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1. Analyticity in space and time for global solutions to the anisotropic Navier–Stokes equations in the critical $L^p(\mathbb{R}^3)$ framework

在临界 $L^p(\mathbb{R}^3)$ 框架下各向异性Navier–Stokes方程整体解在空间与时间上的解析性

作者: Mikihiro Fujii, Yang Li

类别: math.AP

发布日期: 2025-11-25

arXiv ID: 2511.20482v1

链接: http://arxiv.org/abs/2511.20482v1

详细分析

中文标题: 在临界 $L^p(\mathbb{R}^3)$ 框架下各向异性Navier–Stokes方程整体解在空间与时间上的解析性

研究对象和背景

本文研究的是三维不可压缩各向异性Navier–Stokes方程的柯西问题，其控制方程如下：

$$ \begin{cases} \partial_t u - \Delta_h u + (u \cdot \nabla)u + \nabla P = 0, & t > 0,\ x \in \mathbb{R}^3, \\ \mathrm{div}\, u = 0, & t \geqslant 0,\ x \in \mathbb{R}^3, \\ u(0,x) = u_0(x), & x \in \mathbb{R}^3. \end{cases} $$

其中，$ u = (u_h, u_3) : [0,\infty) \times \mathbb{R}^3 \to \mathbb{R}^3 $ 是速度场，$ P $ 是压力函数，$ u_0 $ 是给定的初始速度。这里的关键特征在于粘性项仅为水平拉普拉斯算子 $ \Delta_h := \partial_{x_1}^2 + \partial_{x_2}^2 $，即只在前两个空间变量（水平方向）上具有耗散作用，而垂直方向 $ x_3 $ 上无任何粘性。这种设定源于地球物理流体力学和埃克曼层模型中的实际应用，在这些背景下，水平粘性往往远大于垂直粘性。

由于缺乏垂直方向的耗散机制，该系统的数学分析极具挑战性。特别是非线性项 $ u \cdot \nabla u $ 中包含形如 $ u_3 \partial_{x_3} u_h $ 的耦合项，它涉及对 $ u_3 $ 的一阶导数，但在方程中并无对应的 $ \partial_{x_3}^2 u $ 控制项，从而导致所谓的“导数损失”（derivative loss）现象。这一困难使得标准能量方法难以直接适用。

为克服此障碍，早期工作如 Chemin 等人 [3] 利用散度自由条件 $ \mathrm{div}\, u = 0 $ 推出 $ \partial_{x_3} u_3 = -\mathrm{div}_h u_h $，从而将 $ u_3 $ 的垂直导数转化为水平量的散度，避免了直接估计高阶 $ x_3 $-导数。基于此思想，他们在各向异性 Sobolev 空间 $ H^{0,s}(\mathbb{R}^3) := H^s(\mathbb{R}_{x_3}; L^2(\mathbb{R}^2_{x_h})) $ 中建立了局部适定性和小初值的整体适定性，要求 $ s > 1/2 $。然而，$ s = 1/2 $ 才是尺度不变（scaling-critical）指数，因此临界情形 $ s = 1/2 $ 长期悬而未决。

Paicu [21] 在关键工作中引入了各向异性 Besov 空间 $ \dot{B}^{0,1}_{2,1}(\mathbb{R}^3) := \dot{B}^1_{2,1}(\mathbb{R}_{x_3}; L^2(\mathbb{R}^2_{x_h})) $，并利用 $ \dot{B}^1_{2,1}(\mathbb{R}) \hookrightarrow L^\infty(\mathbb{R}) $ 的嵌入性质，首次在临界空间中证明了整体适定性。此后，[6,27] 将结果推广至更一般的 $ L^p $ 框架下的各向异性 Besov 空间：

$$ \|f\|_{\dot{B}^{s,\sigma}_p} := \sum_{j \in \mathbb{Z}} \left( \sum_{k \geq j-1} 2^{\sigma j + s k} \|\dot{\Delta}_k^h \dot{\Delta}_j^v f\|_{L^p_{x_h} L^2_{x_3}}^2 \right)^{1/2} + \sum_{j \in \mathbb{Z}} 2^{\sigma j} \left\| \sum_{k \leq j-2} 2^{s k} \dot{\Delta}_k^h \dot{\Delta}_j^v f \right\|_{L^2}, $$

但值得注意的是，上述框架在高频垂直区域仍依赖于 $ L^2 $ 范数，以维持能量估计。

相比之下，本文的重大突破在于完全摆脱了 $ L^2 $ 的限制，首次在全 $ L^p(\mathbb{R}^3) $ 基础上的各向异性 Besov 空间中建立整体解的存在性与解析性。作者定义了新的空间：

$$ \|f\|_{\dot{B}^{s,\sigma}_p} := \sum_{k,j \in \mathbb{Z}} 2^{s k + \sigma j} \|\dot{\Delta}_k^h \dot{\Delta}_j^v f\|_{L^p(\mathbb{R}^3)}, $$

其中 $ \dot{\Delta}_k^h $ 和 $ \dot{\Delta}_j^v $ 分别是关于水平频率 $ |\xi_h| $ 和垂直频率 $ |\xi_3| $ 的齐次 Littlewood–Paley 投影。这标志着该领域首次实现对所有变量使用统一的 $ L^p $ 结构。

此外，本文的研究目标不仅是适定性，更是时空联合实解析性（real analyticity in space and time）。所谓实解析性，意味着解在其定义域内可以展开为收敛的泰勒级数。对于 Navier–Stokes 方程，解析性反映了扰动传播的光滑性和稳定性，是比经典 Sobolev 或 Besov 正则性更强的结果。已有大量文献研究各向同性 Navier–Stokes 方程的解析性，例如 Foias–Temam [8] 证明周期情形下解的空间解析性；Bae 等 [1] 将结果推广至临界 Besov 空间 $ \dot{B}^{3/p-1}_{p,q}(\mathbb{R}^3) $。

但对于各向异性情形，由于垂直方向缺乏耗散，解析性的研究更为复杂。最近 Liu–Zhang [17] 在 $ L^2 $-型 Besov 空间中证明了解的空间解析性，但要求初始数据在 $ x_3 $ 方向也解析且满足较强的小性条件。本文则进一步推进：不仅证明了时间和空间上的联合解析性，而且显著放宽了对初始数据的要求——仅需水平初值 $ u_{0,h} $ 在 $ x_3 $ 方向解析且足够小，而对 $ u_{0,3} $ 不作任何小性或解析性假设。

综上所述，本论文的研究对象是一个物理意义明确、数学结构复杂且长期存在技术瓶颈的偏微分方程系统。其特殊之处包括：(i) 区域为全空间 $ \mathbb{R}^3 $；(ii) 边界条件为衰减于无穷远处（隐含在函数空间中）；(iii) 使用全新的基于 $ L^p(\mathbb{R}^3) $ 的各向异性齐次 Besov 空间作为基本框架；(iv) 目标是获得比以往更强的时空解析性结果。

主要定理或主要结果

本文的主要结果由 Theorem 1.1 给出，其完整陈述如下：

设参数 $ p $ 和 $ \theta $ 满足：

$$ 1 \leqslant p < 3, \quad \max\left\{0, 2\left(1 - \frac{1}{p}\right)\right\} < \theta < \min\left\{1, \frac{2}{p}\right\}. $$

则存在正常数 $ \eta = \eta(p,\theta) $, $ C = C(p,\theta) $, $ r = r(p,\theta) $，使得当初始速度 $ u_0 = (u_{0,h}, u_{0,3}) $ 满足 $ \mathrm{div}\, u_0 = 0 $，且

$$ u_{0,h} \in D_h := \dot{B}^{2/p - 1, 1/p}_{p,\theta}(\mathbb{R}^3) \cap \dot{B}^{2/p - 1 + \theta, 1/p - \theta}_{p}(\mathbb{R}^3) \cap \dot{B}^{2/p - 2 + \theta, 1/p - \theta}_{p}(\mathbb{R}^3), $$

$$ u_{0,3} \in D_v := \dot{B}^{2/p - 1, 1/p}_{p,\theta}(\mathbb{R}^3) \cap \dot{B}^{2/p - 2 + \theta, 1/p - \theta}_{p}(\mathbb{R}^3), $$

并且仅水平部分 $ u_{0,h} $ 在 $ x_3 $ 方向实解析且满足小性条件：

$$ \sum_{\beta_3 \in \mathbb{N} \cup \{0\}} \frac{\rho_0^{\beta_3}}{\beta_3!} \|\partial_{x_3}^{\beta_3} u_{0,h}\|_{D_h} \leqslant \eta \exp\left( -C \|u_{0,3}\|_{D_v} \right) $$

对某个 $ \rho_0 > 0 $ 成立时，方程组 (1.1) 存在一个唯一的时空解析整体解 $ u = (u_h, u_3) $，满足以下估计：

对于所有 $ \alpha \in \mathbb{N} \cup \{0\} $, $ \beta = (\beta_h, \beta_3) \in (\mathbb{N} \cup \{0\})^3 $，

$$ \sum_{\alpha, \beta} \frac{r^{\alpha + |\beta_h|} (\rho_0 / 2)^{\beta_3}}{\alpha! \beta!} \|t^{\alpha + |\beta_h|/2} \partial_t^\alpha \partial_x^\beta u_h\|_{S_h} \leqslant C \eta, $$

$$ \sum_{\alpha \in \mathbb{N} \cup \{0\}} \sum_{\beta_h \in (\mathbb{N} \cup \{0\})^2} \frac{r^{\alpha + |\beta_h|}}{\alpha! \beta_h!} \|t^{\alpha + |\beta_h|/2} \partial_t^\alpha \partial_{x_h}^{\beta_h} u_3\|_{S_v} \leqslant C \|u_{0,3}\|_{D_v} + C \eta^2, $$

其中函数空间 $ S_h $ 和 $ S_v $ 定义为：

$$ S_h := \left\{ u \in L^\infty(0,\infty; D_h) \mid \nabla_h^2 u \in L^1(0,\infty; D_h) \right\}, $$

$$ S_v := \left\{ u \in L^\infty(0,\infty; D_v) \mid \nabla_h^2 u \in L^1(0,\infty; D_v) \right\}. $$

此外，当 $ 1 \leqslant p \leqslant 2 $ 时，该解析解在相应空间中是唯一的。

研究方法、关键技术和核心工具

a) 核心分析学工具

本文的核心目标是建立全局先验估计，以控制解的所有时空导数的加权和，从而证明其解析性。为此，作者综合运用了多种现代调和分析与非线性 PDE 技术：

  1. 各向异性 Littlewood–Paley 理论：这是全文的基础工具。通过分别对水平和垂直频率进行分解，构造了齐次投影算子 $ \dot{\Delta}_k^h $ 和 $ \dot{\Delta}_j^v $，并在此基础上定义了新型的各向异性 Besov 范数 $ \|\cdot\|_{\dot{B}^{s,\sigma}_p} $。这种分解允许精确刻画不同方向上的正则性差异，并为后续的乘积估计和双线性估计提供平台。

  2. 最大正则性理论（Maximal Regularity）：用于处理线性化方程 $ \partial_t u - \Delta_h u = f $。作者利用水平热半群 $ e^{t\Delta_h} $ 的良好性质，获得如下类型估计：

    $$ \|u\|_{L^\infty_T(\dot{B}^{s,\sigma}_p)} + \|\nabla_h^2 u\|_{L^1_T(\dot{B}^{s,\sigma}_p)} \lesssim \|u_0\|_{\dot{B}^{s,\sigma}_p} + \|f\|_{L^1_T(\dot{B}^{s,\sigma}_p)}. $$

    这种估计对于控制线性演化部分至关重要。

  3. Bony 双线性分解与乘积估计：为了处理非线性项 $ (u \cdot \nabla)u $，作者采用经典的 Bony 分解 $ fg = T_f g + T_g f + R(f,g) $，结合各向异性版本的乘积不等式（见 Lemma 2.2），将非线性相互作用按频率高低分类估计。例如，当水平频率主导时，利用 $ \dot{B}^{s,\sigma}_p \cdot \dot{B}^{2/p,1/p}_p \hookrightarrow \dot{B}^{s,\sigma}_p $ 类型的嵌入。

  4. Gevrey 类方法与加权导数估计：不同于使用解析算子（如 $ e^{\rho|\partial_{x_3}|} $）的传统方法，本文采取直接估计高阶导数的策略。通过引入带权 $ \frac{r^{\alpha + |\beta_h|} \rho^{\beta_3}}{\alpha! \beta!} $ 的导数范数，将解析性问题转化为证明该加权级数一致有界。这是 Gevrey 类理论的标准手法，适用于处理解析性和强正则性问题。

  5. Osgood 型常微分方程技巧：面对非线性项引起的“导数损失”，作者构造了一个辅助 ODE：

    $$ f'(t) = \|u_3(t)\|_{\dot{B}^{2/p,1/p}_p} + \sum_{\beta_3} \frac{1}{\beta_3!} \|\partial_{x_3}^{\beta_3} u_h(t)\|_{\dot{B}^{2/p+1,1/p}_p} e^{-\lambda \beta_3 f(t)}, \quad f(0)=0, $$

    并通过乘子 $ e^{-\lambda \beta_3 f(t)} $ 来吸收非线性增长。这种方法源自 Osgood 引理的思想，用于控制快速增长的非线性效应。

  6. Duhamel 原理与时间积分分解：对于含时间权重 $ t^{\alpha + |\beta_h|/2} $ 的估计，直接对微分方程求导会产生奇异项 $ t^{-1/2} $。为此，作者转而使用积分形式（Duhamel 公式）：

    $$ u(t) = e^{t\Delta_h} u_0 - \int_0^t e^{(t-\tau)\Delta_h} \mathbb{P}[(u \cdot \nabla)u](\tau) d\tau, $$

    并将时间积分拆分为 $ \int_0^{t/2} + \int_{t/2}^t $，分别利用半群的平滑性和解本身的衰减性进行估计。

b) 解决技术性难题的主要技巧

  1. 应对导数损失 $ u_3 \partial_{x_3} u_h $：
    该项导致 $ \partial_{x_3}^{\beta_3}(u_3 \partial_{x_3} u_h) $ 展开后出现 $ u_3 \partial_{x_3}^{\beta_3+1} u_h $，其 $ x_3 $-导数阶数超过方程控制能力。作者通过引入上述 ODE $ f(t) $，并在估计中使用修正范数 $ \|\partial_{x_3}^{\beta_3} u_h e^{-\lambda \beta_3 f(t)}\| $，使得非线性项的增长被 $ f'(t) $ 吸收。选择足够大的 $ \lambda $ 可使该项被左侧耗散项所控制。

  2. 精确估计压力项 $ \nabla P $：
    压力通过泊松方程 $ -\Delta P = \mathrm{div}((u \cdot \nabla)u) $ 确定。特别地，$ \nabla_h P_2 = 2 \nabla_h \partial_{x_3} (-\Delta)^{-1}(u_h \cdot \nabla_h u_3) $ 在高频下会产生额外导数。作者观察到 $ \nabla_h \partial_{x_3} (-\Delta)^{-1} $ 是一个各向异性 Calderón–Zygmund 算子，其核具有特定奇异性。通过精细分析其作用在 $ \dot{\Delta}_k^h \dot{\Delta}_j^v $ 上的行为，并结合辅助空间 $ \dot{B}^{2/p - 1 + \theta, 1/p - \theta}_p \cap \dot{B}^{2/p - 2 + \theta, 1/p - \theta}_p $ 提供的额外正则性，成功控制了该项。

  3. 处理时间权重奇异性：
    当估计 $ \partial_t^\alpha \partial_x^\beta u $ 且 $ \alpha + |\beta_h|/2 < 1 $ 时，会出现 $ t^{\alpha + |\beta_h|/2 - 1} $ 的奇异性。作者放弃直接微分原方程，改用 Duhamel 表达式，并借助半群 $ e^{t\Delta_h} $ 的衰减估计 $ \|e^{t\Delta_h} f\|_{\dot{B}^{s',\sigma}_p} \lesssim t^{-(s'-s)/2} \|f\|_{\dot{B}^{s,\sigma}_p} $ 来平衡时间奇性。

  4. 分离水平与垂直结构的估计策略：
    由于 $ u_h $ 和 $ u_3 $ 的动力学行为不同（前者受水平耗散影响，后者不受），作者对两者采用不同的函数空间 $ S_h $ 和 $ S_v $，并对它们的非线性耦合项进行不对称估计。例如，$ u_3 \partial_{x_3} u_h $ 中，将 $ u_3 $ 视为低正则系数，$ u_h $ 为高正则目标。

与之前工作的比较

a) 突破与改进

本文相较于已有工作实现了多方面的实质性推进，具体体现在以下几个方面：

  1. 函数框架的根本性突破：
    此前关于各向异性 Navier–Stokes 方程的研究，包括 Paicu [21]、[6,27]、Liu–Zhang [17] 等，均在垂直方向或高频区域依赖于 $ L^2 $ 空间（如 $ L^2_{x_3} $ 或 $ L^2 $ 能量）。这是因为 $ L^2 $ 提供了方便的 Parseval 恒等式和能量估计工具，尤其在处理导数损失时不可或缺。
    本文则是首个在全 $ L^p(\mathbb{R}^3) $ 框架下建立整体适定性与解析性的成果。作者之所以能摆脱 $ L^2 $ 依赖，关键在于采用了解析性估计方法：通过对解进行无限次微分，原本因导数损失而无法控制的项，可通过引入辅助 ODE 和指数衰减因子 $ e^{-\lambda \beta_3 f(t)} $ 被系统吸收。这表明，在追求更强正则性（如解析性）时，反而可以放宽对基础函数空间的限制，是一个深刻的反直觉洞见。

  2. 显著放宽对初始数据的假设：

    • Liu–Zhang [17] 要求初始数据 $ u_0 $ 在 $ x_3 $ 方向整体解析，且需满足复合小性条件（见原文 (1.5)），其中包括 $ u_{0,3} $ 的解析性。
    • 本文仅要求水平初值 $ u_{0,h} $ 在 $ x_3 $ 方向解析且足够小，而对 $ u_{0,3} $ 不要求任何解析性或小性。唯一与 $ u_{0,3} $ 相关的是一个指数修正因子 $ \exp(-C\|u_{0,3}\|) $，这延续了 Zhang [25,26]、Paicu–Zhang [22]、Fujii [10] 等人的“单边小性”传统，但达到了目前最优形式（即 $ u_{0,3} $ 的范数幂次为 1）。
    • 此外，本文移除了 [17] 中所需的额外低频正则性（对应 $ \alpha + \beta > 1 $），取而代之的是一个边界情况 $ \alpha + \beta = 1 $，进一步弱化了条件。

  3. 获得更强的结论：时空联合解析性：
    以往工作大多只证明空间解析性（如 [17] 中 $ e^{\rho_0|\partial_{x_3}|} u \in L^\infty_T(\dot{B}^{0,1}_{2,1}) $）。本文则证明了解在时间和空间上同时解析，即泰勒级数 $ \sum \frac{t^\alpha \partial_t^\alpha u(0)}{\alpha!} $ 收敛。这需要控制所有混合导数 $ \partial_t^\alpha \partial_x^\beta u $，技术难度更高。

  4. 揭示垂直解析半径的动态演化机制：
    由于非线性项 $ u_3 \partial_{x_3} u_h $ 具有“双曲”特性，会拉伸 $ x_3 $ 方向的扰动，导致垂直方向的解析半径随时间缩小。本文精确刻画了这一过程：垂直解析半径从初始的 $ \rho_0 $ 衰减为 $ \rho_0 e^{-\lambda f(t)} $，其中 $ f(t) $ 由非线性累积效应决定。这是对解的精细结构的重要揭示。

综上所述，本文不仅在技术上实现了从 $ L^2 $ 到全 $ L^p $ 框架的跨越，更在科学内涵上推进了人们对各向异性流体中解析性传播机制的理解，特别是在弱耗散方向上如何维持光滑性的问题上提供了新视角。

2. Compressible Euler equations with transport noise

具有输运噪声的可压缩欧拉方程

作者: Richard Boadi, Dominic Breit, Thamsanqa Castern Moyo

类别: math.AP, math.PR

发布日期: 2025-11-25

arXiv ID: 2511.20131v1

链接: http://arxiv.org/abs/2511.20131v1

详细分析

中文标题: 具有输运噪声的可压缩欧拉方程

研究对象和背景

本文研究的是多维空间中带有输运型随机扰动的等熵可压缩欧拉方程组。该系统描述了一个在周期域（即平坦环面 $ \mathbb{T}^N $，其中 $ N = 2,3 $）上流动的可压缩流体的动力学行为。其数学形式如下：

$$ \begin{cases} \partial_t m + \mathrm{div}\left( \dfrac{m \otimes m}{\varrho} \right) = -\nabla p(\varrho) + \dot{\eta}, \\ \partial_t \varrho + \mathrm{div}(m) = \dot{\xi}, \end{cases} $$

其中未知函数为动量场 $ m $ 和密度 $ \varrho $，压力 $ p(\varrho) $ 是一个等熵状态方程，具体取为 $ p(\varrho) = M_a^{-2} \varrho^\gamma $，其中 $ M_a > 0 $ 为马赫数，$ \gamma > 1 $ 为绝热指数。

值得注意的是，文中将问题设定在周期边界条件下，并将物理区域 $ \Omega \subset \mathbb{R}^N $ 视为平坦环面 $ \mathbb{T}^N $，这在分析上简化了边值处理，便于使用傅里叶分析与紧性方法。

更关键的是，系统中的随机项 $ \dot{\eta} $ 和 $ \dot{\xi} $ 并非简单的外力噪声，而是具有特定结构的输运型噪声（transport noise），其定义如下：

$$ \dot{\eta} = \sum_{k=1}^K \mathrm{div}(\sigma_k \otimes m) \circ dW_k, \quad \dot{\xi} = \sum_{k=1}^K \mathrm{div}(\varrho \sigma_k) \circ dW_k, $$

其中 $ W_k $ 是独立的标准布朗运动，$ \sigma_k : \mathbb{T}^N \to \mathbb{R}^N $ 是光滑且无散度（$ \mathrm{div}\,\sigma_k = 0 $）的向量场，$ \circ $ 表示 Stratonovich 积分。这种噪声结构源于几何力学对湍流建模的研究（由 Holm 提出），也因近年来发现其可能具有正则化效应而受到关注。

从物理角度看，这类噪声模拟了由于小尺度湍流引起的宏观量的随机输运过程；从数学角度看，它保留了系统的某些守恒律特性——特别是能量结构比加性噪声更为稳定。

作者强调，尽管带随机强迫（stochastic forcing）的可压缩欧拉方程已有一定研究（如存在性、局部适定性等），但关于输运噪声情形的结果极为稀少。目前仅有 [1] 给出了局部强解的存在性结果，而全局弱解或测度值解的理论尚未建立。

因此，本文旨在填补这一空白，首次系统地研究带有输运噪声的可压缩欧拉方程的全局适定性问题。为此，作者引入了一类新的解概念：耗散型测度值鞅解（dissipative measure-valued martingale solution），这类解允许对非线性项进行“测量”以捕捉近似解序列中的振荡与集中现象，从而克服经典弱解不满足能量守恒与弱-强唯一性的缺陷。

此外，为了保证质量守恒，要求 $ \int_{\mathbb{T}^N} \dot{\xi} dx = 0 $，这自然由 $ \mathrm{div}(\varrho \sigma_k) $ 的积分性质满足（因为 $ \sigma_k $ 无散，且通过分部积分可得总积分为零），故该构造在物理上是自洽的。

函数空间方面，解被定义在以下空间中：

  • 密度 $ \varrho \in C_{\mathrm{loc}}([0,\infty); W^{-4,2}(\mathbb{T}^N)) \cap L^\infty_{\mathrm{loc}}(0,\infty; L^\gamma(\mathbb{T}^N)) $，
  • 动量 $ m \in C_{\mathrm{loc}}([0,\infty); W^{-4,2}(\mathbb{T}^N)) \cap L^\infty_{\mathrm{loc}}(0,\infty; L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^N)) $，

这些空间的选择兼顾了时间连续性和空间可积性，同时适应于分布意义下的偏微分方程求解框架。

特别地，在定义解时，非线性项 $ \frac{m \otimes m}{\varrho} $ 和压力项 $ p(\varrho) $ 被替换为两个参数化的正测度 $ R_{\mathrm{conv}}, R_{\mathrm{press}} $，分别称为浓度-振荡测度，用于描述极限过程中可能出现的奇异性。这使得即使当经典弱解无法定义时，仍可通过测度来表达动量方程的残差部分。

初始数据假设为确定性的：$ \varrho_0 \in L^\gamma(\mathbb{T}^N), m_0 \in L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^N) $，且 $ \varrho_0 \geq 0 $ a.e.，并附加技术性条件 $ \varrho_0^{-1/2} m_0 \in L^2(\mathbb{T}^N) $，以确保初始动能有限。

综上所述，本研究的核心问题是：在存在非线性、缺乏粘性正则化、且受随机输运影响的情况下，如何构建一类足够广泛但仍能保持物理合理性的解，并证明其全局存在性、弱-强唯一性以及马尔可夫选择的存在性。

主要定理或主要结果

定理 2.4（耗散型测度值鞅解的存在性）

设 $ \sigma_k \in C^\infty(\mathbb{T}^N, \mathbb{R}^N) $ 满足 $ \mathrm{div}\,\sigma_k = 0 $，$ k = 1,\dots,K $，且 $ \gamma > \frac{N}{2} $，$ N = 2,3 $。给定确定性初值 $ \varrho_0 \in L^\gamma(\mathbb{T}^N), m_0 \in L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^N) $，满足 $ \varrho_0 \geq 0 $ a.e. 且 $ \varrho_0^{-1/2} m_0 \in L^2(\mathbb{T}^N) $。

则存在一个耗散型测度值鞅解 $ ((\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P}), \varrho, m, R_{\mathrm{conv}}, R_{\mathrm{press}}, W) $，满足 Definition 2.3 中的所有条件，包括：

  • 连续性方程与动量方程在分布意义下成立；
  • 满足如下能量不等式：
$$ -\int_0^\infty \partial_t \psi(t) E_{\mathrm{Euler}}(t) dt \leq \psi(0) E_0, \quad \forall \psi \in C_c^\infty([0,\infty)), \psi \geq 0, $$

其中能量泛函定义为：

$$ E_{\mathrm{Euler}}(t) = \int_{\mathbb{T}^N} \left( \frac{1}{2} \frac{|m(t)|^2}{\varrho(t)} + \frac{a}{\gamma - 1} \varrho(t)^\gamma \right) dx + \frac{1}{2} \int_{\mathbb{T}^N} \mathrm{tr} R_{\mathrm{conv}}(t) dx + \frac{1}{\gamma - 1} \int_{\mathbb{T}^N} R_{\mathrm{press}}(t) dx, $$

$ E_0 $ 为初始总能量。

定理 4.10（弱-强唯一性）

若存在一个局部强路径解 $ (r,v) $ 在某个停时 $ \tau > 0 $ 内存在，则任意耗散型测度值鞅解在其生命时间内与该强解的概率分布一致。换言之，只要强解存在，测度值解就退化为其本身。

定理 4.11（马尔可夫选择的存在性）

对于所有确定性初值 $ (\varrho_0, m_0) $，存在一族解 $ \{ \mathbb{P}_{\varrho_0,m_0} \} $ 构成一个马尔可夫过程，即对未来状态的预测仅依赖于当前状态，而不依赖历史路径。

研究方法、关键技术和核心工具

a) 核心分析工具

  1. 粘性逼近法（Vanishing Viscosity Method）

    作者采用经典的粘性逼近策略：先考虑对应的随机可压缩纳维-斯托克斯方程（Navier–Stokes with transport noise），其粘性项提供额外的紧性，再令粘性系数 $ \mu_n, \lambda_n \to 0 $ 取极限。此方法在确定性可压缩流体力学中已被广泛应用（如 Lions, Feireisl 等人的工作），但在随机情形下需结合概率论工具。

  2. Jakubowski-Skorokhod 紧性定理

    在无限维空间中，传统的 Prokhorov 定理要求空间为波兰空间（Polish space），但 $ L^\infty $ 或测度空间通常不满足。为此，作者采用 Jakubowski 的 Skorokhod 表示定理版本，适用于拓扑空间中拟径向连续函数族上的紧性。这是处理 $ L^\infty_{\mathrm{weak-*}} $ 类空间的关键工具。

    具体而言，作者构造乘积路径空间：

    $$ X := X_\varrho \times X_m \times X_U \times X_C \times X_P \times X_W, $$

    其中各分量分别为密度、动量、应力、压强、对流项及布朗运动的空间，并赋予适当的弱*拓扑。利用估计得到的一致有界性与时间正则性，证明概率律在此空间中是紧的，从而提取收敛子列。

  3. Stratonovich-Itô 转换与交叉变差计算

    由于原始方程采用 Stratonovich 形式（保持经典链式法则），作者将其转化为 Itô 形式以便应用随机积分理论。关键步骤是计算 Itô–Stratonovich 校正项：

    $$ \int_0^t \mathrm{div}(\varrho Q) \circ dW = \int_0^t \mathrm{div}(\varrho Q) dW + \frac{1}{2} \sum_k \int_0^t \mathrm{div}(\sigma_k \otimes \sigma_k \nabla_x \varrho) ds, $$

    类似地对动量项处理。这些校正项是确定性的，且属于 $ W^{-\ell-1,2} $ 空间，可在分布意义下处理。

  4. 能量估计与一致界

    基于 Navier-Stokes 解的能量不等式：

    $$ -\int_0^\infty \partial_t \psi E_{\mathrm{NS}} dt + \int_0^\infty \int_{\mathbb{T}^N} S_n(\nabla u_n):\nabla u_n dx dt \leq \psi(0) E_0, $$

    得到 $ \varrho_n, m_n, \sqrt{\varrho_n} u_n, S_n(\nabla u_n) $ 的一致估计。尤其重要的是粘性耗散项的控制，使得速度梯度在 $ L^2 $ 中有界。

  5. Aldous 条件与时间正则性

    为获得路径空间中的紧性，需验证时间变量的 Hölder 连续性。作者利用 Aldous 准则和 Burkholder-Davis-Gundy 不等式估计随机积分项：

    $$ \mathbb{E} \left\| \int_0^\cdot \mathrm{div}(m_n \otimes Q) dW \right\|_{C^\alpha([0,T]; W^{-k,2})}^q \lesssim \mathbb{E} \int_0^T \|m_n\|_{L^{\frac{2\gamma}{\gamma+1}}}^q dt < \infty, $$

    结合 Sobolev 嵌入 $ L^p \hookrightarrow W^{-1,2} $（当 $ \gamma > N/2 $），最终得到 $ m_n \in C^\alpha([0,T]; W^{-3,2}) $ 的期望有界性。

b) 技术性技巧

  1. 新概率空间上的适应性恢复

    Skorokhod 定理给出的新过程未必在原滤流下适应。为此，作者在新空间上重新定义滤流：

    $$ \tilde{\mathcal{F}}_t = \sigma\left( \sigma_t[\tilde{\varrho}, \tilde{m}, \tilde{W}] \cup \sigma_t[R_{\mathrm{conv}}, R_{\mathrm{press}}] \cup \text{null sets} \right), $$

    使用“限制算子” $ r_t $ 和“随机分布的历史” $ \sigma_t[V] $ 来确保所有变量都是适应的。

  2. 鞅识别法（Elementary Method）

    为验证极限过程仍满足原始方程，作者采用 Breit-Hofman 等人发展的“基本方法”：定义函数泛函

    $$ M_t(\varrho, m, U, C, P) = \langle m, \phi \rangle - \langle m_0, \phi \rangle + \cdots $$

    并证明其在新空间上仍是鞅，且二次变差匹配预期。通过比较旧空间与新空间中期望值相等（利用等分布性与连续映射），推出：

    $$ \mathbb{E} \left[ h(r_s X) (M_t^2 - \Psi) \right] = 0 \Rightarrow \langle\!\langle M \rangle\!\rangle = \Psi, $$

    从而确认极限过程确实满足所需的随机微分关系。

  3. 能量平衡的确定性本质

    关键观察是：输运噪声是能量守恒的，因此极限过程的能量不等式不含随机项，仅为纯确定性不等式。这极大简化了极限过程的能量闭合论证，避免了像随机强迫情形那样需要处理随机积分的下半连续性问题。

与之前工作的比较

a) 突破与改进

  1. 首次建立全局存在性结果

    相较于文献 [1] 仅证明局部强解存在，本文首次构造了全局定义的广义解（测度值鞅解），突破了解只能短时间存在的局限。

  2. 放宽对解正则性的要求

    弱解虽存在，但常不唯一且违反物理原则（如能量守恒）。本文引入的测度值解能自动满足耗散不等式和弱-强唯一性，从而在多重解中选出符合物理直觉的那个。

  3. 处理更一般的噪声结构

    多数现有工作集中在加性或乘性随机强迫（如 $ \Phi(\varrho,m)dW $），而本文处理的是几何来源的输运噪声，其非线性耦合更强，且涉及链式法则修正。这是迈向真实湍流建模的重要一步。

  4. 简化能量闭合论证

    在随机强迫情形（如 [27]），能量不等式包含随机积分项，其下半连续性难以处理。而本文由于输运噪声不增加随机能量，极限过程的能量不等式为完全确定性，大大降低了技术难度。

  5. 首次证明马尔可夫选择的存在性

    对于非唯一解的问题，能否选出一组具有马尔可夫性的解是一个深刻问题。此前结果均针对随机强迫模型（如 [25, 5, 29]），本文是首个针对输运噪声情形建立 Markov selection 的工作，表明即使在复杂噪声下，仍可构建“最优”的解族。

  6. 推广维度与初始数据范围

    虽然技术上假设 $ \gamma > N/2 $，但这可通过其他逼近方式去除。相比早期工作对光滑初值的要求，本文允许 $ L^\gamma $ 初值，更具实用性。

综上，本文不仅拓展了解的存在性理论至全新的噪声类型，还在解的质量（唯一性、马尔可夫性）上取得实质性进展，为后续研究随机双曲系统提供了范式。

了解领域论文（摘要翻译）

1. Control and stabilization problem for a class of fourth-order nonlinear Schrödinger equation on boundaryless compact manifold

无边紧致流形上一类四阶非线性薛定谔方程的控制与镇定问题

作者: Yilin Song, Jiqiang Zheng, Ruihan Zhou

类别: math.AP

发布日期: 2025-11-25

arXiv ID: 2511.19890v1

链接: http://arxiv.org/abs/2511.19890v1

摘要翻译

中文标题: 无边紧致流形上一类四阶非线性薛定谔方程的控制与镇定问题

摘要翻译: 本文研究了一类定义在无边紧致流形上的四阶非线性薛定谔方程的控制与镇定问题，其中流形的维数 $d\in[1,5]$，方程形式如下： \begin{align*} i\partial_t u + (\Delta_g^2 - \beta\Delta_g)u = |u|^{2k}u, \end{align*} 其中 $k\in\mathbb{N}$。当 $1\leq d\leq4$ 且 $k\geq1$ 时，我们结合Loyola提出的方法与半经典分析，在仅满足几何控制条件（GCC）的情况下证明了镇定结果，从而去除了Capistrano-Filho-Pampu [Math. Z. (2022)] 中所需的唯一延拓性假设。对于 $d=5$ 的情形，我们聚焦于一个特殊情形，即单位球面 $\mathbb{S}^5$。通过在Bourgain空间中建立奇性的传播性质，我们在能量空间中证明了与低维情形类似的控制与镇定结果，推广了Laurent [SIAM J. Math. Anal. (2009)] 的结论。

不相关论文（基本信息）

1. Asymptotic analysis of small energy breathers for the nonlinear Klein-Gordon equation

非线性Klein-Gordon方程的小能量呼吸子的渐近分析

作者: Michał Kowalczyk, Yvan Martel

类别: math.AP

发布日期: 2025-11-25

arXiv ID: 2511.20553v1

链接: http://arxiv.org/abs/2511.20553v1

摘要: For a class of nonlinear Klein-Gordon equations, we prove that in the small energy limit, any sequence of breathers decomposes into a finite sum of decoupled, periodically modulated canonical solitons. Each of these solitons is asymptotically equal to an explicit sine-Gordon breather and the distance between them grows to infinity as the energy decreases to 0. Finally we prove that none of these breathers is centered in a bounded set provided that a certain non resonance condition holds.

2. Finite time blow up solutions for heat equations with Neumann boundary conditions on $\mathbb{R}_{+}^{4}$

$\mathbb{R}_{+}^{4}$ 上具诺伊曼边界条件的热方程的有限时间爆破解

作者: Xiang Fang, Juncheng Wei, Youquan Zheng

类别: math.AP

发布日期: 2025-11-25

arXiv ID: 2511.20451v1

链接: http://arxiv.org/abs/2511.20451v1

摘要: We consider the nonlinear heat equations with Neumann boundary conditions

$$ \begin{cases} u_{t}=Δu & \text{in}\ \mathbb{R}_{+}^{4} \times(0, T) ,\\ -\frac{d u}{d x_{4}}(\tilde{x}, 0, t) \ =u^2(\tilde{x}, 0, t)& \text{in}\ \mathbb{R}^{3} \times(0, T). \end{cases} $$

We establish the existence of a finite-time blow-up solution. Specifically, for any sufficiently small $T>0$ and any $k$ distinct points $q_{1},\dots,q_{k}\in \mathbb{R}^{3}$, there exists an initial datum $u_{0}$ such that the corresponding solution $u(x,t)$ blows up exactly at $q_{1},\dots,q_{k}$ as $t\nearrow T$. Furthermore, when $t\nearrow T$, the solution admits the asymptotic profile

$$ u(x,t)=\sum_{j=1}^{k}U_{μ_{j}(t),ξ_{j}(t)}(x)+Z_0^*(x)+o(1)\quad \text{as}~ t\nearrow T, $$

where

$$ U_{μ_{j}(t),ξ_{j}(t)}(x):=μ_{j}^{-1}(t) U\left(\frac{x-ξ_{j}(t)}{μ_{j}(t)}\right),~ x\in \mathbb{R}_{+}^{4}, $$

and $Z_{0}^{*}\in C_{0}^{\infty}(\mathbb{R}_{+}^{4})$ satisfying

$$ Z_{0}^{*}(q_{j},0)<0\quad \text{for all}\ j=1,\dots,k. $$

Here, $U(y)$ denotes the harmonic extension to $\mathbb{R}_{+}^{4}$ of the positive radially symmetric solution $\widetilde{U}$ to the fractional Yamabe problem $(-Δ)^{\frac{1}{2}} \widetilde{U} = \widetilde{U}^{2}$ in $\mathbb{R}^{3}$. For some constants $β_{j}>0$, the scaling parameters $μ{j}(t)$ and the translation parameters $ξ_{j}(t)$ satisfy

$$ μ_{j}(t)=β_{j}\frac{|\log 2T|(T-t)}{|\log(T-t)|^{2}}(1 + o(1)) \to 0,~ξ_{j}(t)\to (q_{j},0)\quad \text{as} ~t\nearrow T. $$

3. The periodic Camassa-Holm equation by the Riemann-Hilbert problem approach

基于黎曼-希尔伯特问题方法的周期Camassa-Holm方程

作者: Anne Boutet de Monvel, Iryna Karpenko, Dmitry Shepelsky, Lech Zielinski

类别: math.AP, nlin.SI

发布日期: 2025-11-25

arXiv ID: 2511.20442v1

链接: http://arxiv.org/abs/2511.20442v1

摘要: This work addresses the development of the Riemann-Hilbert problem (RHP) formalism (the Fokas method) for the Camassa-Holm equation under periodic boundary conditions. Particularly, we present a representation of the solution to this problem in terms of the solution of the associated Riemann-Hilbert problem, the data for which are determined by the initial data for the problem in terms of the associated spectral functions.

4. Extension and neural operator approximation of the electrical impedance tomography inverse map

电学阻抗断层扫描逆映射的延拓与神经算子逼近

作者: Maarten V. de Hoop, Nikola B. Kovachki, Matti Lassas, Nicholas H. Nelsen

类别: math.NA, cs.LG, math.AP

发布日期: 2025-11-25

arXiv ID: 2511.20361v1

链接: http://arxiv.org/abs/2511.20361v1

摘要: This paper considers the problem of noise-robust neural operator approximation for the solution map of Calderón’s inverse conductivity problem. In this continuum model of electrical impedance tomography (EIT), the boundary measurements are realized as a noisy perturbation of the Neumann-to-Dirichlet map’s integral kernel. The theoretical analysis proceeds by extending the domain of the inversion operator to a Hilbert space of kernel functions. The resulting extension shares the same stability properties as the original inverse map from kernels to conductivities, but is now amenable to neural operator approximation. Numerical experiments demonstrate that Fourier neural operators excel at reconstructing infinite-dimensional piecewise constant and lognormal conductivities in noisy setups both within and beyond the theory’s assumptions. The methodology developed in this paper for EIT exemplifies a broader strategy for addressing nonlinear inverse problems with a noise-aware operator learning framework.

5. A sharp Sobolev inequality on the Caffarelli-Kohn-Nirenberg hyperbolic space

关于Caffarelli-Kohn-Nirenberg双曲空间上的一个精确Sobolev不等式

作者: Baptiste Devyver, Louis Dupaigne, Pierre-Damien Thizy

类别: math.AP

发布日期: 2025-11-25

arXiv ID: 2511.20271v1

链接: http://arxiv.org/abs/2511.20271v1

摘要: In the Euclidean space $\mathbb{R}^d$, the sharp classical Sobolev inequality is equivalent by conformal invariance to a Sobolev inequality on the hyperbolic space $\mathbb{H}^d$. This inequality is sharp in dimension $d\geq 4$, but it is not in dimension $d=3$ by results of Benguria, Frank and Loss, as well as Mancini and Sandeep. In this article, we investigate a similar phenomenon for the Caffarelli-Kohn-Nirenberg inequality and its hyperbolic analogue. In our setting, the condition for improving the inequality reads $n\in [3,4)$, where $n$ is an ``effective dimension’'.

6. Symmetry and uniqueness of the positive solution for the critical Hartree equation on the Heisenberg group

Heisenberg群上临界Hartree方程正解的对称性与唯一性

作者: Shuijin Zhang, Jialin Wang, Yu Zheng, Xiang Li, Jijie Xu

类别: math.AP

发布日期: 2025-11-25

arXiv ID: 2511.20264v1

链接: http://arxiv.org/abs/2511.20264v1

摘要: We apply the moving plane method in integral forms to classify the positive solutions of the critical Hartree equation on Heisenberg group \begin{equation}\label{0.1} -Δ_{\mathbb{H}}u=\left(\int_{\mathbb{H}^{n}}\frac{|u(ξ)|^{Q^{\ast}μ}}{|ζ^{-1}ξ|^μ}\mathrm{d}ξ\right)|u|^{Q^{\ast}μ-2}u,~~~ζ,ξ\in\mathbb{H}^{n}, \end{equation} where $Δ_{\mathbb{H}}$ denotes the Kohn Laplacian, $u(ξ)$ is a real-valued function, $Q=2n+2$ is the homogeneous dimension of $\mathbb{H}^{n}$, $μ\in (0,Q)$ is a real parameter and $Q^{\ast}_μ=\frac{2Q-μ}{Q-2}$ is the upper critical exponent associated with the Hardy-Littlewood-Sobolev inequality on the Heisenberg group. By introducing the $\mathbb{H}$-reflection, we prove that the solutions of (\ref{0.1}) are cylindrical, upto Heisenberg translation and suitable scaling of function \begin{equation*}\label{0.2} u{0}(ζ)=u{0}(z,t)=\left((1+|z|^{2})^{2}+t^{2}\right)^{-\frac{Q-2}{4}},~~~ζ=(z,t)\in \mathbb{H}^{n}. \end{equation*} Furthermore, we show that these positive solutions are also CR inversion-symmetric with respect to the unit CC sphere. Consequently, we establish the uniqueness of positive solutions to equation (\ref{0.1}).

7. Extremals for sharp Poincaré-Sobolev inequalities: periodically perforated sets and beyond

尖锐Poincaré-Sobolev不等式的极值函数：周期性穿孔集及其推广

作者: Lorenzo Brasco, Luca Briani, Francesca Prinari

类别: math.AP, math.FA

发布日期: 2025-11-25

arXiv ID: 2511.20260v1

链接: http://arxiv.org/abs/2511.20260v1

摘要: We consider periodically perforated unbounded open sets and prove existence of extremals for the relevant sharp Poincaré-Sobolev embedding constant. The existence result holds no matter the shape or the regularity of the hole: it is sufficient that the latter is a compact set with positive capacity. We also show how to apply the main result in order to get a similar existence statement, for sets which are periodic in some directions and bounded in all the others.

8. Classification of positive solutions to a class of Laplace equations with a gradient term

一类含梯度项拉普拉斯方程正解的分类

作者: Jingbo Dou, Benfeng Shi, Tian Wu, Hua Zhu

类别: math.AP

发布日期: 2025-11-25

arXiv ID: 2511.20205v1

链接: http://arxiv.org/abs/2511.20205v1

摘要: In this paper, we investigate positive solutions to a class of Laplace equations with a gradient term on a complete, connected, and noncompact Riemannian manifold $(M^n,g)$ with nonnegative Ricci curvature, namely

$$ -Δu = f(u)|\nabla u|^q\quad\text{in }~M^n, $$

where $n\geqslant 3$, $q>0,$ and $f$ is a positive continuous function. We prove some Liouville theorems employing a key differential identity derived via the invariant tensor technique. In particular, for $f(u)=u^{\frac{2-q}{n-2}(n+\frac{q}{1-q})-1}$ is the second critical case in dimension $n=3,4,5$, without any additional conditions, such as integrable conditions on $u$, we show the rigidity for the ambient manifold and classification result of positive solutions. To our knowledge, this is the first rigidity result for equations with gradient terms in the second critical case. Moreover, this result confirms that all solutions must be of the form found in \cite{BV-GH-V2019}.

9. A Calderón Problem for the Dirac operator with chiral boundary conditions

具有手征边界条件的狄拉克算子的卡尔德隆问题

作者: Carlos Valero

类别: math.AP, math.DG

发布日期: 2025-11-25

arXiv ID: 2511.20176v1

链接: http://arxiv.org/abs/2511.20176v1

摘要: We consider on a spin manifold with boundary a Dirac operator $D_A$ with chiral boundary conditions, twisted by a unitary connection $A$. When $m$ is not in the chiral spectrum of $D_A$, we define an analogue of the Dirichlet-to-Neumann map for the Dirac equation $D_A - m$, which we call the boundary conjugation map, and show that it is a pseudodifferential operator of order $0$ on the boundary. We show that in dimension greater than 2, its symbol determines the Taylor series of the metric and connection modulo gauge on the boundary when $m \neq 0$ and $m^2$ is not in the Dirichlet spectrum of $D_A^2$. We go on to show that a real-analytic Riemannian manifold and twisted spinor bundle with twisted spin connection can be recovered from its boundary conjugation map. Under further hypotheses, one can recover the unitary connection up to global gauge equivalence and the complex spinor bundles. Similar results hold in dimension $2$ when the auxiliary bundle and connection are absent.

10. Rate independent capillary motion on a narrow Wilhelmy plate

窄型威尔赫利板上非速率相关的毛细运动

作者: Carson Collins, William M Feldman

类别: math.AP

发布日期: 2025-11-25

arXiv ID: 2511.20146v1

链接: http://arxiv.org/abs/2511.20146v1

摘要: We study a rate independent energetic model of the Wilhelmy plate experiment in capillarity. The evolution is driven by vertical motions of the plate. We show stability of energy solutions to the evolution, in the sense used in the rate-independent systems literature, as the ratio between container width and plate width goes to infinity. In particular, we show that the volume-constraint for the finite-ratio problem disappears in the limit. This leads to a volume-unconstrained Dirichlet-forced evolution, a setting where monotonicity, uniqueness, and contact line regularity properties have been established in previous literature. Our result is based on using comparison principle techniques for the prescribed mean curvature equation with capillary contact angle condition that characterizes the liquid surface at equilibrium. Through barrier arguments, we are able to develop asymptotics for the energy which give us control independent of the container-to-plate ratio.

11. Asymptotic behavior of least energy solutions to the nonlinear Hartree equation near critical exponent

非线性Hartree方程在临界指数附近的最小能量解的渐近行为

作者: Silvia Cingolani, Minbo Yang, Shunneng Zhao

类别: math.AP

发布日期: 2025-11-25

arXiv ID: 2511.20136v1

链接: http://arxiv.org/abs/2511.20136v1

摘要: In this paper, we study that the nearly critical nonlocal problem \begin{equation*} \left\lbrace \begin{aligned} &-Δu=(|x|^{-{(n-2)}}\ast u^{p-ε})u^{p-1-ε} \quad \mbox{in}\quad Ω, &u>0\quad \mbox{in}\quad\hspace{1mm} Ω, &u=0\quad \mbox{on}\hspace{2.5mm}\partialΩ, \end{aligned} \right. \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^n$ for $n=3,4,5$, $\ast$ denotes the standard convolution, $ε>0$ is a small parameter and $p=\frac{n+2}{n-2}$ is energy-critical exponent. We study the asymptotic behavior of least energy solutions as $ε\rightarrow0$. These solutions are shown to blow-up at exactly one point $x_0$ and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied. Finally, in order to further locate the blowing-up point $x_0$, we prove that $x_0$ is a global maximum point of the Robin’s function of $Ω$.

与黎曼ζ函数相关的薛定谔型方程的柯西问题

作者: Bensaid Mohamed

类别: math.AP, math-ph

发布日期: 2025-11-25

arXiv ID: 2511.20128v1

链接: http://arxiv.org/abs/2511.20128v1

摘要: We study the Cauchy problem in the space $H^1(Σ)$ for a nonlinear damped Schrödinger equation of the form

$$ i u_t + Δu + i λu \, ζ(|u|+1) = 0, \quad u(0,x) = u_0, $$

where $ζ$ denotes the Riemann zeta function. This equation involves a damping term inspired by a fundamental object from number theory. We first establish the uniqueness of solutions in the sense of distributions. Then, by considering a regularized problem, we prove the existence of a global solution in $H^1(Σ)$, using uniform energy estimates and compactness arguments. Finally, we show that the limiting solution indeed satisfies the original equation in the weak sense,In the last part of this work, we provide an explicit estimate for the mass associated with the equation \begin{equation}\tag{E0}\label{5454} i v_t + Δv + iλ\frac{v}{|v|} = 0, \qquad v(0,x)=v_0.

13. A not-so-strange term coming from somewhere

一个并不那么奇怪的术语的由来

作者: Giacomo Canevari, Kirill Cherednichenko, Arghir Zarnescu

类别: math.AP

发布日期: 2025-11-25

arXiv ID: 2511.20071v1

链接: http://arxiv.org/abs/2511.20071v1

摘要: We consider Laplace’s equation in a periodically perforated domain, with Robin boundary conditions on the holes and a Robin coefficient inversely proportional to the total surface area of the holes. We show that, in a certain critical regime, the homogenised equation contains an additional zero-order term, which depends nonlinearly on the Robin coefficient. This additional term is defined in terms of solutions to a Steklov-type spectral problem, where the eigenvalue appears both in the equation and the boundary conditions. The strange term'' for the problem with Robin conditions converges to zero as the Robin parameter tends to zero and, at least under suitable assumptions, it converges to the capacitary strange term’’ for the Dirichlet problem as the Robin parameter tends to infinity.

14. Equi-integrable approximation of Sobolev mappings between manifolds

流形间Sobolev映射的等度可积逼近

作者: Jean Van Schaftingen

类别: math.AP, math.CA

发布日期: 2025-11-25

arXiv ID: 2511.20064v1

链接: http://arxiv.org/abs/2511.20064v1

摘要: We show that limits of sequences of smooth maps between compact Riemannian manifolds with equi-integrable $W^{1, p}$-Sobolev energy can always be strongly approximated by smooth maps, giving a counterpart of Hang’s density result in $W^{1, 1}$ for the Sobolev space $W^{1, p}$ with integer $p \ge 2$. Our result extends to higher-order Sobolev spaces and is straightforward in fractional Sobolev spaces. We also provide a proof based on the weak continuity of Jacobians in the cases where the cohomological criterion of Bethuel, Demengel, Colon and Hélein applies.

15. The Boltzmann equation in an infinite layer: spectrum and asymptotics toward the heat equation

无限薄层中的玻尔兹曼方程：谱分析及向热方程的渐近行为

作者: Hongxu Chen, Renjun Duan, Shuangqian Liu

类别: math.AP

发布日期: 2025-11-25

arXiv ID: 2511.19950v1

链接: http://arxiv.org/abs/2511.19950v1

摘要: In the paper, we develop spectral theory to analyze the sharp asymptotic behavior of solutions to the Boltzmann equation around global Maxwellians in a three-dimensional infinite layer $\mathbb{R}^2\times (-1,1)$. The isothermal diffuse reflection boundary condition is imposed on two parallel infinite planes at $x_3=\pm 1$. The main difficulties lie in the fact that the direct Fourier transform is not applicable to the vertical $x_3$-variable, and the linear collision operator $K$ loses its compactness on $L^2((-1,1)\times \R^3_v)$ although it is compact on $L^2(\R^3_v)$. By introducing a regularization operator $K_n$ via the finite-dimensional Fourier series truncation in $L^2(-1,1)$, we study the spectrum of the linearized initial-boundary value approximation problem, establish the resolvent estimates, and identify the leading diffusive eigenvalue. This spectral structure governs the sharp asymptotic dynamics of the original linear problem as $n\to \infty$, enabling us to construct the large-time behavior for the nonlinear problem and rigorously prove that the solution converges with a faster rate toward that of the two-dimensional heat equation in the horizontal direction.

16. From the quantum Boltzmann operator to the quantum Landau operator

从量子玻尔兹曼算子到量子朗道算子

作者: Maria Pia Gualdani, Nataša Pavlović, Justin Toyota, Dominic Wynter

类别: math.AP, math-ph

发布日期: 2025-11-25

arXiv ID: 2511.19823v1

链接: http://arxiv.org/abs/2511.19823v1

摘要: In this manuscript we derive the quantum Landau operator as the weak-coupling limit of the quantum Boltzmann operator (also known as the Uehling-Uhlenbeck operator). We consider both Fermi-Dirac and Bose-Einstein statistics. Our approach is inspired by the work by Benedetto and Pulvirenti, where the classical Landau operator was derived from the quantum Boltzmann operator. To capture the ternary term in the quantum Landau operator, we introduce a new two-parameter scaling that preserves the quantum effects in the limit. Furthermore, we provide an explicit rate of convergence that depends on the regularity of the interaction potential.

17. On the Schiffer and Berenstein conjectures for centrally symmetric convex domains in the plane

关于平面上中心对称凸区域的Schiffer猜想与Berenstein猜想

作者: Guowei Dai, Yingxin Sun, Juncheng Wei, Yong Zhang

类别: math.AP

发布日期: 2025-11-25

arXiv ID: 2511.19819v1

链接: http://arxiv.org/abs/2511.19819v1

摘要: Let $Ω$ be a bounded, convex, centrally symmetric in $\mathbb{R}^{2}$ with a connected $C^{2,ε}$ ($ε\in(0,1)$) boundary. We show that, if the following overdetermined elliptic problem \begin{equation} -Δu=αu,, \text{in},,Ω, ,, u=0,,\text{on},, \partialΩ,,,\frac{\partial u}{\partial n} =c,,\text{on},,\partialΩ\nonumber \end{equation} has a nontrivial solution corresponding to a sufficiently large eigenvalue $α$, then $Ω$ is a disk, which is the partially affirmative answer to the Berenstein conjecture. Similarly, we show that, if $Ω$ has a Lipschitz connected boundary and the following overdetermined elliptic problem \begin{equation} -Δu=αu,, \text{in},,Ω, ,, \frac{\partial u}{\partial n}=0,,\text{on},, \partialΩ,,,u =c,,\text{on},,\partialΩ\nonumber \end{equation} has a nontrivial solution corresponding to a sufficiently large eigenvalue $α$, then $Ω$ is also a disk, which is the partially affirmative answer to the Schiffer conjecture.

18. Structured Continuity Equations in Fibred Wasserstein Spaces

纤维Wasserstein空间中的结构化连续性方程

作者: Benoît Bonnet-Weill, Nastassia Pouradier Duteil

类别: math.AP, math.PR

发布日期: 2025-11-24

arXiv ID: 2511.19784v1

链接: http://arxiv.org/abs/2511.19784v1

摘要: In this article, we develop a comprehensive ODE-theory for structured continuity equations in fibred probability spaces, which represent a class of heterogeneous PDEs arising as the meanfield limit nonexchangeable particle systems. After investigating in depth the topologies induced by the so-called fibred and classical Wasserstein metrics on such probability spaces, we establish quantitative Cauchy-Lipschitz and qualitative Carathéodory-Peano well-posedness results for structured continuity equations, along with precise correspondences between this class of evolutions, classical Lagrangian dynamics, and continuity equations. In keeping with what has long been known for exchangeable dynamics, we derive a general meanfield approximation result by solutions of nonexchangeable particle systems, along with a quantitative variant thereof under practically reasonable regularity assumptions on the driving field and initial data.

19. Viability Theory in the $1$-Wasserstein Space

1-维瑟斯坦空间中的生存性理论

作者: Benoît Bonnet-Weill, Alberto Domínguez Corella, Hélène Frankowska

类别: math.AP

发布日期: 2025-11-24

arXiv ID: 2511.19779v1

链接: http://arxiv.org/abs/2511.19779v1

摘要: In this article, we establish necessary and sufficient viability conditions for continuity inclusions over the 1-Wasserstein space. Depending on the regularity properties of the dynamics, we derive two results which are based on fairly different proof strategies. When the admissible velocities are Lipschitz in the measure variable, we show that it is necessary and sufficient for viable solutions to exist that the latter intersect the graphical derivative of the constraints. On the other hand, when the admissible velocities are merely upper semicontinuous in the measure variable, we provide a sufficient condition for viability involving the infinitesimal behaviour of their Aumann integral over a neighbouring set of measures.

20. Higher integrability for parabolic PDEs with generalized Orlicz growth

具有广义Orlicz增长的抛物型偏微分方程的高阶可积性

作者: Peter Hästö, Jihoon Ok

类别: math.AP

发布日期: 2025-11-24

arXiv ID: 2511.19758v1

链接: http://arxiv.org/abs/2511.19758v1

摘要: We prove higher integrability of the gradient of weak solutions to nonlinear parabolic systems whose prototype is

$$ \partial_t u-\mathrm{div}\Big(\frac{\varphi'(z, |\nabla u|)}{|\nabla u|}\nabla u\Big) =0, \qquad u=(u^1,\dots,u^N), $$

where $\varphi$ is a generalized Young function. Special cases of our main theorem include previously known results for the $p$-growth, the variable exponent and the double phase growth. Also included are previously unknown borderline double phase growth and perturbed variable exponent growth, among others. The problem is controlled by a natural requirement of comparison of $\varphi$ between points in intrinsic parabolic cylinders via an (A1)-condition, which unifies disparate conditions from the special cases. Moreover, we handle both the singular and degenerate cases at the same time, providing a simple proof of a reverse Hölder type inequality, which is new even in the $p$-growth case.

21. On the microlocal phase space concentration of Schrödinger evolutions

关于薛定谔演化在微局域相空间中的集中性研究

作者: Gianluca Giacchi, Davide Tramontana

类别: math.AP, math.QA, quant-ph

发布日期: 2025-11-24

arXiv ID: 2511.19733v1

链接: http://arxiv.org/abs/2511.19733v1

摘要: In this work, we investigate the microlocal properties of the evolutions of Schrödinger equations using metaplectic Wigner distributions. So far, only restricted classes of metaplectic Wigner distributions, satisfying particular structural properties, have allowed the analysis of microlocal properties. We first extend the microlocal results to all metaplectic Wigner distributions, including the well-known Kohn-Nirenberg quantization, and examine these findings in the framework of Fourier integral operators with quadratic phase. Finally, we analyze the phase space concentration of the (cross) Wigner distribution arising from the interaction of two states, with particular attention to interactions generated by certain Schrödinger evolutions. These contributions enable a more refined study of the so-called ghost frequencies.

22. On the Fujita Phenomenon for a Forced Spatio-Temporal Fractional Diffusion Equation

关于一类受迫时空分数阶扩散方程的Fujita现象研究

作者: Rihab Ben Belgacem, Mohamed Majdoub

类别: math.AP

发布日期: 2025-11-24

arXiv ID: 2511.19424v1

链接: http://arxiv.org/abs/2511.19424v1

摘要: We study the Cauchy problem for a semilinear fractional diffusion equation with a time-dependent forcing term:

$$ \partial_t^αu + (-Δ)^{\mathsf{s}} u = |u|^p + t^σ\,\mathbf{w}(x), \quad (t,x) \in (0,\infty) \times \mathbb{R}^N, $$

with parameters $α, \mathsf{s} \in (0,1)$, $σ> -α$, and a continuous function $\mathbf{w}$. The operator $\partial_t^α$ denotes the Caputo fractional derivative. Our main contributions are threefold. First, we prove the local existence of mild solutions and demonstrate a finite-time blow-up in the subcritical regime, provided $\int_{\mathbb{R}^N} \mathbf{w}(x)\,dx > 0$. Second, for the supercritical case with $-α< σ< 0$, we establish the global existence for sufficiently small initial data and a forcing term, and we determine the critical exponent to be

$$ p_F=\frac{Nα-2\mathsf{s}σ}{Nα-2\mathsf{s}(α+σ)}. $$

Finally, for this supercritical range, we prove a more robust global existence result under assumptions requiring only local smallness and controlled growth of the data.

23. Chernoff-Mehler Approximation for Lévy Processes with Drift

带漂移的Lévy过程的Chernoff-Mehler逼近

作者: Max Nendel

类别: math.PR, math.AP

发布日期: 2025-11-24

arXiv ID: 2511.19414v1

链接: http://arxiv.org/abs/2511.19414v1

摘要: In this paper, we study an approximation scheme for Lévy processes with drift in terms of a representation that is akin to the celebrated Mehler formula for Lévy-Ornstein-Uhlenbeck processes. The approximation scheme is based on a variant of the Chernoff product formula on the space of bounded continuous functions. In a first step, we provide sufficient and necessary conditions for arbitrary families of probability measures, indexed by positive real numbers, to give rise to a convolution semigroup via a Chernoff approximation on the space of bounded continuous functions, equipped with the mixed topology. In this context, we provide explicit criteria both for the convergence of subsequences and the entire family, and discuss fine properties related to the domain of the associated generator of the Lévy process and the infinitesimal behavior of the approximating family of measures. In a second step, we enrich the family of measures by a deterministic component and derive explicit conditions that ensure both the convergence of subsequences and the entire family to a Lévy process with drift under a Chernoff approximation. In a series of examples, we show that our general conditions on the dynamics are satisfied, for example, by flows of Lipschitz ordinary differential equations, Euler schemes, and arbitrary Runge-Kutta methods, and that the Central Limit Theorem can be subsumed under our framework.

24. Wigner and Gabor phase-space analysis of propagators for evolution equations

演化方程传播子的Wigner与Gabor相空间分析

作者: Elena Cordero, Gianluca Giacchi, Luigi Rodino

类别: math.AP, math-ph, quant-ph

发布日期: 2025-11-24

arXiv ID: 2511.19400v1

链接: http://arxiv.org/abs/2511.19400v1

摘要: We study the Wigner kernel and the Gabor matrix associated with the propagators of a broad class of linear evolution equations, including the complex heat, wave, and Hermite equations. Within the framework of time-frequency analysis, we derive explicit expressions for the Wigner kernels of Fourier multipliers and establish quantitative decay estimates for the corresponding Gabor matrices. These results are obtained under symbol regularity conditions formulated in the Gelfand-Shilov scale and ensure exponential off-diagonal decay or quasi-diagonality of the matrix representation. We believe this approach can be extended to more general symbols in the pseudodifferential setting, improving the existing results in terms of their Gabor matrix decay. For the complex heat equation, we obtain closed-form formulas exhibiting both dissipative and oscillatory behavior governed respectively by the real and imaginary parts of the diffusion parameter. The modulus of the Gabor matrix is shown to display Gaussian decay and temporal spreading consistent with diffusion phenomena. In contrast, the complex Hermite equation is analyzed via Hörmander’s metaplectic semigroup, where the propagator decomposes as the product of a real Hermite semigroup and a fractional Fourier transform. In this setting, the Gabor matrix retains its Gaussian shape while undergoing a pure rotation on the time-frequency plane, reflecting the symplectic structure of the underlying flow. The analysis provides a unified operator-theoretic and phase-space perspective on parabolic and hyperbolic evolution equations, linking the geometry of their symbols with the sparsity and localization properties of their Gabor representations. Explicit formulas are given in a form suitable for numerical computation and visualization of phase-space dynamics.