边界，并通过单位分解拼接局部构造的边界层修正项，形成全局光滑的修正函数。

b) 技术性难题的处理技巧

  • 边界切向速度不匹配的处理：在出流边界 $\Gamma_-$ 上，NSE 要求全速度给定，而 EE 仅知法向速度，因此切向部分出现不匹配。作者通过构造基于旋度势（vector potential）的边界层修正项 $\varphi_0 = \mathrm{curl}\,\psi_0$ 来精确补偿这一差异，同时保持不可压条件 $\mathrm{div}\,\varphi_0 = 0$。

  • 高阶收敛的实现：为了获得 $\epsilon^2$ 精度的 $L^\infty L^2$ 收敛，需引入一阶外展项 $v^1$ 和相应的一阶边界层修正 $\varphi_1$。这需要更高的初始兼容性条件（如 $(2.6)$），确保初始时刻误差也达到更高阶小量。

  • 指数小项控制（exponentially small terms, e.s.t.）：文中引入符号 $e.s.t.$ 表示形如 $Ce^{-c/\epsilon}$ 的项及其导数，这类项在任何代数阶意义下均可忽略，用于简化误差估计。

  • 向量势构造法：为了避免直接求解复杂的向量边界层方程，作者巧妙地通过构造标量势函数 $\psi_0$ 并取其旋度来生成满足不可压性和边界条件的修正项，极大简化了构造过程。

4. 与之前工作的比较

a) 突破与改进

本文相较于先前工作取得了以下几个重要进展：

  • 推广了几何结构与流动方向：以往关于流入/流出边界下粘性消失极限的研究（如 Gie-Hamouda-Temam [37] 和 Lombardo-Sammartino [66]）大多局限于正交注入或简单几何（如周期通道）。本文首次在一般光滑有界三维区域中处理斜向流入/流出方向，只要注入角不退化（即 $\alpha \cdot n \geq \alpha_0 > 0$），便能保证收敛性。这是对原有框架的重大推广。

  • 统一处理边界层结构：尽管此前已有工作考虑斜流（如 [66] 处理二维 Oseen 方程），但本文在非线性 NSE 框架下建立了完整的 Prandtl 型边界层理论，并显式构造了适用于任意光滑边界的局部修正项，结合单位分解实现全局一致性。

  • 恢复最优收敛率：尽管流动方向更一般，但作者仍证明了与正交注入情形相同的收敛速率：$L^\infty_t L^2_x$ 下为 $O(\epsilon)$，$L^2_t H^1_x$ 下为 $O(\epsilon^{1/2})$，并通过高阶展开提升至 $O(\epsilon^2)$ 和 $O(\epsilon^{3/2})$。这说明斜向流动并未恶化收敛性，只要注入强度有下界。

  • 明确边界层尺度：通过渐近分析指出，此类问题的边界层厚度为 $O(\epsilon)$，而非经典无滑移情形的 $O(\sqrt{\epsilon})$，意味着边界效应更局域化，稳定性更强，这也解释了实验中观察到的“注入稳定边界层”现象。

  • 放宽对全局坐标的依赖：以往许多边界层分析依赖于基于主曲率的全局坐标系统，但在三维中脐点的存在会破坏这种结构。本文采用局部法向坐标加单位分解的方法，更具普适性和鲁棒性。

综上所述，本文提供了一个统一、稳健且可计算的框架，用于分析具有流入/流出边界的粘性消失极限，显著拓展了适用范围，并为数值模拟和工程应用提供了坚实的理论基础。

5. Global convergence rates in the relaxation limits for the compressible

Euler and Euler-Maxwell systems in Sobolev spaces

Sobolev空间中可压缩Euler与Euler-Maxwell系统的松弛极限下的全局收敛速率

作者: Timothée Crin-Barat, Yue-Jun Peng, Ling-Yun Shou

类别: math.AP

发布日期: 2025-06-04

ArXiv ID: 2506.04103v2

链接: http://arxiv.org/abs/2506.04103v2

详细分析

中文标题: Sobolev空间中可压缩Euler与Euler-Maxwell系统的松弛极限下的全局收敛速率

1. 研究对象和背景

本文研究两类带有松弛项的偏微分方程系统：可压缩Euler方程组与可压缩Euler-Maxwell方程组，二者均属于部分耗散双曲系统（partially dissipative hyperbolic systems）的范畴。研究区域为全空间 $\mathbb{R}^d$（对Euler系统 $d \geq 1$，对Euler-Maxwell系统 $d=3$），初始数据允许为“非适定”（ill-prepared）情形，即不满足特定的渐近相容性条件。

（1）可压缩Euler系统（带阻尼）

该系统描述了具有摩擦力的可压缩流体动力学。原始系统为：

$$ \begin{cases} \partial_t \rho + \mathrm{div}(\rho v) = 0, \\ \partial_t (\rho v) + \mathrm{div}(\rho v \otimes v) + \nabla p(\rho) = -\frac{\rho v}{\varepsilon}, \end{cases} $$

其中 $\rho > 0$ 为密度，$v$ 为速度，$p(\rho)$ 为压力函数，$\varepsilon > 0$ 为小的松弛时间参数。考虑平衡态 $(\bar{\rho}, \bar{v}) = (1, 0)$ 附近的小扰动解。

通过扩散重标度：

$$ \rho^\varepsilon(t,x) := \rho\left(\frac{t}{\varepsilon}, x\right), \quad u^\varepsilon(t,x) := \frac{1}{\varepsilon} v\left(\frac{t}{\varepsilon}, x\right), $$

得到重标度后的系统：

$$ \begin{cases} \partial_t \rho^\varepsilon + \mathrm{div}(\rho^\varepsilon u^\varepsilon) = 0, \\ \varepsilon^2 \partial_t (\rho^\varepsilon u^\varepsilon) + \varepsilon^2 \mathrm{div}(\rho^\varepsilon u^\varepsilon \otimes u^\varepsilon) + \nabla p(\rho^\varepsilon) = -\rho^\varepsilon u^\varepsilon, \end{cases} $$

初始条件为 $(\rho^\varepsilon, u^\varepsilon)(0,x) = (\rho_0^\varepsilon, u_0^\varepsilon)(x)$。

当 $\varepsilon \to 0$ 时，形式极限 $(\rho_*, u_*)$ 满足达西定律（Darcy’s law）与过滤方程：

$$ \partial_t \rho_* - \Delta p(\rho_*) = 0, \quad \text{即 porous medium equation（多孔介质方程）}. $$

（2）可压缩Euler-Maxwell系统

该系统描述等离子体中可压缩流体与自洽电磁场的相互作用，三维情形下为：

$$ \begin{cases} \partial_t \rho + \mathrm{div}(\rho v) = 0, \\ \partial_t (\rho v) + \mathrm{div}(\rho v \otimes v) + \nabla p(\rho) = -\rho (E + v \times B) - \frac{\rho v}{\varepsilon}, \\ \partial_t E - \nabla \times B = \rho v, \\ \partial_t B + \nabla \times E = 0, \\ \mathrm{div} E = 1 - \rho, \quad \mathrm{div} B = 0. \end{cases} $$

未知量包括密度 $\rho$、速度 $v$、电场 $E$ 和磁场 $B$，压力 $p = p(\rho)$ 满足光滑性和正压条件 $p'(\bar{\rho}) > 0$。

同样进行扩散重标度后，系统变为：

$$ \begin{cases} \partial_t \rho^\varepsilon + \mathrm{div}(\rho^\varepsilon u^\varepsilon) = 0, \\ \varepsilon^2 \partial_t (\rho^\varepsilon u^\varepsilon) + \varepsilon^2 \mathrm{div}(\rho^\varepsilon u^\varepsilon \otimes u^\varepsilon) + \nabla p(\rho^\varepsilon) = -\rho^\varepsilon (E^\varepsilon + \varepsilon u^\varepsilon \times B^\varepsilon) - \rho^\varepsilon u^\varepsilon, \\ \varepsilon \partial_t E^\varepsilon - \nabla \times B^\varepsilon = \varepsilon \rho^\varepsilon u^\varepsilon, \\ \varepsilon \partial_t B^\varepsilon + \nabla \times E^\varepsilon = 0, \\ \mathrm{div} E^\varepsilon = 1 - \rho^\varepsilon, \quad \mathrm{div} B^\varepsilon = 0. \end{cases} $$

当 $\varepsilon \to 0$ 时，其形式极限满足漂移-扩散模型（drift-diffusion model）：

$$ \begin{cases} \partial_t \rho_* - \mathrm{div}(\nabla p(\rho_*) + \rho_* \nabla \phi_*) = 0, \\ \Delta \phi_* = 1 - \rho_*, \end{cases} $$

其中电势 $\phi_* = \Lambda^{-2}(\rho_* - 1)$，$\Lambda^{-2} = (-\Delta)^{-1}$。

2. 主要定理或主要结果

定理 2.1（可压缩Euler系统在非适定初值下的全局收敛率）
设 $d \geq 1$，$m \geq [d/2] + 2$，且压力函数 $p \in C^\infty(\mathbb{R}_+)$ 满足 $p'(1) > 0$。设 $(\rho^\varepsilon, q^\varepsilon)$ 和 $(\rho_*, q_*)$ 分别是重标度Euler系统（1.4）与过滤方程（1.7）在初值 $(\rho_0^\varepsilon, q_0^\varepsilon)$ 和 $\rho_{*0}$ 下的经典解（由Proposition 2.1 与 2.2 保证存在）。则存在常数 $C > 0$ 独立于 $\varepsilon$，使得以下误差估计成立：

$$ \sup_{t \in \mathbb{R}_+} \|(\rho^\varepsilon - \rho_*)(t)\|_{H^{m-1}}^2 + \int_0^\infty \|\nabla(\rho^\varepsilon - \rho_*)(t)\|_{H^{m-1}}^2 dt \leq C(\|\rho_0^\varepsilon - \rho_{*0}\|_{H^{m-1}}^2 + \varepsilon^2), $$

$$ \int_0^\infty \|(q^\varepsilon - q_* - q_I^\varepsilon)(t)\|_{H^{m-1}}^2 dt \leq C(\|\rho_0^\varepsilon - \rho_{*0}\|_{H^{m-1}}^2 + \varepsilon^2), $$

其中 $q_I^\varepsilon(t,x) := e^{-t/\varepsilon^2} q_0^\varepsilon(x)$ 是一个用于消除初始层奇性的修正项。若进一步假设 $\rho_0^\varepsilon - \rho_{*0} \in \dot{H}^{-1}$，则还有：

$$ \int_0^\infty \|(\rho^\varepsilon - \rho_*)(t)\|_{L^2}^2 dt \leq C(\|\rho_0^\varepsilon - \rho_{*0}\|_{\dot{H}^{-1}}^2 + \varepsilon^2). $$

定理 2.2（可压缩Euler系统在适定初值下的二阶收敛率）
在定理2.1的假设基础上，进一步假设初值满足适定性条件：

$$ \|\rho_0^\varepsilon - \rho_{*0} - \varepsilon \rho_{1,0}\|_{H^{m-2}} \leq \varepsilon^2, \quad \|q_0^\varepsilon - q_*\|_{H^{m-1}} \leq \varepsilon, $$

其中 $\rho_{1,0} \in H^m$，$q_* = -\nabla p(\rho_{*0})$。定义一阶渐近展开：

$$ \rho_a^\varepsilon := \rho_* + \varepsilon \rho_1, \quad q_a^\varepsilon := q_* + \varepsilon q_1, $$

其中 $(\rho_1, q_1)$ 满足线性系统：

$$ \partial_t \rho_1 + \mathrm{div} q_1 = 0, \quad q_1 = -\nabla(p'(\rho_*) \rho_1). $$

则存在常数 $C > 0$ 独立于 $\varepsilon$，使得：

$$ \sup_{t \in \mathbb{R}_+} \|(\rho^\varepsilon - \rho_* - \varepsilon \rho_1)(t)\|_{H^{m-2}}^2 + \int_0^\infty \|\nabla(\rho^\varepsilon - \rho_* - \varepsilon \rho_1)(t)\|_{H^{m-2}}^2 dt \leq C \varepsilon^4, $$

$$ \int_0^\infty \|(q^\varepsilon - q_*)(t)\|_{H^{m-2}}^2 dt \leq C \varepsilon^2, \quad \sup_{t \in \mathbb{R}_+} \|(q^\varepsilon - q_* - \varepsilon q_1)(t)\|_{H^{m-2}}^2 \leq C \varepsilon^2, $$

$$ \int_0^\infty \|(q^\varepsilon - q_* - \varepsilon q_1)(t)\|_{H^{m-2}}^2 dt \leq C \varepsilon^4. $$

定理 2.3（可压缩Euler-Maxwell系统在非适定初值下的强收敛）
设 $d=3$，$m \geq 3$，压力函数 $p$ 满足光滑性条件。设 $(\rho^\varepsilon, q^\varepsilon, E^\varepsilon, B^\varepsilon)$ 和 $(\rho_*, q_*, E_*)$ 分别是重标度Euler-Maxwell系统（1.12）与漂移-扩散模型（1.15）的解。假设 $\rho_{*0} - 1 \in \dot{H}^{-1}$，则存在常数 $C > 0$ 独立于 $\varepsilon$，使得：

$$ \sup_{t \in \mathbb{R}_+} \left( \|(\rho^\varepsilon - \rho_*)(t)\|_{H^{m-1}}^2 + \|(E^\varepsilon - E_*)(t)\|_{H^{m-1}}^2 + \|B^\varepsilon(t) - B_e\|_{H^{m-1}}^2 \right) $$

$$

  • \int_0^\infty \left( |(\rho^\varepsilon - \rho_)(t)|{H^m}^2 + |(E^\varepsilon - E)(t)|{H^m}^2 + |\nabla B^\varepsilon(t)|{H^{m-2}}^2 \right) dt
    $$ $$
    \leq C \left( |\rho_0^\varepsilon - \rho_{*0}|{H^{m-1}}^2 + |E_0^\varepsilon - E{0}|{H^{m-1}}^2 + |B_0^\varepsilon - B_e|{H^{m-1}}^2 + \varepsilon^2 \right), $ > 且对于动量误差有： > $ \int_0^\infty |(q^\varepsilon - q_ - q_I^\varepsilon)(t)|{H^{m-2}}^2 dt \leq C \left( |\rho_0^\varepsilon - \rho{*0}|{H^{m-1}}^2 + |E_0^\varepsilon - E{*0}|{H^{m-1}}^2 + |B_0^\varepsilon - B_e|{H^{m-1}}^2 + \varepsilon^2 \right), $$ 其中 $q_I^\varepsilon(t,x) := q_0^\varepsilon(x) e^{-t/\varepsilon^2}$。

3. 研究方法、关键技术和核心工具

a) 核心分析学工具：

  • 高维流函数技术（high-dimensional stream function techniques）：这是本文的核心创新之一。作者引入了一个新的辅助变量（称为“流函数”）来处理密度差 $\rho^\varepsilon - \rho_*$ 的时间演化。例如，在Euler系统中定义：

    $$ N^\varepsilon(t,x) := -\int_0^t (q^\varepsilon - q_*)(s,x) ds - \Lambda^{-2} \nabla(\rho^\varepsilon - \rho_*)(0,x), $$

    它满足 $\partial_t N^\varepsilon = -(q^\varepsilon - q_*)$ 且 $\mathrm{div} N^\varepsilon = \rho^\varepsilon - \rho_*$，从而将密度误差控制转化为对 $N^\varepsilon$ 的能量估计，进而获得 $L^2(\mathbb{R}_+; L^2)$ 类型的误差界。

  • 渐近展开法（asymptotic expansion method）：用于获得更高阶的收敛率（如 $\mathcal{O}(\varepsilon^2)$）。通过对解进行形式展开 $\rho^\varepsilon = \rho_* + \varepsilon \rho_1 + \cdots$，构造近似解，并对误差项进行精细的能量估计。

  • 一致在时间上的能量估计（uniform-in-time error estimates）：结合Sobolev空间中的经典能量方法，利用系统本身的耗散结构（特别是压力梯度项带来的抛物性）建立对误差的一致控制。

b) 关键技术难点与应对技巧：

  • 初始层奇异性：在非适定初值下，初始时刻的速度可能为 $O(1/\varepsilon)$ 量级，导致经典能量估计失效。作者引入指数衰减的初始层修正项 $q_I^\varepsilon(t) = q_0^\varepsilon e^{-t/\varepsilon^2}$，将其从动量中分离出来，使剩余误差在 $t=0$ 处具有良好的正则性。

  • 低正则性挑战：极限解 $\partial_t q_*$ 的正则性有限（仅为 $H^{m-2}$），而误差方程中涉及 $\varepsilon^2 \partial_t q_*$ 项。作者通过在 $H^{m-1}$ 空间进行估计，并巧妙利用已知解的先验界（来自Proposition 2.1–2.4）来吸收该项。

  • 避免频率依赖工具：以往工作多依赖Littlewood-Paley分解和Besov空间的精细频率分析。本文完全在经典Sobolev空间 $H^m$ 中完成证明，不使用任何频域工具，提高了方法的普适性，适用于有界域或半空间等问题。

4. 与之前工作的比较

a) 突破与改进：

  • 首次在经典Sobolev空间中建立全局强收敛率：此前关于高维松弛极限的工作（如[10,11,26]）大多依赖临界Besov空间或周期域设定，并使用Littlewood-Paley理论。本文首次在标准Sobolev空间 $H^m(\mathbb{R}^d)$ 中建立了显式的收敛率 $\mathcal{O}(\varepsilon)$ 和 $\mathcal{O}(\varepsilon^2)$，回答了引言中提出的关键问题。

  • 获得更强的收敛拓扑：本文证明了误差在 $L^2(\mathbb{R}_+; L^2)$ 中的收敛性，这比弱收敛或局部强收敛更强，且无法从前人基于紧性或Besov空间的结果直接推出。

  • 适用于非适定初值：许多早期工作要求初值为“适定”（well-prepared），即满足某种相容性条件。本文在非适定初值下仍能建立 $\mathcal{O}(\varepsilon)$ 收敛率，仅需添加一个简单的指数衰减修正项 $q_I^\varepsilon$。

  • 统一处理Euler与Euler-Maxwell系统：尽管两个系统的耗散机制不同（后者为非对称松弛），作者发展了一套统一的框架，结合流函数思想（对Euler）与电场作为“流函数”的观察（对Euler-Maxwell），实现了方法的推广。

  • 方法更具普适性：所发展的流函数+渐近展开方法不依赖频域分析，因此有望推广到边界问题、离散格式的连续极限分析等更广泛场景，拓展了现有理论的应用范围。

了解领域论文（摘要翻译）

1. Linear damping estimates for periodic roll wave solutions of the

inviscid Saint-Venant equations and related systems of hyperbolic balance laws

无黏圣维南方程及相关的双曲型平衡律系统的周期滚波解的线性阻尼估计

作者: L. Miguel Rodrigues, Kevin Zumbrun

类别: math.AP, 35B35, 35L67, 15A18, 15A42, 35Q35, 35P15, 15A60, 15A63

发布日期: 2025-04-17

ArXiv ID: 2504.12804v3

链接: http://arxiv.org/abs/2504.12804v3

摘要翻译

中文标题: 无黏圣维南方程及相关的双曲型平衡律系统的周期滚波解的线性阻尼估计

摘要翻译: 在显著推广作者此前关于黏性情形下光滑解的已有结果的基础上，本文建立了无黏圣维南方程及相关的双曲型平衡律系统中周期滚波解的线性阻尼估计。此类阻尼估计由 $H^s$ 能量估计构成，可导出高阶导数范数被低阶导数范数指数式支配的结果，在超定或部分抛物型系统的行波非线性稳定性分析中起着关键作用——无论是在获得高频预解估计方面，还是在闭合非线性迭代过程中（在此类迭代中，现有的线性化稳定性估计似乎会损失正则性）均至关重要。本文针对规模为 $n \leq 6$ 的系统建立了一个类Lyapunov定理，指出只要满足严格的高频谱稳定性，则此类能量估计必然存在；而对于维数7及以上的情形，高频谱稳定性与本文所发展的这类能量估计的存在性之间可能存在一般性的差距。其中一项关键工具是一个依赖于维度的线性代数引理，该引理在形式上类似于常微分方程中的Lyapunov引理，据我们所知，此引理是全新的。

2. The centered maximal operator removes the non-concave Cantor part from

the gradient

中心极大算子消除了梯度中的非凹Cantor部分

作者: Panu Lahti, Julian Weigt

类别: math.CA, math.AP, 42B25, 26B30

发布日期: 2025-10-02

ArXiv ID: 2510.01936v1

链接: http://arxiv.org/abs/2510.01936v1

摘要翻译

中文标题: 中心极大算子消除了梯度中的非凹Cantor部分

摘要翻译: 我们研究定义在 $\mathbb{R}^d$（其中 $d \in \mathbb{N}$）上的有界变差函数 $f$ 的中心Hardy–Littlewood极大函数 $M f$ 的正则性。特别地，我们证明：在 $|D^c f|$-几乎处处使得 $f$ 具有非凹爆破的点 $x$ 处，均有 $M f(x) > f^*(x)$。进一步，由此我们推导出：若 $f$ 的变差测度不含跳跃部分，且其Cantor部分具有非凹爆破，则 $M f$ 的有界变差正则性可提升为Sobolev正则性。

3. On sharp Strichartz estimate for hyperbolic Schrödinger equation on

$\mathbb{T}^3$

关于三维环面 $\mathbb{T}^3$ 上双曲薛定谔方程的Sharp Strichartz估计

作者: Baoping Liu, Xu Zheng

类别: math.AP, math.CA, 35Q55

发布日期: 2025-10-02

ArXiv ID: 2510.01886v1

链接: http://arxiv.org/abs/2510.01886v1

摘要翻译

中文标题: 关于三维环面 $\mathbb{T}^3$ 上双曲薛定谔方程的Sharp Strichartz估计

摘要翻译: 本文通过结合几何测度论中的相交几何方法，证明了三维环面 $\mathbb{T}^3$ 上双曲薛定谔方程的Sharp Strichartz估计。作为应用，我们得到了非线性双曲薛定谔方程在局部适定性方面的最优结果。

4. Strichartz and dispersive estimates for quantum bouncing ball model:

exponential sums and Van der Corput methods in 1d semi-classical Schrödinger equations

量子反弹球模型的Strichartz与色散估计：一维半经典Schrödinger方程中的指数和与Van der Corput方法

作者: Oana Ivanovici

类别: math.AP, math.NT

发布日期: 2025-10-02

ArXiv ID: 2510.01779v1

链接: http://arxiv.org/abs/2510.01779v1

摘要翻译

中文标题: 量子反弹球模型的Strichartz与色散估计：一维半经典Schrödinger方程中的指数和与Van der Corput方法

摘要翻译: 本文研究定义在半直线上、具有线性势和Dirichlet边界条件的一维半经典Schrödinger方程。我们的主要目标是为该模型建立改进的色散估计和Strichartz估计，以刻画解的空间-时间行为。我们利用Van der Corput型导数检验方法，证明了更为精细的Strichartz估计，克服了以往结果中存在的1/4阶损失。此外，若假设某些指数和具有精确的上界估计，则我们的结果表明，这些损失有望进一步降至对任意$\epsilon>0$均成立的$1/6 + \epsilon$阶，这一结果将是尖锐的。我们还推测，在高维情形下，类似的Strichartz估计应在Friedlander模型区域内成立。

5. On dispersive decay for the generalized Korteweg–de Vries equation

关于广义Korteweg–de Vries方程的色散衰减性研究

作者: Matthew Kowalski, Minjie Shan

类别: math.AP, 35Q53

发布日期: 2025-10-02

ArXiv ID: 2510.01728v1

链接: http://arxiv.org/abs/2510.01728v1

摘要翻译

中文标题: 关于广义Korteweg–de Vries方程的色散衰减性研究

摘要翻译: 我们证明了广义Korteweg–de Vries（gKdV）方程解在时间上的逐点色散估计。特别地，对于质量临界情形的模型，我们仅假设初值数据属于 $\dot{H}^{\frac{1}{4}} \cap \dot{H}^{-\frac{1}{12}}$，并证明了解在 $L^\infty$ 范数下的衰减速率为 $|t|^{-\frac{1}{3}}$。为实现这一目标，我们建立了gKdV方程解的负正则性的保持性质，并将Lorentz–Strichartz估计推广至混合范数情形。

6. Bifurcation of global energy minimizers for a diffusion-aggregation

model on sphere

球面上扩散-聚集模型的全局能量极小解的分支现象

作者: Razvan C. Fetecau, Hansol Park, Vishnu Vaidya

类别: math.AP

发布日期: 2025-02-14

ArXiv ID: 2502.10337v2

链接: http://arxiv.org/abs/2502.10337v2

摘要翻译

中文标题: 球面上扩散-聚集模型的全局能量极小解的分支现象

摘要翻译: 我们考虑定义在单位球面 $\mathbb{S}^d$ 上概率密度上的自由能泛函，并研究其全局极小解。该能量由两部分构成：熵项与非局部相互作用项，分别倾向于促进扩散行为和聚集行为。我们找到了吸引相互作用强度的一个阈值，并在每种情况下确定了全局能量极小解。进一步，我们研究了在此阈值处发生的分支现象。此外，我们将结果推广至由任意多个球面构成的空间（例如平坦环面 $\mathbb{S}^1 \times \mathbb{S}^1$）。

7. Symmetry analysis and new partially invariant solutions for the gas

dynamics system with a special equation of state

具有特殊物态方程的气体动力学系统的对称性分析及新的部分不变解

作者: Dilara Siraeva, Irina A. Kogan

类别: math.AP, 35B06, 35A30

发布日期: 2025-10-01

ArXiv ID: 2510.01415v1

链接: http://arxiv.org/abs/2510.01415v1

摘要翻译

中文标题: 具有特殊物态方程的气体动力学系统的对称性分析及新的部分不变解

摘要翻译: 本文属于Ovsyannikov（1994年）提出的“子模型”（podmodeli）研究纲领框架下对气体动力学系统进行对称性分析的成果之一。我们考虑一种特殊的物态方程情形，即压力被规定为熵与密度任意函数之和。在此情形下，该系统具有一个12维的李对称代数。本工作在Siraeva（2024）研究的基础上，进一步推进了对该系统四维子代数的研究。针对Siraeva（2014）所给出最优列表中一大类此前未被研究且互不同构的四维子代数，我们计算出其完整的生成不变量集合。对于其中一个子代数，构造了相应的部分对称约化系统（即子模型），并显式求解了该约化后的系统。由此获得了原系统的若干新族显式解，并对这些解的轨迹进行了分析。此外，本文还将所考虑的每一个子代数与其同构类一一对应，为今后研究约化系统的层级结构奠定了基础。

不相关论文（基本信息）

1. Uniqueness in the Plateau problem for calibrated currents

标定链的Plateau问题解的唯一性

作者: Bryan Dimler, Chen-Kuan Lee

类别: math.DG, math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.02299v1

链接: http://arxiv.org/abs/2510.02299v1

摘要: We show that every compactly supported calibrated integral current with connected $C^{3,\alpha}$ boundary is the unique solution to the oriented Plateau problem for its boundary data. This is proved as a consequence of the boundary regularity theory for area-minimizing currents and classical unique continuation principles adapted to the minimal surface system.

2. Optimal Lieb-Thirring type inequalities for Schrödinger and Jacobi

operators with complex potentials

复势下Schrödinger算子与Jacobi算子的最佳Lieb-Thirring型不等式

作者: Sabine Bögli, Sukrid Petpradittha

类别: math.SP, math-ph, math.AP, math.MP, 47B36, 34L40, 47A10, 47A75

发布日期: 2025-10-02

ArXiv ID: 2510.02288v1

链接: http://arxiv.org/abs/2510.02288v1

摘要: We prove optimal Lieb-Thirring type inequalities for Schr"odinger and Jacobi operators with complex potentials. Our results bound eigenvalue power sums (Riesz means) by the $L^p$ norm of the potential, where in contrast to the self-adjoint case, each term needs to be weighted by a function of the ratio of the distance of the eigenvalue to the essential spectrum and the distance to the endpoint(s) thereof. Our Lieb-Thirring type bounds only hold for integrable weight functions. To prove optimality, we establish divergence estimates for non-integrable weight functions. The divergence rates exhibit a logarithmic or even polynomial gain compared to semiclassical methods (Weyl asymptotics) for real potentials.

3. Transfer of Stability from the Classical to the Fractional Anisotropic

Calderón Problem

稳定性从经典各向异性Calderón问题到分数阶各向异性Calderón问题的传递

作者: Hendrik Baers, Angkana Rüland

类别: math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.02242v1

链接: http://arxiv.org/abs/2510.02242v1

摘要: We discuss two spectral fractional anisotropic Calder'on problems with source-to-solution measurements and their quantitative relation to the classical Calder'on problem. Firstly, we consider the anistropic fractional Calder'on problem from [FGKU25]. In this setting, we quantify the relation between the local and nonlocal Calder'on problems which had been deduced in [R25] and provide an associated stability estimate. As a consequence, any stability result which holds on the level of the local problem with source-to-solution data has a direct nonlocal analogue (up to a logarithmic loss). Secondly, we introduce and discuss the fractional Calder'on problem with source-to-solution measurements for the spectral fractional Dirichlet Laplacian on open, bounded, connected, Lipschitz sets on $\mathbb{R}^n$. Also in this context, we provide a qualitative and quantitative transfer of uniqueness from the local to the nonlocal setting. As a consequence, we infer the first stability results for the principal part for a fractional Calder'on type problem for which no reduction of Liouville type is known. Our arguments rely on quantitative unique continuation arguments. As a result of independent interest, we also prove a quantitative relation between source-to-solution and Dirichlet-to-Neumann measurements for the classical Calder'on problem.

4. On Lieb-Thirring inequalities for multidimensional Schrödinger

operators with complex potentials

关于具有复势的多维薛定谔算子的Lieb-Thirring不等式研究

作者: Sabine Bögli, Sukrid Petpradittha, František Štampach

类别: math.SP, math-ph, math.AP, math.MP, 35P15, 47A10, 47A75, 81Q12

发布日期: 2025-10-02

ArXiv ID: 2510.02192v1

链接: http://arxiv.org/abs/2510.02192v1

摘要: We solve the open problem by Demuth, Hansmann, and Katriel announced in [Integr. Equ. Oper. Theory 75 (2013), 1-5] by a counter-example construction. The problem concerns a possible generalisation of the Lieb-Thirring inequality for Schr"odinger operators in to the case of complex-valued potentials. A counter-example has already been found for the one-dimensional case by the first and third authors in [J. Spectr. Theory 11 (2021), 1391-1413]. Here we generalise the counter-example to higher dimensions.

5. Sharp remainder terms and stability of weighted Hardy-Poincaré and

Heisenberg-Pauli-Weyl inequalities related to the Baouendi-Grushin operator

与Baouendi-Grushin算子相关的加权Hardy-Poincaré和Heisenberg-Pauli-Weyl不等式的精确余项及其稳定性

作者: Yerkin Shaimerdenov, Nurgissa Yessirkegenov, Amir Zhangirbayev

类别: math.AP, 26D10, 35J70

发布日期: 2025-08-22

ArXiv ID: 2508.16380v2

链接: http://arxiv.org/abs/2508.16380v2

摘要: In this paper, we obtain sharp remainder terms for the Hardy-Poincar'e inequalities with general non-radial weights in the setting of Baouendi-Grushin vector fields (see Theorem 2.5). It is worth emphasizing that all of our results are new both in the Baouendi-Grushin and standard Euclidean settings. The method employed allows us to not only unify, but also improve the results of Kombe and Yener [KY18] for any $1<p<\infty$ while holding true for complex-valued functions and providing explicit constants (Corollary 2.7). As a result, we are able to obtain sharp remainder terms to many known weighted Hardy-type inequalities (see Section 3.1). Aside from weighted Hardy-type inequalities, we also recover a sharp remainder formula for the $L^{p}$-Poincar'e inequality (Corollary 3.5). In the special case of radial weights, we are naturally able to introduce the notion of Baouendi-Grushin $p$-Bessel pairs (see Definition 2.9). Finally, we apply the technique to establish the sharp remainder term of the Heisenberg-Pauli-Weyl inequality in $L^{p}$ for $1<p<\infty$ (Corollary 3.13), which includes the sharp constant. This makes it possible to obtain the $L^{p}$-analogue for $2\leq p < n$ (Theorem 3.17) of a stability result by Cazacu, Flynn, Lam and Lu [CFLL24].

6. Equivalence between solvability of the Dirichlet and Regularity problem

under an $L^1$ Carleson condition on $\partial_t A$

在 $\partial_t A$ 满足 $L^1$ Carleson 条件下 Dirichlet 问题与正则性问题可解性的等价性

作者: Martin Ulmer

类别: math.AP, 35J25, 35J15

发布日期: 2025-09-12

ArXiv ID: 2509.10328v2

链接: http://arxiv.org/abs/2509.10328v2

摘要: We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that solvability of the Regularity problem in $\dot{W}^{1,p}$ implies solvability of the adjoint Dirichlet problem in $L^{p'}$. Previously, Shen (2007) established a partial reverse result. In our work, we show that if we assume an $L^1$-Carleson condition on only $|\partial_t A|$ the full reverse direction holds. As a result, we obtain equivalence between solvability of the Dirichlet problem $(D)^*_{p'}$ and the Regularity problem $(R)_p$ under this condition. As a further consequence, we can extend the class of operators for which the $L^p$ Regularity problem is solvable by operators satisfying the mixed $L^1-L^\infty$ condition. Additionally in the case of the upper half plane, this class includes operators satisfying this $L^1$-Carleson condition on $|\partial_t A|$.

7. Refined asymptotics for the Cauchy problem for the fast $p$-Laplace

evolution equation

快速 $p$-Laplace 演化方程柯西问题的精细渐近分析

作者: Matteo Bonforte, Iwona Chlebicka, Nikita Simonov

类别: math.AP

发布日期: 2024-05-08

ArXiv ID: 2405.05405v3

链接: http://arxiv.org/abs/2405.05405v3

摘要: Our focus is on the fast diffusion equation driven by the $p$-Laplacian operator, that is $\partial_t u=\Delta_p u$ with $1<p<2$, posed in the whole space $\mathbb{R}^N$, $N\geq 2$. The nonnegative solutions are expected to converge in time toward a stationary profile. While such convergence had been previously established for $p$ close to $2$, no quantitative rates were known, and the asymptotic behaviour remained poorly understood across the full fast diffusion range. In fact, the long time behaviour of solutions to the $p$-Laplace Cauchy problem drastically change in different subranges of the $p$. Some of them are analysed here for the first time. In this work, we provide the convergence rates for nonnegative, integrable solutions in the so-called good fast diffusion range, $p_c=\tfrac{2N}{N+1} <p<2$, where mass is conserved. We prove that solutions converge to a self-similar profile with matching mass, with explicit rates measured in relative error. Our constructive proof is based on a new entropy method that remains effective even when the entropy is not displacement convex – where optimal transport techniques fail. In the very fast diffusion range $1<p<p_c$, we give the first asymptotic analysis near the extinction time. We uncover new critical exponents – especially in high dimensions – that give rise to markedly different qualitative behaviour depending on the value of $p$. We also establish convergence rates for the gradients of radial solutions in the good fast diffusion range, again measured in relative error. Finally, we analyze the structural properties required for the entropy method to apply, thereby opening a broader investigation into the basin of attraction of Barenblatt-type profiles, particularly in the singular case of $p$ close to $1$.

8. Blow up dynamics for the 3D energy-critical Nonlinear Schrödinger

equation

三维临界能量非线性薛定谔方程的爆破动力学

作者: Tobias Schmid

类别: math.AP

发布日期: 2023-08-03

ArXiv ID: 2308.01883v3

链接: http://arxiv.org/abs/2308.01883v3

摘要: We construct a two-parameter continuum of type II blow up solutions for the energy-critical focusing NLS in dimension $ d = 3$. The solutions collapse to a single energy bubble in finite time, precisely they have the form $ u(t,x) = e^{i \alpha(t)}\lambda(t)^{\frac{1}{2}}W(\lambda(t) x) + \eta(t, x )$, $ t \in[0, T)$, $ x \in \mathbb{R}^3$, where $ W( x) = \big( 1 + \frac{|x|^2}{3}\big)^{-\frac{1}{2}}$ is the ground state solution, $\lambda(t) = (T-t)^{- \frac12 - \nu} $ for suitable $ \nu > 0 $, $ \alpha(t) = \alpha_0 \log(T - t)$ and $ T= T(\nu, \alpha_0) > 0 $. Further $ \|\eta(t) - \eta_T\|_{\dot{H}^1 \cap \dot{H}^2} = o(1)$ as $ t \to T^-$ for some $ \eta_T \in \dot{H}^{1} \cap~ \dot{H}^2$.

9. Low regularity Sobolev well-posedness for Vlasov–Poisson

Vlasov–Poisson方程的低正则性Sobolev适定性

作者: In-Jee Jeong, Sangwook Tae

类别: math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.02112v1

链接: http://arxiv.org/abs/2510.02112v1

摘要: We consider the Vlasov–Poisson equation on $\mathbb{R}^n \times \mathbb{R}^n$ with $n \ge 3$. We prove local well-posedness in $H^{s}(\mathbb{R}^n \times \mathbb{R}^n)$ with $s> n/2-1/4$, for initial distribution $f_{0} \in H^{s}(\mathbb{R}^n \times \mathbb{R}^n)$ having compact support in $v$. In particular, data not belonging to $L^p(\mathbb{R}^n \times \mathbb{R}^n)$ for large $p$ are allowed.

10. Numerical Approximation of the Critical Value of Eikonal Hamilton-Jacobi

Equations on Networks

网络上程函型哈密顿-雅可比方程临界值的数值逼近

作者: Valentina Coscetti, Marco Pozza

类别: math.NA, cs.NA, math.AP, 35R02, 65M15, 49L25, 35B40

发布日期: 2025-02-28

ArXiv ID: 2502.20993v2

链接: http://arxiv.org/abs/2502.20993v2

摘要: The critical value of an eikonal equation is the unique value of a parameter for which the equation admits solutions and is deeply related to the effective Hamiltonian of a corresponding homogenization problem. We study approximation strategies for the critical value of eikonal equations posed on networks. They are based on the large time behavior of corresponding time-dependent Hamilton-Jacobi equations. We provide error estimates and some numerical tests, showing the performance and the convergence properties of the proposed algorithms.

11. Perturbation theory for the parabolic Regularity and Neumann problem

抛物型正则性与诺伊曼问题的摄动理论

作者: Martin Ulmer

类别: math.AP, 35K10, 35K20

发布日期: 2024-08-22

ArXiv ID: 2408.12529v2

链接: http://arxiv.org/abs/2408.12529v2

摘要: We show small and large Carleson perturbation results for the parabolic Regularity boundary value problem with boundary data in $\dot{L}_{1,1/2}^p$ and small Carelson perturbation results for the Neumann problem with boundary data in $L^p$. The operator we consider is $L:=\partial_t -\mathrm{div}(A\nabla\cdot)$ and the domains are parabolic cylinders $\Omega=\mathcal{O}\times\mathbb{R}$, where $\mathcal{O}$ is a Lipschitz domain.

12. Geometry of wave damping on the torus

环面上的波阻尼几何

作者: Kiril Datchev, Perry Kleinhenz, Antoine Prouff

类别: math.AP, math-ph, math.MP

发布日期: 2025-09-05

ArXiv ID: 2509.05239v2

链接: http://arxiv.org/abs/2509.05239v2

摘要: Energy decay rates of damped waves on the torus depend on the behavior of the damping near the undamped region and on the geometry of the damped set. In this paper we refine these geometric considerations, by introducing the concept of order of a glancing undamped point, and estimating decay rates in terms of this order. The proof is based on generalizing an averaging argument due to Sun. We also show that damping sets which attain these improvements are generic among polygons and smooth curves.

13. Perturbations of Minkowski spacetime with regular conformal

compactification

具有正则共形紧致化的闵可夫斯基时空的扰动

作者: Andrea Nützi

类别: gr-qc, math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.01964v1

链接: http://arxiv.org/abs/2510.01964v1

摘要: We construct perturbations of Minkowski spacetime in general relativity, when given initial data that decays inverse polynomially to initial data of a Kerr spacetime towards spacelike infinity. We show that the perturbations admit a regular conformal compactification at null and timelike infinity, where the degree of regularity increases linearly with the rate of decay of the initial data to Kerr initial data. In particular, the compactification is smooth if the initial data decays rapidly to Kerr initial data. This generalizes results of Friedrich, who constructed spacetimes with a smooth conformal compactification in the case when the initial data is identical to Kerr initial data on the complement of a compact set. Our results rely on a novel formulation of the Einstein equations about Minkowski spacetime introduced by the author, that allows one to formulate the dynamic problem as a quasilinear, symmetric hyperbolic PDE that is regular at null infinity and with null infinity being at a fixed locus. It is not regular at spacelike infinity, due to the asymptotics of Kerr. Thus the main technical task is the construction of solutions near spacelike infinity, using tailored energy estimates. To accomplish this, we organize the equations according to homogeneity with respect to scaling about spacelike infinity, which identifies terms that are leading, respectively lower order, near spacelike infinity, with contributions from Kerr being lower order.

14. Mixed Christoffel-Minkowski problems for bodies of revolution

旋转体的混合Christoffel-Minkowski问题

作者: Leo Brauner, Georg C. Hofstätter, Oscar Ortega-Moreno

类别: math.MG, math.AP, math.DG, 52A39, 52A20, 52B45

发布日期: 2025-08-13

ArXiv ID: 2508.09794v3

链接: http://arxiv.org/abs/2508.09794v3

摘要: The mixed Christoffel-Minkowski problem asks for necessary and sufficient conditions for a Borel measure on the Euclidean unit sphere to be the mixed area measure of some convex bodies, one of which, appearing multiple times, is free and the rest are fixed. In the case where all bodies involved are symmetric around a common axis, we provide a complete solution to this problem, without assuming any regularity. In particular, we refine Firey’s classification of area measures of figures of revolution. In our argument, we introduce an easy way to transform mixed area measures and mixed volumes involving axially symmetric bodies, and we significantly improve Firey’s estimate on the local behavior of area measures. As a secondary result, we obtain a family of Hadwiger type theorems for convex valuations that are invariant under rotations around an axis.

15. Subwavelength resonances in two-dimensional elastic media with high

contrast

高对比度二维弹性介质中的亚波长共振

作者: Yuanchun Ren, Yixian Gao

类别: math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.01911v1

链接: http://arxiv.org/abs/2510.01911v1

摘要: This paper employs layer potential techniques to investigate wave scattering in two-dimensional elastic media exhibiting high contrasts in both Lam'{e} parameters and density. Our contributions are fourfold. First, we construct an invertible operator based on the kernel spaces of boundary integral operators, which enables the characterization of resonant frequencies through an orthogonality condition. Second, we use asymptotic analysis to derive the equation governing the leading-order terms of these resonant frequencies. Third, we analyze the scattered field in the interior domain for incident frequencies across different regimes and characterize the longitudinal and transverse far-field patterns in the exterior domain. Finally, we examine the subwavelength bandgap in the phononic crystal with a dilute structure.

16. Reconstruction in the Calderón problem on a fixed partition from

finite and partial boundary data

基于有限且部分边界数据在固定划分下的Calderón问题的重构

作者: Henrik Garde

类别: math.AP, cs.NA, math.NA, 35R30, 35Q60, 35R05, 47H05

发布日期: 2025-07-25

ArXiv ID: 2507.19410v2

链接: http://arxiv.org/abs/2507.19410v2

摘要: This short note modifies a reconstruction method by the author (Comm.~PDE, 45(9):1118–1133, 2020), for reconstructing piecewise constant conductivities in the Calder'on problem (electrical impedance tomography). In the former paper, a layering assumption and the local Neumann-to-Dirichlet map were needed since the piecewise constant partition also was assumed unknown. Here I show how to modify the method in case the partition is known, for general piecewise constant conductivities and only a finite number of partial boundary measurements. Moreover, no lower/upper bounds on the unknown conductivity are needed.

17. A note on the recovery sequence in the double gradient model for phase

transitions

关于相变双梯度模型中恢复序列的一点注记

作者: Jakob Deutsch

类别: math.AP, Primary: 49J45, Secondary: 35Q74, 49Q20, 74N99

发布日期: 2025-10-02

ArXiv ID: 2510.01893v1

链接: http://arxiv.org/abs/2510.01893v1

摘要: We investigate the $\limsup$ inequality in the double gradient model for phase transitions governed by a Modica–Mortola functional with a double-well potential in two dimensions. Specifically, we consider energy functionals of the form

$$ E_\varepsilon(u, \Omega) = \int_\Omega \left( \frac{1}{\varepsilon} W(\nabla u) + \varepsilon |\nabla^2 u|^2 \right) dx $$

for maps $ u \in H^2(\Omega; \mathbb{R}^2) $, where $ W $ vanishes only at two wells. Assuming a bound on the optimal profile constant – namely the cell problem on the unit cube – in terms of the geodesic distance between the two wells, we characterise the limiting interfacial energy via periodic recovery sequences as $\varepsilon \to 0^+$.

18. Monotonicity and Liouville-type theorems for semilinear elliptic

problems in the half space

半空间中半线性椭圆问题的单调性与刘维尔型定理

作者: Berardino Sciunzi, Domenico Vuono

类别: math.AP, 35B06, 35B51, 35J61

发布日期: 2025-10-02

ArXiv ID: 2510.01865v1

链接: http://arxiv.org/abs/2510.01865v1

摘要: We consider classical solutions to $-\Delta u = f(u)$ in half-spaces, under homogeneous Dirichlet boundary conditions. We prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary, provided that it is directionally bounded on finite strips. As a corollary, we deduce a new Liouville-type theorem for the Lane-Emden equation.

19. Efficient manifold evolution algorithm using adaptive B-Spline

interpolation

基于自适应B样条插值的高效流形演化算法

作者: Muhammad Ammad, Leevan Ling

类别: math.NA, cs.NA, math.AP, math.DG

发布日期: 2025-10-02

ArXiv ID: 2510.01790v1

链接: http://arxiv.org/abs/2510.01790v1

摘要: This paper explores an efficient Lagrangian approach for evolving point cloud data on smooth manifolds. In this preliminary study, we focus on analyzing plane curves, and our ultimate goal is to provide an alternative to the conventional radial basis function (RBF) approach for manifolds in higher dimensions. In particular, we use the B-Spline as the basis function for all local interpolations. Just like RBF and other smooth basis functions, B-Splines enable the approximation of geometric features such as normal vectors and curvature. Once properly set up, the advantage of using B-Splines is that their coefficients carry geometric meanings. This allows the coefficients to be manipulated like points, facilitates rapid updates of the interpolant, and eliminates the need for frequent re-interpolation. Consequently, the removal and insertion of point cloud data become seamless processes, particularly advantageous in regions experiencing significant fluctuations in point density. The numerical results demonstrate the convergence of geometric quantities and the effectiveness of our approach. Finally, we show simulations of curvature flows whose speeds depend on the solutions of coupled reaction–diffusion systems for pattern formation.

20. Notes on Schauder estimates by scaling for elliptic PDEs in divergence

form

关于散度形式椭圆型偏微分方程通过尺度变换得到的Schauder估计的注记

作者: Stefano Vita

类别: math.AP, 35B65, 35B44, 35B45, 35B53, 35B30, 35B08

发布日期: 2025-10-02

ArXiv ID: 2510.01765v1

链接: http://arxiv.org/abs/2510.01765v1

摘要: These are the notes of a part of the PhD course Regularity for free boundary problems and for elliptic PDEs, held in Pavia in the spring of 2025. The aim is to provide a comprehensive and self-contained treatment of classical interior and local Schauder estimates for second-order linear elliptic PDEs in divergence form via scaling in the spirit of Simon’s work. The main techniques presented here are geometric in nature and were primarily developed in the study of geometric problems such as minimal surfaces. The adopted approach relies on compactness and blow-up arguments, combined with rigidity results (Liouville theorems), and shares many features with the one used in the study of free boundary problems, which was the main topic of the other part of the PhD course.

21. Nonlinear Forward-Backward Problems

非线性前向-后向问题

作者: Anne-Laure Dalibard, Frédéric Marbach, Jean Rax

类别: math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.01732v1

链接: http://arxiv.org/abs/2510.01732v1

摘要: We prove the existence and uniqueness of strong solutions to the equation $u u_x - u_{yy} = f$ in the vicinity of the linear shear flow, subject to perturbations of the source term and lateral boundary conditions. Since the solutions we consider have opposite signs in the lower and upper half of the domain, this is a quasilinear forward-backward parabolic problem, which changes type across a critical curved line within the domain. In particular, lateral boundary conditions can be imposed only where the characteristics are inwards. There are several difficulties associated with this problem. First, the forward-backward geometry depends on the solution itself. This requires to be quite careful with the approximation procedure used to construct solutions. Second, and more importantly, the linearized equations solved at each step of the iterative scheme admit a finite number of singular solutions, of which we provide an explicit construction. This is similar to well-known phenomena in elliptic problems in nonsmooth domains. Hence, the solutions to the equation are regular if and only if the source terms satisfy a finite number of orthogonality conditions. A key difficulty of this work is to cope with these orthogonality conditions during the nonlinear fixed-point scheme. In particular, we are led to prove their stability with respect to the underlying base flow. To tackle this deceivingly simple problem, we develop a methodology which we believe to be both quite natural and adaptable to other situations in which one wishes to prove the existence of regular solutions to a nonlinear problem for suitable data despite the existence of singular solutions at the linear level. This paper is a shorter version of [3].

22. The weighted isoperimetric inequality and Sobolev inequality outside

convex sets

凸集外部的加权等周不等式与索博列夫不等式

作者: Lu Chen, Jiali Lan

类别: math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.01647v1

链接: http://arxiv.org/abs/2510.01647v1

摘要: In this paper, we establish a weighted capillary isoperimetric inequality outside convex sets using the $\lambda_w$-ABP method. The weight function $w$ is assumed to be positive, even, and homogeneous of degree $\alpha$, such that $w^{1/\alpha}$ is concave on $\R^n$. Based on the weighted isoperimetric inequality, we develop a technique of capillary Schwarz symmetrization outside convex sets, and establish a weighted P'{o}lya-Szeg"{o} principle and a sharp weighted capillary Sobolev inequality outside convex domain. Our result can be seen as an extension of the weighted Sobolev inequality in the half-space established by Ciraolo-Figalli-Roncoroni in \cite{CFR}.

23. On the attainment of boundary data in variational problems with linear

growth

关于线性增长变分问题中边界数据的可达性

作者: David Meyer

类别: math.AP

发布日期: 2025-10-01

ArXiv ID: 2510.01515v1

链接: http://arxiv.org/abs/2510.01515v1

摘要: It is well-known that convex variational problems with linear growth and Dirichlet boundary conditions might not have minimizers if the boundary condition is not suitably relaxed. We show that for a wide range of integrands, including the least gradient problem and the non-parametric Plateau problem, and under suitable mean-convexity conditions of the boundary, minimizers of the relaxed problem attain the boundary data in the trace sense if it lies in $BV$ or $W^{\alpha,p}$ with $\alpha p\geq 2$ without any kind of continuity assumption. Unlike previous works, our methods are also able to treat systems under a certain quasi-isotropy assumption on the integrand. We further show that without this quasi-isotropy assumption, smooth counterexamples on uniformly convex domains exist. Further applications to the uniqueness of minimizers and to open problems about the ROF functional with Dirichlet boundary conditions, and to the trace space of functions of least gradient are given.

24. Correlation estimates for Brownian particles with singular interactions

具有奇异相互作用的布朗粒子的相关性估计

作者: Mitia Duerinckx, Pierre-Emmanuel Jabin

类别: math.AP, math-ph, math.MP, math.PR

发布日期: 2025-10-01

ArXiv ID: 2510.01507v1

链接: http://arxiv.org/abs/2510.01507v1

摘要: We study particle systems with singular pairwise interactions and non-vanishing diffusion in the mean-field scaling. A classical approach to describing corrections to mean-field behavior is through the analysis of correlation functions. For bounded interactions, the optimal estimates on correlations are well known: the $m$-particle correlation function is $G_{N,m}=O(N^{1-m})$ for all $m$. Such estimates, however, have remained out of reach for more singular interactions. In this work, we develop a new framework based on linearized correlation functions, which allows us to derive robust bounds for systems with merely square-integrable interaction kernels, providing the first systematic control of correlations in the singular setting. Although at first not optimal, our estimates can be partially refined a posteriori using the BBGKY hierarchy: in the case of bounded interactions, our method recovers the known optimal estimates with a simplified argument. As key applications, we establish the validity of the Bogolyubov correction to mean field and prove a central limit theorem for the empirical measure, extending these results beyond the bounded interaction regime for the first time.

25. Localized Pattern Formation and Oscillatory Instabilities in a

Three-component Gierer Meinhardt Model

三组分Gierer-Meinhardt模型中的局域斑图形成与振荡不稳定性

作者: Chunyi Gai, Fahad Al Saadi

类别: math.AP, math.DS

发布日期: 2025-10-01

ArXiv ID: 2510.01401v1

链接: http://arxiv.org/abs/2510.01401v1

摘要: In this paper, we introduce a three-component Gierer-Meinhardt model in the semi-strong interaction regime, characterized by an asymptotically large diffusivity ratio. A key feature of this model is that the interior spike can undergo Hopf bifurcations in both amplitude and position, leading to rich oscillatory dynamics not present in classical two-component systems. Using asymptotic analysis and numerical path-following, we construct localized spike equilibria and analyze spike nucleation that occurs through slow passage beyond a saddle-node bifurcation. Moreover, stability of spike equilibrium is analyzed by introducing time-scaling parameters, which reveal two distinct mechanisms: amplitude oscillations triggered by large-eigenvalue instabilities and oscillatory spike motion associated with small eigenvalues. Numerical simulations illustrate these dynamics and their transition regimes. This dual mechanism highlights richer spike behavior in three-component systems and suggests several open problems for future study.

26. Mean curvature flow through singularities

通过奇点的平均曲率流

作者: Robert Haslhofer

类别: math.DG, math.AP

发布日期: 2025-10-01

ArXiv ID: 2510.01355v1

链接: http://arxiv.org/abs/2510.01355v1

摘要: We first give a general introduction to the mean curvature flow, and then discuss fundamental results established over the last 10 years that yield a precise theory for the flow through singularities in $\mathbb{R}^3$. With the aim of developing a satisfying theory in higher dimensions, we then describe our recent classification of all noncollapsed singularities in $\mathbb{R}^4$. Finally, we provide a detailed discussion of open problems and conjectures.

27. The noncommutative KP hierarchy and its solution via descent algebra

非交换KP层次结构及其通过下降代数的求解

作者: Gordon Blower, Simon J. A. Malham

类别: nlin.SI, math-ph, math.AP, math.MP

发布日期: 2025-10-01

ArXiv ID: 2510.01352v1

链接: http://arxiv.org/abs/2510.01352v1

摘要: We give the solution to the complete noncommutative Kadomtsev–Petviashvili (KP) hierarchy. We achieve this via direct linearisation which involves the Gelfand–Levitan–Marchenko (GLM) equation. This is a linear integral equation in which the scattering data satisfies the linearised KP hierarchy. The solution to the GLM equation is then shown to coincide with the solution to the noncommutative KP hierarchy. We achieve this using two approaches. In the first approach we use the standard Sato-Wilson dressing transformation. In the second approach, which was pioneered by Poppe, we assume the scattering data is semi-additive and by direct substitution, we show that the solution to the GLM equation satisfies the infinite set of field equations representing the noncommutative KP hierarchy. This approach relies on the augmented pre-Poppe algebra. This is a representative algebra that underlies the field equations representing the hierarchy. It is nonassociative and isomorphic to a descent algebra equipped with a grafting product. While we perform computations in the nonassociative descent algebra, the final result which establishes the solution to the complete hierarchy, resides in the natural associative subalgebra. The advantages of this second approach are that it is constructive, explicit, highlights the underlying combinatorial structures within the hierarchy, and reveals the mechanisms underlying the solution procedure.

28. Existence and Uniqueness of Solutions to Nonlinear Diffusion with Memory

具有记忆效应的非线性扩散方程解的存在性与唯一性

作者: Yixian Chen

类别: math.AP, [2020] 35K55 35B45 35D30 35R60

发布日期: 2025-07-22

ArXiv ID: 2507.16659v2

链接: http://arxiv.org/abs/2507.16659v2

摘要: This paper studies a nonlinear diffusion equation with memory:

$$ u_t=\nabla\cdot \big( D(x)\cdot\int_0^t K(t-s) \nabla\cdot\Phi(u(x,s))ds \big)+f(x,t) $$

Where $K$ is memory Kernel and $D(x)$ is bounded. Under monotonicity and growth conditions on $\Phi$, the existence and uniqueness of weak solution is established. The analysis employs Orthogonal approximation, energy estimates, and monotone operator theory. The convolution structure is handled within variational frameworks. The result provides a basis for studying memory-type diffusion.

29. The $L_p$ chord Minkowski problem for super-critical exponent

超临界指数的 $L_p$ 弦Minkowski问题

作者: Shibing Chen, Qi-Rui Li, Yuanyuan Li

类别: math.AP, 35A15, F.2.2; G.1.8

发布日期: 2025-10-01

ArXiv ID: 2510.01084v1

链接: http://arxiv.org/abs/2510.01084v1

摘要: The $L_p$ chord Minkowski problem was recently introduced by Lutwak, Xi, Yang and Zhang, which seeks to determine the necessary and sufficient conditions for a given finite Borel measure such that it is the $L_p$ chord measure of a convex body. In this paper, we solve the $L_p$ chord Minkowski problem for the super-critical exponents by combining a nonlocal Gauss curvature flow introduced in \cite{HHLW exi} and a topological argument developed in \cite{GLW2022}. Notably, we provide a simplified argument for the topological part.

30. The inhomogeneous fractional Dirichlet problem

非齐次分数阶狄利克雷问题

作者: Florian Grube

类别: math.AP, 47G20, 35B65, 35S15, 35R09, 60G52

发布日期: 2025-10-01

ArXiv ID: 2510.01055v1

链接: http://arxiv.org/abs/2510.01055v1

摘要: We study boundary regularity for the inhomogeneous Dirichlet problem for $2s$-stable operators in generalized H"older spaces. Moreover, we provide explicit counterexamples that showcase the sharpness of our results. Our approach directly addresses the inhomogeneous Dirichlet problem, rather than subtracting an appropriate extension of the exterior data. Even for the fractional Laplacian, our result is new.

31. Funnel control for passive infinite-dimensional systems

被动无穷维系统的漏斗控制

作者: Thavamani Govindaraj, Anthony Hastir, Lassi Paunonen, Timo Reis

类别: math.OC, math.AP, math.FA

发布日期: 2025-10-01

ArXiv ID: 2510.01027v1

链接: http://arxiv.org/abs/2510.01027v1

摘要: We consider funnel control for linear infinite-dimensional systems that are impedance passive, meaning that they satisfy an energy balance in which the stored energy equals the squared norm of the state and the supplied power is the inner product of input and output. For the analysis we employ the system node approach, which offers a unified framework for infinite-dimensional systems with boundary and distributed control and observation. The resulting closed-loop dynamics are governed by a nonlinear evolution equation; we establish its solvability and hence the applicability of funnel control to this class. The applicability is illustrated by an Euler-Bernoulli beam, which is studied in two distinct scenarios: once with boundary control and once with distributed control.

32. Calderón-Zygmund estimates for double phase problems with matrix

weights

具有矩阵权的双相问题的Calderón-Zygmund估计

作者: Sun-Sig Byun, Yumi Cho, Seungjin Ryu

类别: math.AP, 35B65, 35J70, 35J75

发布日期: 2025-05-27

ArXiv ID: 2505.20856v2

链接: http://arxiv.org/abs/2505.20856v2

摘要: We establish an optimal Calder'{o}n-Zygmund theory for nonuniformly elliptic double phase problems with matrix weights. For $1<p<q<\infty$, $a(\cdot)\in C^{0,\alpha}(\Omega)$ ($0<\alpha\le1$), and a symmetric, almost everywhere positive definite matrix weight $\M$ with $|\M(x)|\,|\M(x)^{-1}|\le\Lambda$ for some constant $\Lambda\ge 1$ and small $|\log \M|_{\mathrm{BMO}}$, we prove, for every $\gamma>1$,

$$ (|\M F|^p+a(x)|\M F|^q)\in L^\gamma_{\mathrm{loc}} \;\Longrightarrow\; (|\M Du|^p+a(x)|\M Du|^q)\in L^\gamma_{\mathrm{loc}}. $$

Our argument combines a freezing of the logarithm of the matrix field, $\log \M$, with a fractional maximal-operator method governed by the Muckenhoupt-Wheeden $\mathcal{A}_{p,s}$ classes (where $1/s=1/p-\alpha/(nq)$). This yields scale-invariant comparison and level-set estimates and precludes Lavrentiev gaps at the sharp threshold $q/p\le 1+\alpha/n$. Our result recovers the identity case $\,\M\equiv {\rm I}_n\,$, i.e., the classical (unweighted) Calder'{o}n-Zygmund theory for double-phase problems.

33. A Unified Hölder Lebesgue Framework for Caffarelli Kohn Nirenberg

Inequalities

Caffarelli-Kohn-Nirenberg不等式的统一Hölder-Lebesgue框架

作者: Mengxia Dong

类别: math.AP, math.FA

发布日期: 2025-10-01

ArXiv ID: 2510.00949v1

链接: http://arxiv.org/abs/2510.00949v1

摘要: We develop a unified H"older Lebesgue scale $X^p$ and its weighted, higher order variants $X^{k,p,a}$ to extend the Caffarelli Kohn Nirenberg (CKN) inequality beyond the classical Lebesgue regime. Within this framework we prove a two parameter interpolation theorem that is continuous in the triplet $(k,1/p,a)$ and bridges integrability and regularity across the Lebesgue H"older spectrum. As a consequence we obtain a generalized CKN inequality on bounded punctured domains $\Omega\subset\mathbb{R}^n\setminus\{0\}$; the dependence of the constant on $\Omega$ is characterized precisely by the (non)integrability of the weights at the origin. At the critical endpoint $p=n$ we establish a localized, weighted Brezis Wainger type bound via Trudinger Moser together with a localized weighted Hardy lemma, yielding an endpoint CKN inequality with a logarithmic loss. Sharp constants are not pursued; rather, we prove existence of constants depending only on the structural parameters and coarse geometry of $\Omega$. Several corollaries, including a unified Hardy–Sobolev inequality, follow from the same interpolation mechanism.

34. Stochastic flows for Hölder drifts and transport/continuity equations

with noise

针对Hölder漂移及带噪声的输运/连续性方程的随机流

作者: Magnus C. Ørke

类别: math.PR, math.AP, 60H10, 60H15, 35B65

发布日期: 2025-01-03

ArXiv ID: 2501.01759v2

链接: http://arxiv.org/abs/2501.01759v2

摘要: We prove existence of a stochastic flow of diffeomorphisms generated by SDEs with drift in $L^q_t C^{0, \alpha}_x$ for any $q \in [2, \infty)$ and $\alpha \in (0, 1)$. This result is achieved using a Zvonkin-type transformation for the SDE. As a key intermediate step, well-posedness and optimal regularity for a class of parabolic PDEs related to the transformation is established. Using the existence of a differentiable stochastic flow, we prove well-posedness of $BV_\text{loc}$-solutions of stochastic transport equations and weak solutions of stochastic continuity equations with so-called transport noise and velocity fields in $L^q_t C^{0, \alpha}_x$. For both equations, solutions may fail to be unique in the deterministic setting.

35. Exponential Dichotomies in Higher-Dimensional Spatial Dynamics for

Elliptic Partial Differential Equations

椭圆型偏微分方程在高维空间动力学中的指数二分性

作者: Margaret Beck, Ryan Goh, Alanna Haslam-Hyde

类别: math.AP, math.DS, 35A24, 34D09, 37L10, 35B36

发布日期: 2025-10-01

ArXiv ID: 2510.00917v1

链接: http://arxiv.org/abs/2510.00917v1

摘要: Exponential dichotomies, when they exist, provide powerful information about the structure of bounded solutions even in the case of an ill-posed evolutionary equation. The method of spatial dynamics, in which one views a spatial variable as a time-like evolutionary variable, allows for the use of classical dynamical systems techniques, such as exponential dichotomies, in broader contexts. This has been utilized to study stationary, traveling wave, time-periodic, and spiral wave solutions of PDEs on spatial domains with a distinguished unbounded direction (e.g. the real line or a channel of the form $\mathbb{R}\times\Omega$). Recent work has shown how to extend the spatial dynamics framework to elliptic PDEs posed on general multi-dimensional spatial domains. In this paper, we show that, in the same context, exponential dichotomies exist, thus allowing for their use in future analyses of coherent structures, such as spatial patterns in reaction-diffusion equations on more general domains.

36. Existence of multiple normalized solutions to a critical growth Choquard

equation involving mixed operator

涉及混合算子的临界增长Choquard方程多个归一化解的存在性

作者: Nidhi Nidhi, K. Sreenadh

类别: math.AP

发布日期: 2025-10-01

ArXiv ID: 2510.00893v1

链接: http://arxiv.org/abs/2510.00893v1

摘要: In this paper we study the normalized solutions of the following critical growth Choquard equation with mixed local and non-local operators: \begin{equation*} \begin{array}{rcl} -\Delta u +(-\Delta)^s u & = & \lambda u +\mu |u|^{p-2}u +(I_{\alpha}|u|^{2^{\alpha}})|u|^{2^*{\alpha}-2}u \text{ in } \mathbb{R}^N;;; \left| u \right|_2 & = & \tau, \end{array} \end{equation*} here $N\geq 3$, $\tau>0$, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (0,N)$, $2^*_{\alpha}=\frac{N+\alpha}{N-2}$ is the critical exponent corresponding to the Hardy Littlewood Sobolev inequality, $(-\Delta)^s$ is the non-local fractional Laplacian operator with $s\in (0,1)$, $\mu>0$ is a parameter and $\lambda$ appears as Lagrange multiplier. We have shown the existence of atleast two distinct solutions in the presence of mass subcritical perturbation, $\mu |u|^{p-2}u$ with $2<p<2+\frac{4s}{N}$ under some assumptions on $\tau$.

37. Compactness of conformal metrics with constant $Q$-curvature of higher

order

高阶常 $Q$-曲率共形度量的紧性

作者: Saikat Mazumdar, Bruno Premoselli

类别: math.AP, math.DG

发布日期: 2025-10-01

ArXiv ID: 2510.00888v1

链接: http://arxiv.org/abs/2510.00888v1

摘要: Let $k\ge1$ be a positive integer and let $P_g$ be the GJMS operator $P_{g}$ of order $2k$ on a closed Riemannian manifold $(M,g)$ of dimension $n>2k$. We investigate the compactness of the set of conformal metrics to $g$ with prescribed constant positive $Q$-curvature of order $2k$- or, equivalently, of the set of positive solutions for the $2k$-th order $Q$-curvature equation. Under a natural positivity-preserving condition on $P_{g}$ we establish compactness, for an arbitrary $1 \le k < \frac{n}{2}$, under the following assumptions: $(M,g)$ is locally conformally flat and $P_g$ has positive mass in $M$, or $2k+1 \le n \le 2k+5$ and $P_g$ has positive mass in $M$, or $n \ge 2k+4$ and $|\text{W}_g|_g >0$ in $M$. For an arbitrary $1 \le k < \frac{n}{2}$, the expression of $P_g$ is not explicit, which is an obstacle to proving compactness. We overcome this by relying on Juhl’s celebrated recursive formulae for $P_g$ to perform a refined blow-up analysis for solutions of the $Q$-curvature equation and to prove a Weyl vanishing result for $P_g$. This is the first compactness result for an arbitrary $1 \le k < \frac{n}{2}$ and the first successful instance where Juhl’s formulae are used to yield compactness. Our result also hints that the threshold dimension for compactness for the $2k$-th order $Q$-curvature equation diverges as $k \to + \infty$.

38. Minimality of level sets in phase transitions

相变中水平集的极小性

作者: Dimitrios Gazoulis

类别: math.AP, math.DG

发布日期: 2025-03-04

ArXiv ID: 2503.02604v3

链接: http://arxiv.org/abs/2503.02604v3

摘要: In this work we study the level sets of entire solutions of the Allen-Cahn equation and we prove minimality of the zero level set with respect to a certain perimeter functional with density. This provides a direct relationship between phase transition type problems and minimal surfaces with some weight. In addition, we obtain that the zero level set of entire solutions of the Allen-Cahn equation has zero mean curvature. As an application, we establish the De Giorgi conjecture in it’s original statement, without the limiting assumption of O. Savin, by reducing it to classical Bernstein type results for minimal graphs, thus directly linking it to the geometric problem.