arXiv论文分析报告

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1. Nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes Equation

无外力作用下三维不可压缩Navier-Stokes方程Leray-Hopf解的非唯一性

作者: Thomas Hou, Yixuan Wang, Changhe Yang

类别: math.AP

发布日期: 2025-09-29

ArXiv ID: 2509.25116v1

链接: http://arxiv.org/abs/2509.25116v1

详细分析

中文标题: 无外力作用下三维不可压缩Navier-Stokes方程Leray-Hopf解的非唯一性

1. 研究对象和背景

本文研究的是无外力作用下的三维不可压缩Navier-Stokes方程（3D incompressible Navier-Stokes equations without forcing）在全空间 $\mathbb{R}^3$ 上的弱解，特别是 Leray–Hopf 解的非唯一性问题。

该方程组如下：

$$ \begin{cases} \partial_t u + u \cdot \nabla u - \Delta u + \nabla p = 0, \\ \mathrm{div}\, u = 0, \\ u(t=0, x) = u_{\mathrm{in}}(x). \end{cases} \tag{1.1} $$

其中 $u: \mathbb{R}^3 \times [0,T) \to \mathbb{R}^3$ 是速度场，$p: \mathbb{R}^3 \times [0,T) \to \mathbb{R}$ 是压力项，初始数据 $u_{\mathrm{in}}$ 为散度自由（divergence-free），且具有紧支集，属于局部 $L^q$ 空间（对任意 $q < 3$），特别地包含 $L^2_{\mathrm{loc}}$。

Leray–Hopf 解是满足能量不等式的一类全局弱解，定义在函数空间：

$$ u \in L^\infty_{\mathrm{loc}}([0,T); L^2_\sigma(\mathbb{R}^3)) \cap L^2_{\mathrm{loc}}([0,T); H^1_\sigma(\mathbb{R}^3)), $$

并满足分布意义下的动量方程与能量不等式：

$$ \frac{1}{2} \|u(t)\|_{L^2}^2 + \int_0^t \|\nabla u(s)\|_{L^2}^2 ds \leq \frac{1}{2} \|u_{\mathrm{in}}\|_{L^2}^2. $$

值得注意的是，这类解的存在性由 Leray 和 Hopf 建立，但其唯一性仍是数学流体力学中的核心开放问题之一。本文关注的是：是否存在多个不同的 Leray–Hopf 解对应于相同的初始数据？

此外，文中考虑的解恰好处于经典适定性理论的边界——即它们不满足 Prodi–Serrin 条件：

$$ u \in L^s([0,1]; L^q(\mathbb{R}^3)), \quad \text{其中 } 3 < q \leq \infty, \; 2 < s \leq \infty, \quad \frac{3}{q} + \frac{2}{s} \leq 1, $$

从而避免了强解的唯一性保证。事实上，所构造的解属于 $L^s([0,1]; L^q)$ 对所有 $q > 2$ 成立，且 $\frac{3}{q} + \frac{2}{s} > 1$，因此“刚好错过”Prodi–Serrin 条件。

关键结构假设是采用自相似变量（self-similar variables）。设：

$$ u(t,x) = \frac{1}{\sqrt{t}} U\left( \frac{x}{\sqrt{t}} \right), \quad \xi = \frac{x}{\sqrt{t}}, $$

则原方程转化为关于 $U(\xi)$ 的稳态系统：

$$ -\frac{1}{2} U - \frac{1}{2} \xi \cdot \nabla U + \Pi(U \cdot \nabla U) - \Delta U = 0, \quad \mathrm{div}\, U = 0, \tag{1.6} $$

其中 $\Pi = I - \nabla \Delta^{-1} \mathrm{div}$ 为 Leray 投影。对应的初始数据形式为：

$$ u_{\mathrm{in}}(x) = \frac{1}{|x|} A\left( \frac{x}{|x|} \right), \tag{1.4} $$

这是一种齐次为 $-1$ 的奇异初值，在原点处有 $O(|x|^{-1})$ 奇异性。

2. 主要定理或主要结果

定理 1（Leray–Hopf 解的非唯一性）存在一个具有紧支集、散度自由的初始数据 $u_{\mathrm{in}} \in L^2_{\mathrm{loc}}(\mathbb{R}^3)$，使得对应的无外力三维不可压缩Navier-Stokes方程 (1.1) 在 $\mathbb{R}^3 \times [0,1]$ 上存在无穷多个互不相同的合适（suitable）Leray–Hopf 解。这些解都属于空间 $L^s([0,1]; L^q(\mathbb{R}^3))$，其中 $q > 2$，$\frac{3}{q} + \frac{2}{s} > 1$，即刚好不满足 Prodi–Serrin 唯一性条件。

定理 2（自相似系统的弱解存在性）存在一个三元组 $(U, v, \lambda)$，满足如下耦合系统：

$$ \begin{cases} -\frac{1}{2} U - \frac{1}{2} \xi \cdot \nabla U + \Pi(U \cdot \nabla U) - \Delta U = 0, & \mathrm{div}\, U = 0, \\ -\frac{1}{2} v - \frac{1}{2} \xi \cdot \nabla v + \Pi(U \cdot \nabla v + v \cdot \nabla U) - \Delta v = \lambda v, & \mathrm{div}\, v = 0, \end{cases} \tag{1.12} $$

其中 $U \in H^1_{\mathrm{loc}}$, $v \in H^1_{\mathrm{loc}}$，且 $\lambda < 0$。这表示线性化算子 $\mathcal{L}_U$ 存在一个负实部的特征值（即不稳定模态），从而支持非唯一性的产生。

命题 1（精确自相似剖面的存在性）若近似剖面 $(U, P, v, q, \lambda)$ 满足小残差假设（见 Assumption 1），则存在一个小扰动 $U_c \in H^1$，使得 $U_e := U + U_c$ 是方程 (1.15) 的分布解，并精确满足第一个方程（自相似Navier-Stokes）和散度自由条件。

命题 2（精确不稳定特征对的存在性）在相同假设下，存在修正的特征值 $\lambda_c$ 和特征函数 $v_c \in H^1$，使得 $(\lambda_e, v_e) := (\lambda + \lambda_c, v + v_c)$ 是线性化算子 $\mathcal{L}_{U_e}$ 的特征对，满足第二个方程，且 $v_e$ 非平凡、正交于原特征函数，同时误差估计 $\|v_c\|_{L^2} + |\lambda_c| \lesssim \varepsilon_v$ 成立。

3. 研究方法、关键技术和核心工具

a) 核心分析学工具

  • 计算机辅助证明框架（computer-assisted proof）：结合高精度数值计算与严格的误差分析，通过区间算术验证关键算子的可逆性。
  • 不动点论证（fixed-point argument）：基于 Schauder 不动点定理，用于从近似解出发构造精确解。
  • 线性算子的强制-紧致分解（coercive-compact decomposition）：将线性化算子 $\mathcal{L}_U$ 分解为一个强制部分（coercive part）与一个紧扰动（compact perturbation），后者再被有限秩算子逼近。
  • 椭圆正则性提升（elliptic bootstrapping）：利用稳态方程的结构和强制性，逐步提升解的正则性至光滑级别。
  • 加权 Sobolev 估计与二进展开（dyadic rescaling）：用于控制远场衰减行为，证明 $U_{\mathrm{near}}(\xi) = O(|\xi|^{-3/2})$。
  • Riesz 投影与稳定-不稳定分解（stable-unstable decomposition）：将演化半群 $\mathrm{e}^{\tau \mathcal{L}_U}$ 分解为稳定与不稳定子空间，前者可用于收缩映射，后者提供自由参数导致非唯一性。

b) 技术性难题的处理技巧

  • 非线性项的严格估计：通过对近似解使用谱插值（spectral interpolation）到精确散度自由基底上，确保非线性项（如 $U \cdot \nabla U$）在 $L^2$ 中可精确计算残差，避免舍入误差累积。
  • 奇异性处理：虽然数值解在原点可能非光滑，但通过远近场分解（near-far field decomposition），将整体剖面写成 $U = U_{\mathrm{near}} + U_{\mathrm{far}}$，其中 $U_{\mathrm{far}}$ 构造为光滑并匹配渐近行为，再利用强制性和椭圆估计提升 $U_{\mathrm{near}}$ 的正则性。
  • 无限维算子的有限维验证：将紧算子部分用有限秩逼近（rank-25 approximation），并通过区间算术验证该低维广义特征值问题的可逆性，从而推导出整个无限维线性算子的局部可逆性。
  • 残差与逼近误差的定量控制：借助三角多项式张量积表示（trigonometric tensor-product representation）和解析积分，实现残差 $E_U, E_v$ 的无累积误差评估；同时对 Bessel 函数相关的内积使用四阶求积公式并给出 $W^{4,\infty}$ 误差界。

4. 与之前工作的比较

a) 突破与改进

本文的主要突破在于：

  • 首次给出无外力情形下 Leray–Hopf 解非唯一的严格计算机辅助证明。此前的工作如 Jia–Sverak [29,30] 和 Guillod–Sverak [25] 提出了基于自相似剖面不稳定性的猜想，并提供了数值证据，但未完成严格证明；而 Albritton–Brue–Colombo [1] 虽然完成了非唯一性证明，但依赖于人为添加的奇异外力项，使其背景解不必精确满足Navier-Stokes方程。

相比之下，本文的关键进步体现在：

  1. 无需外力项：所构造的自相似剖面 $U$ 精确满足无外力Navier-Stokes方程（即 (1.6)），这是真正物理意义上的解，而非带强迫的人造解。

  2. 更自然的初始数据：最终得到的 Leray–Hopf 解来自具有紧支集的 $L^2_{\mathrm{loc}}$ 初始数据，符合标准弱解理论框架。

  3. 完整闭环的计算机验证：不同于 [25] 中仅提供数值证据，本文发展了一套完整的计算机辅助验证流程，包括：

    • 高精度有限元+谱方法混合求解；
    • 强制-紧致分解以降低无限维问题复杂度；
    • 使用低维（25×25）广义特征值问题完成稳定性验证；
    • 区间算术保障所有浮点运算的可靠性。

  4. 更强的结果：不仅证明了解的非唯一性，而且得出存在无穷多个不同的 Leray–Hopf 解，源于不稳定特征方向上的自由参数选择。

综上所述，本文在物理真实性（无外力）、数学严格性（完全证明）和技术可行性（高效验证）三个方面均超越了前人工作，标志着对Navier-Stokes方程弱解非唯一性这一长期开放问题的重大进展。

2. A posteriori existence of strong solutions to the Navier-Stokes equations in 3D

三维Navier-Stokes方程强解的后验存在性

作者: Aaron Brunk, Jan Giesselmann, Tabea Tscherpel

类别: math.NA, cs.NA, math.AP, 35Q30, 35B44, 65M15, 65M60, 76D05

发布日期: 2025-09-29

ArXiv ID: 2509.25105v1

链接: http://arxiv.org/abs/2509.25105v1

详细分析

中文标题: 三维Navier-Stokes方程强解的后验存在性

1. 研究对象和背景

本文研究的是三维不可压缩Navier-Stokes方程在周期边界条件下的强解存在性问题。所考虑的方程组为：

$$ \partial_t u + (u \cdot \nabla)u - \nu \Delta u + \nabla \pi = f, \quad \mathrm{div}\, u = 0 \quad \text{在 } [0,T] \times \mathbb{T}^3, $$

其中 $ u $ 是速度场，$ \pi $ 是压力，$ f $ 是外力项，$ \nu > 0 $ 是粘性系数，$ \mathbb{T}^3 = [0,1]^3 $ 表示三维环面（即周期边界条件）。初始条件为 $ u(0,\cdot) = u_0 $。

该系统的解分为两类：Leray-Hopf弱解和强解。弱解全局存在但正则性未知；而强解具有更高的空间正则性（如 $ u \in L^\infty(0,T;W^{1,2}) $），局部存在且唯一，但是否能在有限时间内爆破（blow-up）仍是著名的千禧年难题之一。

本文关注的是强解的后验存在性验证，即通过数值解来严格排除某个时间区间内发生爆破的可能性，从而证明强解的存在性。为此，作者利用了关键的爆破准则（blow-up criterion）——Iskauriaza、Seregin 和 Šverák 在2003年提出的 $ L^\infty(L^3) $ 准则：若强解在某时刻前保持 $ \|u(t)\|_{L^3} $ 有界，则可延拓至更长时间。

研究中采用的函数空间包括：

  • $ L^p(\mathbb{T}^3) $、$ W^{k,p}(\mathbb{T}^3) $：标准Lebesgue与Sobolev空间；
  • $ L^p_{\mathrm{div}}(\mathbb{T}^3) $、$ W^{1,p}_{\mathrm{div}}(\mathbb{T}^3) $：散度自由子空间；
  • Bochner空间如 $ L^\infty(0,T;L^3(\mathbb{T}^3)) $，用于描述时空混合范数。

特别地，文中强调使用“临界空间” $ L^\infty(L^3) $，因其满足Navier-Stokes方程的尺度不变性，是分析爆破行为的最佳范数选择。

2. 主要定理或主要结果

定理1（强解的后验存在性）
设 $ T > 0 $，$ \nu > 0 $ 给定。设 $ u $ 是对应于初值数据 $ (u_0,f) \in W^{1,2}_{\mathrm{div}}(\mathbb{T}^3) \times L^2(0,T;L^2(\mathbb{T}^3)^3) $ 的Leray–Hopf弱解。设函数 $ \hat{u} \in C([0,T];W^{1,2}(\mathbb{T}^3)) $ 满足 $ \hat{u}(0) = \hat{u}_0 $，并令其残差为 $ r[\hat{u}] $，使得 $ \hat{u} $ 是对应于数据 $ (\hat{u}_0, f + r[\hat{u}]) $ 的Leray–Hopf弱解。假设以下两个条件成立：

（i）残差 $ r = r[\hat{u}] \in L^2(0,T;L^2(\mathbb{T}^3)^3) $；

（ii）对某个 $ T' \in (0,T] $，误差 $ e = u - \hat{u} $ 和 $ e_0 = u_0 - \hat{u}_0 $ 满足条件（42）（见原文），该条件涉及 $ \hat{u} $ 的 $ L^\infty(L^2) \cap L^4(L^6) $ 范数、残差在负Sobolev空间 $ L^2(W^{-1,2}) \cap L^3(W^{-1,3}) $ 中的范数以及初始误差。

则 $ u $ 是 $[0,T']$ 上的强解。

定理2（完全可计算的存在性判据）
存在一个数值格式（混合有限元空间离散 + 隐式Euler时间离散），使得对于某个时间 $ T > 0 $，其数值解 $ u_{\tau h} \in L^\infty(0,T;W^{1,2}(\mathbb{T}^3)) $ 可构造一个理论重构解 $ \hat{u} $，满足定理1中的所有条件，并且定理1中的稳定性条件（42）可以通过仅依赖于 $ u_{\tau h} $ 和已知数据的后验误差估计完全计算出来，无需任何关于真实解 $ u $ 的先验信息。

3. 研究方法、关键技术和核心工具

a) 核心分析学工具

  • 条件稳定性估计（Conditional Stability Estimate）：文章的核心在于建立一个关于误差 $ e = u - \hat{u} $ 的稳定性不等式，其中右端仅依赖于近似解 $ \hat{u} $ 和残差 $ r[\hat{u}] $，而不显式依赖于真实解 $ u $。这种估计结合了 $ L^2 $ 和 $ L^3 $ 范数的能量型控制。

  • Gronwall型引理的推广形式：用于从积分不等式中推出一致有界性。具体而言，作者引用了一个广义Gronwall引理（见[BGL25, Lem. 3.4]），允许处理非线性增长项（如 $ |e|^3 $），从而得出误差在 $ L^\infty(L^3) $ 中的小性条件。

  • Helmholtz-Leray投影及其稳定性：通过Leray投影将动量方程投影到散度自由空间，消去压力项。文中明确使用了该投影在 $ L^p $ 和 $ W^{1,p} $ 空间中的有界性（见Lemma 3），这对控制非线性项至关重要。

  • Bochner空间框架下的弱解理论：基于Leray-Hopf弱解的定义，使用三线性形式 $ b(u,v,w) = -\langle u \otimes v, \nabla w \rangle $ 来处理对流项，并利用其反对称性质进行能量估计。

b) 技术性技巧

  • 重构技术（Reconstruction Technique）：由于直接使用的数值解（分段常数或分段线性插值）不够光滑，无法作为定理中的 $ \hat{u} $，作者采用类似于椭圆重构的方法（参考[MN03]、[KM07]），通过求解辅助Stokes问题构造一个高正则性的“重构解” $ \hat{u} $，使其满足所需的正则性和方程扰动结构。

  • 后验误差估计的应用：为了使判据完全可计算，作者利用已有的后验误差估计技术（如[Ver13]）来量化数值解与其重构之间的差距，并估计残差在负Sobolev范数 $ \|r\|_{L^2(W^{-1,2})} $ 和 $ \|r\|_{L^3(W^{-1,3})} $ 下的大小，这些范数出现在最终的可计算判据中。

  • 临界范数下的稳定性分析：不同于以往工作依赖 $ L^\infty(W^{1,2}) $ 这类超临界范数，本文首次在临界空间 $ L^\infty(L^3) $ 框架下建立条件稳定性，这要求精细估计非线性项 $ (u \cdot \nabla)u $ 在 $ L^3 $ 框架下的行为，借助嵌入不等式（如Sobolev嵌入 $ W^{1,2} \hookrightarrow L^6 $）和Hölder不等式完成。

4. 与之前工作的比较

a) 突破与改进

本文相较于已有后验存在性工作的主要突破体现在以下几个方面：

  • 使用更优的爆破准则：此前的工作（如[DR08]、[MP08]）多基于较弱的爆破准则（如 $ L^\infty(W^{1,2}) $ 范数爆破），而本文采用了临界空间 $ L^\infty(L^3) $ 的Iskauriaza-Seregin-Šverák准则，这是目前最接近最优尺度的爆破判据，因而能提供更强的存在性结论。

  • 显著放宽对数值解正则性的要求：以往方法要求数值解满足 $ \hat{u} \in L^4(W^{1,2}) \cap L^2(W^{2,2}) $ 或更高正则性，这对于标准有限元方法难以实现（因缺乏二阶弱导数）。本文仅要求 $ \hat{u} \in L^\infty(W^{1,2}) \cap L^4(L^6) $，这一条件可通过混合有限元自然满足，大大增强了方法的适用性。

  • 降低对初始数据和外力项的要求：本文只需 $ u_0 \in W^{1,2}_{\mathrm{div}} $、$ f \in L^2(L^2) $，而此前一些工作还额外要求 $ f \in L^1(W^{1,2}) $ 或类似的高正则性假设。

  • 判据完全可计算且无需额外假设：通过引入重构技术和后验误差估计，作者成功将原本依赖于未知常数或高阶范数的条件转化为仅依赖于数值解和数据的全可计算表达式，无需任何关于真实解的先验估计或正则性假设。

  • 适用于更一般的数值方法：虽然文中以混合有限元+隐式Euler为例，但指出只要满足inf-sup稳定性的任意有限元对均可使用，表明该框架具有良好的扩展性。

综上所述，本文不仅将后验存在性理论推进到了临界正则性框架，而且极大提升了其在实际数值计算中的可行性与严谨性，为未来通过高性能计算验证Navier-Stokes方程长期强解存在性提供了新的路径。

3. $H^\infty$-calculus for Stokes operators on rough and on unbounded domains

粗糙域和无界域上Stokes算子的$H^\infty$-演算

作者: Peer Christian Kunstmann, Patrick Tolksdorf

类别: math.AP, math.FA, 35Q30, 76D07, 47A60, 47D06

发布日期: 2025-09-29

ArXiv ID: 2509.24501v1

链接: http://arxiv.org/abs/2509.24501v1

详细分析

中文标题: 粗糙域和无界域上Stokes算子的$H^\infty$-演算

1. 研究对象和背景

本文研究的是在粗糙域（如Lipschitz域）和无界但光滑的域上的Stokes算子的$H^\infty$-演算的有界性。Stokes算子是流体力学中描述粘性不可压缩流体运动的基础算子，其对应的稳态Stokes方程为：

$$ \begin{cases} -\Delta u + \nabla \phi = f & \text{在 } \Omega \text{ 中}, \\ \mathrm{div}(u) = 0 & \text{在 } \Omega \text{ 中}, \end{cases} $$

其中$u$表示速度场，$\phi$表示压力函数，$f$为外力项。

文章重点考虑以下两类边界条件：

  • 无滑移边界条件（no-slip boundary conditions）：$u = 0$ on $\partial\Omega$，常见于有界Lipschitz域；
  • Neumann型边界条件：$\{Du + \mu[Du]^\top\}n - \phi n = 0$ on $\partial\Omega$，其中$n$为外法向量，$\mu \in (-1,1)$为参数。

研究的空间为solenoidal $L^p$空间，即散度为零的向量场空间：

$$ L^p_\sigma(\Omega) := \left\{ f \in L^p(\Omega)^d : \mathrm{div}(f) = 0 \right\}, \quad 1 < p < \infty. $$

特别地，文章关注的区域包括：

  • 有界Lipschitz域（允许角点等非光滑结构）；
  • 凸有界域（用于Neumann型边界条件的新结果）；
  • 无界一致$C^{1,1}$或$C^3$域（如外部域、层状域等）。

此外，核心工具依赖于Helmtoltz投影（即Leray投影）将$L^p(\Omega)^d$分解为无散部分与梯度部分之直和：

$$ L^p(\Omega)^d = L^p_\sigma(\Omega) \oplus \nabla W^{1,p}(\Omega), $$

从而实现Stokes算子与Laplace算子之间的比较。

2. 主要定理或主要结果

定理 3.1（抽象比较原理）
设$(\Omega, \mu)$为$\sigma$-有限测度空间，$p_0 \in (1,\infty)\setminus\{2\}$，令$I := [\min\{2,p_0\}, \max\{2,p_0\}]$。设$(B_p)_{p\in I}$是一族在$L^p(\Omega)$上的一致可解的扇形算子，且每个$B_p$在对应空间上有有界的$H^\infty$-演算。
设$(X_p)_{p\in I}$是$L^p(\Omega)$中的闭线性子空间，$(R_p), (S_p)$为一族一致有界的线性算子，满足对每个$p\in I$，$R_p: L^p(\Omega) \to X_p$，$S_p: X_p \to L^p(\Omega)$，且$R_p S_p = I_{X_p}$。设$(A_p)_{p\in I}$为一族一致可解的线性算子，每个$A_p$在$X_p$上是$R$-扇形的。
假设存在$\alpha_1 > 0 > \alpha_2$和$\beta_1 > 0 > \beta_2$，使得以下包含关系和估计成立：

$$ R_2(D(B_2^{\alpha})) \subseteq D(A_2^{\alpha}), \quad \|A_2^{\alpha} R_2 f\| \lesssim \|B_2^{\alpha} f\|, \quad f \in D(B_2^{\alpha}),\ \alpha \in \{\alpha_1, \alpha_2\}, $$

$$ S_2(D(A_2^{\beta})) \subseteq D(B_2^{\beta}), \quad \|B_2^{\beta} S_2 f\| \lesssim \|A_2^{\beta} f\|, \quad f \in D(A_2^{\beta}),\ \beta \in \{\beta_1, \beta_2\}. $$


则对于所有$p \in I \setminus \{p_0\}$，算子$A_p$在$X_p$上具有有界的$H^\infty$-演算。

推论 3.5（自伴投影情形下的简化版本）
在定理3.1的设定下，若进一步假设：存在一族一致投影$P_p \in \mathcal{L}(L^p(\Omega))$使得$X_p = \mathrm{ran}(P_p)$，且$P_2$在$L^2(\Omega)$中是自伴的；同时$0 \in \rho(A_2) \cap \rho(B_2)$，并且存在$\alpha, \beta > 0$使得

$$ P_2(D(B_2^\alpha)) = D(A_2^\alpha) = D(B_2^\alpha) \cap X_2, $$

$$ P_2(D((B_2^*)^\beta)) = D((A_2^*)^\beta) = D((B_2^*)^\beta) \cap \mathrm{ran}(P_2), $$


则对所有$p \in I \setminus \{p_0\}$，$A_p$在$X_p$上具有有界的$H^\infty$-演算。

定理 4.2（Neumann型Stokes算子的$R$-扇形性）
设$\Omega \subseteq \mathbb{R}^d$（$d \geq 2$）为有界凸域，$\theta \in [0,\pi)$，$\mu \in (-1, \sqrt{2}-1)$，且满足

$$ \left|\frac{1}{p} - \frac{1}{2}\right| < \frac{1}{d}. $$


则如下算子族在相应空间上是$R$-有界的：

$$ \left\{ \lambda(\lambda + A_\mu)^{-1} : \lambda \in \Sigma_\theta \right\} \subset \mathcal{L}(L^p_\sigma(\Omega)), $$

$$ \left\{ |\lambda|^{1/2} \nabla (\lambda + A_\mu)^{-1} : \lambda \in \Sigma_\theta \right\} \subset \mathcal{L}(L^p_\sigma(\Omega), L^p(\Omega)^{d\times d}), $$

$$ \left\{ |\lambda|^{1/2} \Phi_\lambda : \lambda \in \Sigma_\theta \right\} \subset \mathcal{L}(L^p_\sigma(\Omega), L^p(\Omega)) \quad (\text{当 } p \geq 2), $$


其中$\Phi_\lambda f$为对应的压力函数。由此结合Dore-Venni定理可得，该Stokes算子在$L^p_\sigma(\Omega)$上具有有界的$H^\infty$-演算。

3. 研究方法、关键技术和核心工具

a) 核心分析学工具

  • $H^\infty$-函数演算理论：作为谱定理在Banach空间中的推广，用于定义非整数幂、证明最大正则性。
  • $R$-有界性与最大$L^q$-正则性：通过Weis定理将$R$-扇形性与抛物问题的最大正则性联系起来，这是现代进化方程理论的核心。
  • 复插值理论（complex interpolation）：用于从$L^2$上的分数次域空间比较推广到$L^p$情形，是连接不同$p$值之间性质的关键桥梁。
  • Littlewood-Paley理论与平方函数估计：用于控制振荡积分和建立向量值不等式，特别是在处理$R$-有界性时转化为$\ell^2$-值空间中的估计。
  • Shen的$L^p$外推定理（Theorem 4.3）：一种强有力的局部化技术，允许从弱$(1,1)$和强$(2,2)$估计推出更强的$L^p$估计，适用于非平滑区域。

b) 技术性难题的解决技巧

  • 利用Helmtoltz投影进行算子比较：通过将Stokes算子的分数域空间与Laplace算子在全空间中的分数域空间进行比较，借助投影算子$P$（即Leray投影）建立嵌入和等价性。
  • 抽象比较原则的应用：避免直接构造伪微分算子或使用局部化+扰动的传统方法，而是通过在$L^2$上验证分数次域的“匹配”来间接获得$H^\infty$-有界性。
  • 向量化与Rademacher随机变量：将$R$-有界性的验证转化为向量值空间中的平方函数估计，再利用Shen型外推定理统一处理。
  • 正则化逼近策略：对于凸域这类可能不具备$C^2$边界的区域，采用光滑凸域序列$(\Omega_\ell)$逼近原域，并在极限下保持估计的一致性。
  • 压力项的精细估计：通过对压力函数引入适当的测试函数（如$\nabla \Delta_D^{-1} \phi$），结合Dirichlet Laplacian的正则性，获得独立于$\lambda$的压力控制。

4. 与之前工作的比较

a) 突破与改进之处

  • 放宽几何正则性要求：
    以往关于Stokes算子$H^\infty$-演算的结果多要求区域为$C^\infty$或$C^{2,1}$（如Giga、Prüss等），而本文将结果推广至有界Lipschitz域，显著降低了边界光滑性要求。

  • 简化证明路径：
    对于无界域的情形（如外部域或渐近平坦层），传统方法依赖局部化+扰动+傅里叶乘子（如Noll-Saal、Abels），过程复杂。本文通过抽象比较原理提供了更简洁、统一的证明框架，无需处理局部坐标变换或模型问题。

  • 填补文献空白与修复漏洞：

    • 文献[18]中关于无界$C^{1,1}$域上Stokes算子的结果曾存在证明缺陷（见[9, Rem. 1.2]），本文通过改进的方法重新证明了该结果。
    • 首次建立了Neumann型边界条件下Stokes算子在有界凸域上的$H^\infty$-演算有界性，这是一个全新的结果，此前未见报道。

  • 统一处理多种边界条件与区域类型：
    同一抽象框架可用于处理无滑移边界、Neumann型边界以及不同几何结构（有界、无界、凸、Lipschitz）的区域，体现了方法的普适性和结构性优势。

  • 更强的功能性结论：
    获得$H^\infty$-演算有界性后，立即可推出：

    • 算子具有有界的虚数幂（bounded imaginary powers）；
    • 分数次域等于复插值空间：$D(A^\alpha) = [X, D(A)]_\alpha$；
    • 在UMD空间（如$L^p$）中蕴含最大$L^q$-正则性（$1<q<\infty$）；
    • 可应用于随机演化方程和数值格式的稳定性分析。

综上所述，本文不仅在结果上有所拓展，更重要的是提供了一种基于分数域比较的系统性方法，为后续研究非光滑或复杂几何背景下流体算子的泛函演算奠定了基础。

了解领域论文（摘要翻译）

1. Weak $(1,1)$ estimate for maximal truncated rough singular integral operator

极大截断粗糙奇异积分算子的弱 $(1,1)$ 型估计

作者: Xudong Lai

类别: math.CA, math.AP

发布日期: 2025-08-25

ArXiv ID: 2508.17737v3

链接: http://arxiv.org/abs/2508.17737v3

摘要翻译

中文标题: 极大截断粗糙奇异积分算子的弱 $(1,1)$ 型估计

摘要翻译: 在他们的开创性工作（Amer. J. Math. 78: 289–309, 1956）中，Calderón 和 Zygmund 引入了极大截断粗糙奇异积分算子，并建立了其在 $1 < p < \infty$ 范围内的 $L^p$ 有界性。然而，端点情形 $p = 1$ 长期以来一直是一个未解决的问题。本文解决了这一问题。更准确地说，我们证明了极大截断粗糙奇异积分算子具有弱型 $(1,1)$ 性质。

2. BIBO stability of 1-D hyperbolic boundary control systems

一维双曲型边界控制系统的有界输入有界输出稳定性

作者: Felix L. Schwenninger, Alexander A. Wierzba

类别: math.OC, math.AP, math.FA, 93C20, 93D25, 93C05, 35L04, 37K45

发布日期: 2024-10-16

ArXiv ID: 2410.12697v2

链接: http://arxiv.org/abs/2410.12697v2

摘要翻译

中文标题: 一维双曲型边界控制系统的有界输入有界输出稳定性

摘要翻译: 本文研究了一类一维双曲型边界控制系统的有界输入有界输出（BIBO）稳定性问题，该类系统特别包含了分布式端口哈密顿系统。通过利用此类系统传递函数的特定结构，我们推导出了若干BIBO稳定性的充分条件。

不相关论文（基本信息）

1. Steady Gradient Ricci Solitons with $O(p)\times O(q)$ Symmetry

具有 $O(p)\times O(q)$ 对称性的定常梯度Ricci孤立子

作者: Lucas Lavoyer, Luke T. Peachey

类别: math.DG, math.AP

发布日期: 2025-09-29

ArXiv ID: 2509.25081v1

链接: http://arxiv.org/abs/2509.25081v1

摘要: We find new examples of steady gradient Ricci solitons with positive curvature operator in dimensions four and above. Utilising a procedure first introduced by Lai, we construct examples with $O(p) \times O(q)$ symmetry in dimension $p+q$ for any pair of integers $p,q \geq 2$ and discuss their asymptotic geometry.

2. On domains of elliptic operators with distributional coefficients

关于具有分布系数的椭圆算子的定义域

作者: Immanuel Zachhuber

类别: math.AP, math.FA, math.PR

发布日期: 2025-09-29

ArXiv ID: 2509.24950v1

链接: http://arxiv.org/abs/2509.24950v1

摘要: In this note we show how one can use recently gained insights from the study of singular SPDEs, more particularly the study of singular operators via the theory of Paracontrolled Distributions, to construct domains for (singular) elliptic operators. Formally we consider [ A (u) \text{"$=$’’} (1 - \Delta) u

  • \nabla V \cdot \nabla u + \xi u + {{div} (\rho u)}, ] where $V \in \mathcal{C}^{\delta}$, $\xi \in \mathcal{C}^{- 2 + \delta}$, $\rho \in \mathcal{C}^{- 1 + \delta},{div}\rho = 0$ and which satisfy a structural assumption that is notably satisfied when $\xi$ is a “sub-critical noise”. We also show that under this assumption, one can construct a continuous change of variables $\Theta$ which satisfies
    $$ A \Theta - (1 - \Delta) \in \mathcal{L} (H^2 ; H^{\delta'}) $$
    which allows us to define $A$ rigorously and parametrise a domain. Moreover, for suitably regularised operators
    $$ A_{\varepsilon} (u) := (1 - \Delta) u + \nabla V_{\varepsilon} \cdot \nabla u + (\xi_{\varepsilon} + c_{\varepsilon}) u + {{div} (\rho_{\varepsilon} \cdot u)}, $$
    we show that for a strongly converging regularised change of variables $\Theta_{\varepsilon} \rightarrow \Theta$ we have
    $$ A_{\varepsilon} \Theta_{\varepsilon} \rightarrow A \Theta \text{ in } \mathcal{L} (H^2 ; L^2) $$
    which in particular implies norm resolvent convergence to a limiting closed operator.

3. Sharp behavior of semilinear damped wave equations driven by mixed local-nonlocal operators

由局部-非局部混合算子驱动的半线性阻尼波动方程的精确性态研究

作者: Wenhui Chen, Tuan Anh Dao

类别: math.AP, 35L71, 35B33, 35A01, 35B44, 35B40

发布日期: 2025-09-29

ArXiv ID: 2509.24940v1

链接: http://arxiv.org/abs/2509.24940v1

摘要: This paper investigates the Cauchy problem for the semilinear damped wave equation $u_{tt}+\mathcal{L}_{a,b}u+u_t=|u|^p$ with the mixed local-nonlocal operator $\mathcal{L}_{a,b}:=-a\Delta+b(-\Delta)^{\sigma}$, where $a,b\in\mathbb{R}_+$ and $\sigma\in(0,1)\cup (1,+\infty)$. We determine the critical exponent for this problem being $p_{\mathrm{crit}}=1+\frac{2\min\{1,\sigma\}}{n}$, which sharply separates global in-time existence and finite-time blow-up of solutions. Furthermore, for the super-critical case $p>p_{\mathrm{crit}}$, we establish the asymptotic profiles of global in-time solutions, showing the anomalous diffusion when $\sigma\in(0,1)$ and the classical diffusion when $\sigma\in(1,+\infty)$, together with the sharp decay estimates. For solutions blowing up in finite time when $1<p\leqslant p_{\mathrm{crit}}$, we derive the sharp estimates for upper and lower bounds of lifespan. Our results reveal the crucial influence of mixed operators on the qualitative properties of solutions, fundamentally governing their critical phenomena, large-time behavior and blow-up dynamics, via $\max\{1,\sigma\}$ or $\min\{1,\sigma\}$.

4. Traveling wave solutions for the generalized Burgers-Fisher equation

广义Burgers-Fisher方程的行波解

作者: Razvan Gabriel Iagar, Ariel Sánchez

类别: math.AP, math.DS

发布日期: 2025-09-29

ArXiv ID: 2509.24909v1

链接: http://arxiv.org/abs/2509.24909v1

摘要: Traveling wave solutions, in the form $u(x,t)=f(x+ct)$, to the generalized Burgers-Fisher equation

$$ \partial_tu=u_{xx}+k(u^n)_x+u^p-u^q, \quad (x,t)\in\mathbb{R}\times(0,\infty), $$

with $n\geq2$, $p>q\geq1$ and $k>0$, are classified with respect to their speed $c\in(-\infty,\infty)$ and the behavior at $\pm\infty$. The existence and uniqueness of traveling waves with any speed $c\in\mathbb{R}$ is established and their behavior as $x\to\pm\infty$ is described. In particular, it is shown that there exists a unique $c^*\in(0,\infty)$ such that there exists a unique soliton $f^*$ with speed $c^*$ and such that

$$ \lim\limits_{\xi\to-\infty}f^*(\xi)=\lim\limits_{\xi\to\infty}f^*(\xi)=0, \quad \xi=x+ct. $$

Moreover, if $n<p+q+1$ then $c^*<kn$ and if $n>p+q+1$ then $c^*>kn$. For $c<\min\{c^*,kn\}$, any traveling wave with speed $c$ satisfies $\lim\limits_{\xi\to-\infty}f(\xi)=0$ and $\lim\limits_{\xi\to\infty}f(\xi)=1$, while for $c>\max\{c^*,kn\}$ any traveling wave with speed $c$ satisfies $\lim\limits_{\xi\to-\infty}f(\xi)=1$ and $\lim\limits_{\xi\to\infty}f(\xi)=0$. In particular, for any speed $c\in(0,c^*)$, there are traveling wave solutions $u$ with speed $c$ such that $u(x,t)\to1$ as $t\to\infty$, in contrast to the non-convective case $k=0$.

5. A coarse-graining theory for elliptic operators and homogenization in high contrast

椭圆算子在高对比度下的粗粒化理论与均匀化

作者: Scott Armstrong, Tuomo Kuusi

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

发布日期: 2025-09-29

ArXiv ID: 2509.24887v1

链接: http://arxiv.org/abs/2509.24887v1

摘要: We review a coarse-graining theory for divergence-form elliptic operators. The construction centers on a pair of coarse-grained matrices defined on spatial blocks that encode a scale-dependent notion of ellipticity, transmit precise information from small to large scales, and yield coarse-grained counterparts of standard elliptic estimates. Under simplifying assumptions, we give a complete proof of the result of [arXiv:2405.10732] that homogenization is reached within at most $C\log^2(1+\Theta)$ dyadic length scales in the high-contrast regime, where $\Theta$ is the ellipticity contrast. We argue that this scale-local notion of ellipticity is genuinely iterable across arbitrarily many scales, providing a framework for a rigorous renormalization group analysis.

6. Existence, regularity and stability in a strongly degenerate nonlinear

diffusion and haptotaxis model of cancer invasion

一种强退化非线性扩散与趋触性癌症侵袭模型的存在性、正则性与稳定性研究

作者: Benoit Perthame, Chiara Villa

类别: math.AP, 35K65, 35K55, 92C50

发布日期: 2024-12-24

ArXiv ID: 2412.18261v2

链接: http://arxiv.org/abs/2412.18261v2

摘要: We consider a mathematical model of cancer cell invasion of the extracellular matrix (ECM), comprising a strongly degenerate parabolic partial differential equation for the cell volume fraction, coupled with an ordinary differential equation for the ECM volume fraction ($0 \leq {\phi} \leq 1$). The model captures the intricate link between the dynamics of invading cancer cells and the surrounding ECM. First, migrating cells undergo haptotaxis, i.e., movement up ECM density gradients. Secondly, cancer cells degrade the ECM fibers by means of membrane-bound proteases. Finally, their migration speed is modulated by the ECM pore sizes, resulting in the saturation or even interruption of cell motility both at high and low ECM densities. The inclusion of the physical limits of cell migration results in two regimes of degeneracy (at ${\phi} = 0$ and ${\phi} = 1$) impacting simultaneously the nonlinear diffusion and haptotaxis terms. We devise specific estimates that are compatible with the strong degeneracy of the equation, focusing on the vacuum present at low ECM densities. Based on these regularity results, and associated local compactness, we prove stability of weak solutions with respect to perturbations on bounded initial data, and global existence of weak solutions in the limit of an approximate non-degenerate problem.

7. Rigidity of critical points of hydrophobic capillary functionals

疏水性毛细泛函临界点的刚性

作者: Antonio De Rosa, Robin Neumayer, Reinaldo Resende

类别: math.AP, math.DG, 49Q10, 49Q20, 53A10

发布日期: 2025-09-26

ArXiv ID: 2509.22532v2

链接: http://arxiv.org/abs/2509.22532v2

摘要: We prove the rigidity, among sets of finite perimeter, of volume-preserving critical points of the capillary energy in the half space, in the case where the prescribed interior contact angle is between $90^\circ$ and $120^\circ$. No structural or regularity assumption is required on the finite perimeter sets. Assuming that the ``tangential’’ part of the capillary boundary is $\mathcal{H}^n$-null, this rigidity theorem extends to the full hydrophobic regime of interior contact angles between $90^\circ$ and $180^\circ$. Furthermore, we establish the anisotropic counterpart of this theorem under the assumption of lower density bounds.

8. A geometrical approach to the sharp Hardy inequality in

Sobolev-Slobodeckiĭ spaces

Sobolev-Slobodeckiĭ空间中Sharp Hardy不等式的几何方法

作者: Francesca Bianchi, Giorgio Stefani, Anna Chiara Zagati

类别: math.AP, math.FA, Primary 49Q20. Secondary 35R11, 46E35

发布日期: 2024-07-11

ArXiv ID: 2407.08373v3

链接: http://arxiv.org/abs/2407.08373v3

摘要: We give a partial negative answer to a question left open in a previous work by Brasco and the first and third-named authors concerning the sharp constant in the fractional Hardy inequality on convex sets. Our approach has a geometrical flavor and equivalently reformulates the sharp constant in the limit case $p=1$ as the Cheeger constant for the fractional perimeter and the Lebesgue measure with a suitable weight. As a by-product, we obtain new lower bounds on the sharp constant in the $1$-dimensional case, even for non-convex sets, some of which optimal in the case $p=1$.

9. Geometric structure of stationary problem for spatial 1D self-diffusion

equation with logistic growth

具有逻辑增长的空间一维自扩散方程定态问题的几何结构

作者: Yu ICHIDA

类别: math.DS, math.AP, math.CA, 34C05, 35B40, 35K65

发布日期: 2025-09-29

ArXiv ID: 2509.24752v1

链接: http://arxiv.org/abs/2509.24752v1

摘要: This paper considers the solution structure of non-trivial, non-constant stationary states of 1D spatial parabolic equations with nonlinear self-diffusion and logistic growth terms. A two-dimensional ordinary differential equation satisfying the stationary problem is derived and all its dynamics, including to infinity, is revealed by the Poincar'e-Lyapunov compactification, one of the compactifications of phase space. The advantage of this method is that it can be used to classify all dynamical systems (especially connecting orbits) of a two-dimensional system including infinity. Therefore, the classification results for the dynamical system including to infinity give the classification results for the non-constant stationary states obtained only from the structure of the original equations. This argument allows us to observe a change in the classification of the non-constant stationary states by an explicit relation between the linear diffusion coefficient and the self-diffusion coefficient, combined with arguments about the symmetries and conserved quantities of the ODEs. This means that changing the self-diffusion coefficient as a bifurcation parameter not only qualitatively changes the dynamical system from a big saddle homoclinic orbit of the ODEs to a heteroclinic orbit that connects the saddle equilibria, but also significantly changes the shape and the properties of the stationary states. It explicitly shows the relationship between linear and self-diffusion, gives a characterization of non-trivial stationary states in terms of dynamical systems, and gives a deep insight into the influence of self-diffusion, one of the nonlinear diffusions.

10. An inductive approach to stochastic estimates for the

$\varphi^{4}_2$-equation with correlated coefficient field

关于具相关系数场的$\varphi^{4}_2$方程随机估计的归纳方法

作者: Nicolas Clozeau

类别: math.AP, math.PR

发布日期: 2025-09-14

ArXiv ID: 2509.11309v3

链接: http://arxiv.org/abs/2509.11309v3

摘要: We develop an inductive approach to obtaining stochastic estimates for the $\varphi^{4}_2$-equation when the coefficient field is correlated with the driving noise. Our method is based on (infinite-dimensional) Gaussian integration by parts with respect to Wick products of Gaussian random variables (more precisely, mollifications of space-time white noise)

11. Half-space decay for linear kinetic equations

线性动力学方程的半空间衰减

作者: Émeric Bouin, Stéphane Mischler, Clément Mouhot

类别: math.AP

发布日期: 2025-07-14

ArXiv ID: 2507.10506v3

链接: http://arxiv.org/abs/2507.10506v3

摘要: We prove that solutions to linear kinetic equations in a half-space with absorbing boundary conditions decay for large times like $t^{-\frac{1}{2}-\frac{d}{4}}$ in a weighted $\sfL^{2}$ space and like $t^{-1-\frac{d}{2}}$ in a weighted $\sfL^{\infty}$ space, i.e. faster than in the whole space and in agreement with the decay of solutions to the heat equation in the half-space with Dirichlet conditions. The class of linear kinetic equations considered includes the linear relaxation equation, the kinetic Fokker-Planck equation and the Kolmogorov equation associated with the time-integrated spherical Brownian motion.

12. Rapid stabilization and finite time stabilization of the bilinear

Schrödinger equation

双线性薛定谔方程的快速镇定与有限时间镇定

作者: Hoai-Minh Nguyen

类别: math.OC, math.AP

发布日期: 2024-05-16

ArXiv ID: 2405.10002v2

链接: http://arxiv.org/abs/2405.10002v2

摘要: We propose a method to establish the rapid stabilization of the bilinear Schr"odinger control system and its linearized system, and the finite time stabilization of the linearized system using the Grammian operators. The analysis of the rapid stabilization involves a new quantity (variable) which is inspired by the adjoint state in the optimal control theory and is proposed in our recent work on control systems associated with strongly continuous group. The analysis of the finite time stabilization follows the strategy introduced by Coron and Nguyen in the study of the finite time stabilization of the heat equation and incorporate a new ingredient involving the estimate of the cost of controls of the linearized system in small time derived in this paper.

13. Scattering of the Vlasov-Riesz system in the three dimensions

三维空间中Vlasov-Riesz系统的散射问题

作者: Wenrui Huang, Hyunwoo Kwon

类别: math.AP, 35F50, 35Q82, 35B40

发布日期: 2024-07-24

ArXiv ID: 2407.16919v3

链接: http://arxiv.org/abs/2407.16919v3

摘要: We consider an asymptotic behavior of solutions to the Vlasov-Riesz system of order $\alpha$ in $\mathbb{R}^3$ which is a kinetic model induced by Riesz interactions. We prove small data scattering when $1/2<\alpha<1$ and modified scattering when $1<\alpha<1+\delta$ for some $\delta>0$. Moreover, we show the existence of (modified) wave operators for such a regime. To the best of our knowledge, this is the first result on the existence of modified scattering with polynomial correction in kinetic models.

14. Observability estimates for the Schrödinger equation on the

equilateral triangle

等边三角形上薛定谔方程的可观测性估计

作者: Paul Alphonse, David Lafontaine

类别: math.AP, math.OC, 35Q41, 35Q93, 52C20

发布日期: 2025-09-29

ArXiv ID: 2509.24642v1

链接: http://arxiv.org/abs/2509.24642v1

摘要: We prove observability estimates for the Schr"odinger equation posed on the equilateral triangle in the plane, under both Neumann and Dirichlet boundary conditions. No geometric control condition is required on the rough localization functions that we consider. This is the first result of this kind on a non-toric domain in the compact setting. Our strategy is to exploit Pinsky’s tiling argument to deduce this result from observability estimates on rational twisted tori. These are obtained via propagation of singularities, adapting arguments from Burq and Zworski. The later require Strichartz estimates on such twisted rational tori, that we derive from Zygmund inequalities in the same geometric setting, also providing the sharp constant. Strichartz estimates on the equilateral triangle are also derived from this analysis.

15. On a Differential Model for Sandpiles Growing in a Silo

关于沙堆在筒仓中增长的微分模型研究

作者: Graziano Crasta, Annalisa Malusa

类别: math.AP, 35C15, 35F30

发布日期: 2025-09-29

ArXiv ID: 2509.24618v1

链接: http://arxiv.org/abs/2509.24618v1

摘要: We discuss some features of a boundary value problem for a system of PDEs that describes the growth of a sandpile in a container under the action of a vertical source. In particular, we characterize the long-term behavior of the profiles, and we provide a sufficient condition on the vertical source that guarantees the convergence to the equilibrium in a finite time. We show by counterexamples that a stable configuration may not be reached in a finite time, in general, even if the source is time-independent. Finally, we provide a complete characterization of the equilibrium profiles.

16. Quantitative regularity for minimizing intrinsic fractional harmonic

maps

极小内蕴分数阶调和映射的定量正则性

作者: Y. -Y. Wang, C. -L. Xiang, G. -F. Zheng

类别: math.AP

发布日期: 2025-09-29

ArXiv ID: 2509.24587v1

链接: http://arxiv.org/abs/2509.24587v1

摘要: In this note, we study compactness and regularity theory of minimizing intrinsic fractional harmonic mappings introduced by Moser and Roberts. Based on the partial regularity theory of Moser and Roberts, we first use the modified Luckhaus lemma of Roberts to deduce compactness of these mappings, and then develop volume estimates of singular sets by the quantitative stratification theory of Cheeger and Naber. Combining these two results lead to a global regularity estimates which, in turn, allow us to obtain an improvement of the dimension estimate of singular sets.

17. On the Serrin’s problem with Robin boundary conditions

关于具Robin边界条件的Serrin问题

作者: Nunzia Gavitone, Riccardo Molinarolo

类别: math.AP, 35N25, 35A23, 35B50, 53A10

发布日期: 2025-09-29

ArXiv ID: 2509.24557v1

链接: http://arxiv.org/abs/2509.24557v1

摘要: Let $\Omega \subset \mathbb{R}^N$, $N\ge 2$, be an open, connected, bounded set with $C^2$ boundary. In this paper we consider the torsion problem with Robin boundary conditions and we study the symmetry of the solutions when suitable extra conditions are imposed on the boundary of $\Omega$. In particular, we prove the Serrin’s rigidity result under suitable assumptions on the domain and on the Robin parameter.

18. Frozen Gaussian approximation for the fractional Schrödinger equation

分数阶薛定谔方程的冻结高斯近似

作者: Lihui Chai, Hengzhun Chen, Xu Yang

类别: math.NA, cs.NA, math.AP

发布日期: 2024-03-27

ArXiv ID: 2403.18287v2

链接: http://arxiv.org/abs/2403.18287v2

摘要: We develop a refined Frozen Gaussian approximation (FGA) for the fractional Schr"odinger equation in the semi-classical regime, where the solution exhibits rapid oscillations as the scaled Planck constant $\varepsilon$ becomes small. Our approach utilizes an integral representation based on asymptotic analysis, offering a highly efficient computational framework for high-frequency wave function evolution. Crucially, we introduce the momentum space representation of the FGA and a regularization parameter $\delta$ to address singularities in the higher-order derivatives of the Hamiltonian flow coefficients, which are typically assumed to be second-order differentiable or smooth in conventional analysis. We rigorously prove convergence of the method to the true solution and provide numerical experiments that demonstrate its precision and robust convergence behavior.

19. Domains with globally exponentially integrable parabolic forward-in-time

BMO

具有全局指数可积抛物向前BMO的区域

作者: Kim Myyryläinen, Tuomas Oikari, Olli Saari

类别: math.CA, math.AP

发布日期: 2025-09-29

ArXiv ID: 2509.24486v1

链接: http://arxiv.org/abs/2509.24486v1

摘要: We characterize those open sets of the space time in which parabolic forward-in-time BMO functions are in a certain forward-in-time exponential integrability class. The characterization holds under qualitative connectivity assumptions on the domain and is formulated in terms of a quantitative growth bound on a forward-in-time version of the classical quasihyperbolic distance.

20. A Source Identification Problem for the Bi-Parabolic Equation Containing

a Poly-harmonic Operator

含有多调和算子的双抛物方程的源识别问题

作者: Dang Duc Trong, Bui Thanh Duy, Nguyen Dang Minh

类别: math.AP

发布日期: 2025-09-29

ArXiv ID: 2509.24470v1

链接: http://arxiv.org/abs/2509.24470v1

摘要: In this paper, we address the source identification problem for the bi-parabolic equation involving a operator. Specifically, we investigate the equation $(\partial_t +\frak A)^2u(t)=\psi(t)f$, where $\frak A$ denotes a poly-harmonic operator. Given the perturbed data of $\psi$ and $u(T)$ (where $T>0$), our objective is to determine $f$. Although several scientific publications have explored regularization techniques for bi-parabolic problems, the existing literature remains limited. By relaxing certain conditions on the function $\psi$ and employing a truncation regularization method while considering the problem on an unbounded domain, we believe our results provide valuable insights.

21. Convergence of graph Dirichlet energies and graph Laplacians on

intersecting manifolds of varying dimensions

变维相交流形上图Dirichlet能量与图Laplacian的收敛性

作者: Leon Bungert, Dejan Slepčev

类别: math.AP, math.SP, stat.ML, 49J55, 49J45, 60D05, 49R50, 68R10, 62G20

发布日期: 2025-09-29

ArXiv ID: 2509.24458v1

链接: http://arxiv.org/abs/2509.24458v1

摘要: We study $\Gamma$-convergence of graph Dirichlet energies and spectral convergence of graph Laplacians on unions of intersecting manifolds of potentially different dimensions. Our investigation is motivated by problems of machine learning, as real-world data often consist of parts or classes with different intrinsic dimensions. An important challenge is to understand which machine learning methods adapt to such varied dimensionalities. We investigate the standard unnormalized and the normalized graph Dirichlet energies. We show that the unnormalized energy and its associated graph Laplacian asymptotically only sees the variations within the manifold of the highest dimension. On the other hand, we prove that the normalized Dirichlet energy converges to a (tensorized) Dirichlet energy on the union of manifolds that adapts to all dimensions simultaneously. We also establish the related spectral convergence and present a few numerical experiments to illustrate our findings.

22. Existence, Nonexistence, and Symmetry of Positive Solutions for

Fractional Laplacian Problems

分数阶拉普拉斯问题正解的存在性、非存在性与对称性

作者: Haipeng Lu, Mei Yu

类别: math.AP

发布日期: 2025-09-29

ArXiv ID: 2509.24454v1

链接: http://arxiv.org/abs/2509.24454v1

摘要: This paper studies the properties of solutions to a class of elliptic and parabolic problems involving the fractional Laplacian. By applying the mountain pass theorem, we prove the existence of bounded classical positive solutions in the subcritical regime. Moreover, using the method of moving planes, we establish that these solutions are symmetric or monotone in the first variable. In contrast, we show that no such solutions exist in the supercritical or negative exponent cases. An analysis of the asymptotic behavior of solutions at infinity provides further insight into their profiles, which supports applications to real-world problems. The approaches developed in this work can also be extended to a wider range of nonlocal elliptic and parabolic equations, including those with more general operators and nonlinearities.

23. Harnack inequality for a class of degenerate fully nonlinear

pseudo-$p$-Laplacian equations

一类退化完全非线性伪$p$-Laplace方程的Harnack不等式

作者: Sun-Sig Byun, Hongsoo Kim

类别: math.AP, 35B65, 35D40, 35J15, 35J70

发布日期: 2025-09-29

ArXiv ID: 2509.24442v1

链接: http://arxiv.org/abs/2509.24442v1

摘要: We prove interior H"older regularity and Harnack inequalities for viscosity solutions of a class of degenerate fully nonlinear pseudo-$p$-Laplacian equations in nondivergence form. Our main approach is an adaptation of the sliding paraboloid method with anisotropic functions tailored to the coordinatewise degeneracy.

24. Finite number of traces for Mumford-Shah minimizers in dimension 2

二维Mumford-Shah极小化子的迹具有有限个数

作者: Camille Labourie, Antoine Lemenant

类别: math.AP

发布日期: 2025-09-29

ArXiv ID: 2509.24426v2

链接: http://arxiv.org/abs/2509.24426v2

摘要: In this short note, we answer a question raised by E. De Giorgi, showing that a Mumford-Shah minimizer in dimension 2 can only admit three maximum limit values as approaching the singular set. This result stems from tools developed in the early 2000’s by G. David, A. Bonnet, and J.-C. L'eger.

25. Discarding Lavrentiev’s Gap in Non-autonomous and Non-Convex Variational

Problems

消除非自治与非凸变分问题中的拉夫连季耶夫间隙

作者: Michał Borowski, Pierre Bousquet, Iwona Chlebicka, Benjamin Lledos, Błażej Miasojedow

类别: math.AP, math.FA

发布日期: 2024-10-19

ArXiv ID: 2410.14995v3

链接: http://arxiv.org/abs/2410.14995v3

摘要: We establish that the Lavrentiev gap between Sobolev and Lipschitz maps does not occur for a scalar variational problem of the form:

$$ \textrm{to minimize} \qquad u \mapsto \int_\Omega f(x,u,\nabla u)\,dx \,, $$

under a Dirichlet boundary condition. Here, $\Omega$ is a bounded Lipschitz open set in $\rn$, $N\geq 1$ and the function $f$ is required to be measurable with respect to the spatial variable, continuous with respect to the second one, and convex with respect to the last variable. Under these assumptions alone, Lavrentiev gaps may occur, as illustrated by classical examples in the literature. We identify an additional natural condition on $f$ to discard such phenomena, that can be interpreted as a balance between the variations with respect to the first variable and the growth with respect to the last one. This unifies most of the structural assumptions that have been introduced so far to prevent the occurence of Lavrentiev gaps. Remarkably, typical assumptions that are usually imposed on $f$ in this setting are dropped here: we do not require $f$ to be bounded or convex with respect to the second variable, nor impose any condition of $\Delta_2$-kind with respect to the last variable.

26. A new minimax principle and application to the p-Laplace equation

一个新的极小极大原理及其在p-拉普拉斯方程中的应用

作者: Xu-Jia Wang, Xinyue Zhang

类别: math.AP

发布日期: 2025-09-29

ArXiv ID: 2509.24268v1

链接: http://arxiv.org/abs/2509.24268v1

摘要: We introduce a new minimax principle to prove the existence of multi-peak solutions to the Neumann problem of the $p$-Laplace equation

$$ -\varepsilon^p \Delta_p u = u^{q-1} - u^{p-1} \ \ \text{in}\ \Omega, $$

where $\Om$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $1<p<n$ and $p<q< \frac{np}{n-p}$. The minimax principle will be applied to the set of peak functions, which is a subset of the Sobolev space $W^{1,p} (\Omega)$. The argument is based on a combination of variational method, topological degree theory, and gradient flow.