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1. 输运噪声下的埃克曼边界层

Ekman Boundary Layers Under Transport Noise

作者: Daniel Goodair, Szymon Sobczak

类别: math.AP, math.PR

发布日期: 2026-09-18

arXiv ID: 2609.22011v1

摘要翻译: 我们考虑位于两块水平板之间的三维旋转 Navier-Stokes 方程，该方程带有输运-拉伸噪声，并研究垂直黏度趋于零与快速旋转的联合极限。我们的驱动噪声仅依赖于水平坐标，且其垂直分量随黏度消失。假设初始数据纯粹是水平的，我们构造鞅弱解，这些解在 $L^2_ωL^

Comment: 35 pages

链接: http://arxiv.org/abs/2609.22011v1

1. 研究对象和背景

本文研究三维旋转 Navier-Stokes 方程在垂直黏度趋于零且旋转速率趋于无穷大时的联合极限行为。具体而言，考虑定义在圆柱域 $\Omega = \mathbb{T}^2 \times (0,1)$ 上的速度场 $u$，其满足如下带有 Stratonovich 型输运-拉伸噪声的随机偏微分方程：

其中 $B(u,u) = P((u \cdot \nabla)u)$ 为非线性对流项，$A_h = -P \sum_{j=1}^2 \partial_j^2$ 为水平 Stokes 算子，$A_z = -P \partial_3^2$ 为垂直 Stokes 算子。参数 $\nu$ 代表湍流垂直黏度，$\varepsilon$ 为 Coriolis 频率的倒数（即 Rossby 数），$P$ 为投影到无散、法向分量为零且水平均值为零向量场上的正交投影。Coriolis 力项由 $e_3 \wedge u / \varepsilon$ 建模，其中 $e_3$ 为垂直单位向量。噪声项 $G_h u_t \circ dW_t$ 和 $\nu^\alpha G_z u_t \circ dW_t$ 遵循 Holm 提出的随机 Lie 输运原理，其中空间相关函数 $\xi_i: \mathbb{T}^2 \to \mathbb{R}^3$ 仅依赖于水平坐标 $(x,y)$，且分解为水平部分 $\xi^h_i$ 和垂直部分 $\xi^z_i$。指数 $\alpha$ 满足 $\alpha > 1/2$（或在特定条件下 $\alpha = 1/2$）。

该问题源于地球物理流体力学中的经典模型。在实际大气或海洋流动中，由于 Coriolis 力的作用，流体倾向于形成 Taylor-Proudman 柱状结构，即速度场独立于垂直方向 $z$。然而，无滑移边界条件 $u|_{\partial \Omega} = 0$ 与 Taylor-Proudman 定理相矛盾，导致在边界附近形成厚度为 $O(\sqrt{\nu \varepsilon})$ 的埃克曼边界层。当 $\nu \to 0$ 且 $\varepsilon \to 0$ 且比值 $\nu/\varepsilon \to \beta$ 时，系统演化至一个二维阻尼随机 Navier-Stokes 方程：

其中 $w: \mathbb{T}^2 \to \mathbb{R}^2$ 为二维无散且零均值速度场，$\sqrt{2\beta}$ 为阻尼系数，源自埃克曼泵吸效应导致的“旋降”现象。

确定性情形下，Grenier 和 Masmoudi 已证明类似收敛性。但在随机情形下，尤其是涉及输运-拉伸噪声时，理论进展较少。现有工作如 [47] 仅考虑加性噪声，而本文处理的是乘性的、具有几何结构的输运-拉伸噪声。主要难点在于：1) 噪声的垂直分量 $\xi^z_i$ 虽然不依赖 $z$，但在边界处不满足 $ abla \cdot abla f = 0$ 的典型兼容性条件；2) 垂直导数项在边界处的奇异性与旋转的正则化效应之间的平衡尚未被充分理解；3) 投影算子 $P$ 的特殊性质（包含零均值约束）使得标准的 Itô-Stratonovich 校正项简化失效。本文旨在填补这一空白，构造弱解并证明其在 $L^2_\omega L^\infty_t L^2_x$ 范数下收敛至二维强解。

2. 主要定理或主要结果

本文的主要结果陈述于定理 3.3。假设初始数据 $u_0$ 是纯水平的（即 $u_0(x,y,z) = (u_{0,1}(x,y), u_{0,2}(x,y), 0)$），且属于适当的 Sobolev 空间。设 $\nu, \varepsilon \to 0$ 且 $\nu/\varepsilon \to \beta \in [0, \infty)$。对于 $\alpha > 1/2$（若 $\beta=0$ 则允许 $\alpha=1/2$），作者构造了方程 (1) 的鞅弱解序列 $\{u^{\nu,\varepsilon}\}$。这些解满足以下收敛性：

其中 $w$ 是二维阻尼随机 Navier-Stokes 方程 (3) 的唯一强解。更具体地，收敛是在概率空间 $\Omega$ 上的 $L^2$ 意义下成立的，即：

这里 $w$ 被视为三维向量场的平凡扩展。阻尼系数 $\beta$ 直接取决于极限比值 $\lim_{\nu,\varepsilon \to 0} \nu/\varepsilon$。特别地，当 $\beta = 0$ 时，极限方程无显式阻尼项，但边界层的消失仍受旋转正则化的影响。当 $\beta > 0$ 时，阻尼项 $-\sqrt{2\beta} w_t$ 显著存在，反映了埃克曼边界层对整体流动的耗散效应。此外，作者证明了所构造的鞅弱解在 Galerkin 近似水平上的一致性，并通过 Skorokhod 定理将概率空间统一，从而避免了每个参数对应不同概率空间的复杂性。

3. 研究方法、关键技术和核心工具

证明的核心障碍在于处理输运-拉伸噪声与旋转项及边界条件的相互作用。主要技术工具和创新点包括：

1. Itô-Stratonovich 校正项的处理： 将 Stratonovich 方程转换为 Itô 形式后，出现二次变差项 $\frac{1}{2} \sum_i P \tilde{G}_i P \tilde{G}_i u_t dt$。由于投影算子 $P$ 不仅包含无散条件还包含零均值条件，标准的恒等式 $P \tilde{G}_i P \tilde{G}_i = P \tilde{G}_i \tilde{G}_i$ 不再成立。作者引入了新的估计技巧，利用输运算子 $T_\phi$ 和拉伸算子 $S_\phi$ 的交换子性质，证明了如下不等式：

   $$ \langle P \tilde{G}_i P \tilde{G}_i u, u \rangle + \| P \tilde{G}_i u \|^2 \leq \| \xi_i \|^2_{W^{1,\infty}} \left( c_\delta \| u \|^2 + \delta \| \nabla_h u \|^2 + \delta \nu^{2\alpha} \| \partial_3 u \|^2 \right) $$

   其中 $c_\delta$ 独立于 $\nu$，高阶导数项可被黏性项吸收。这一估计克服了传统方法中因 $P$ 的非交换性导致的困难。

2. 边界修正项 $B$ 的构造与分析： 借鉴 Grenier-Masmoudi 的方法，引入边界修正项 $B$，使得 $u - w - B$ 在边界处为零。关键创新在于 $B$ 是随机的，且依赖于 $w$ 的演化。作者将 $B$ 视为 $w$ 的一阶线性算子，逐项分析其在 Itô 形式下的演化方程。这要求精确控制 $B$ 的时间导数和空间导数，特别是处理 $B$ 中出现的噪声项及其 Itô-Stratonovich 校正项。

3. 能量估计与随机 Gronwall 不等式： 在 Galerkin 近似层面进行能量估计。由于弱解的不唯一性，概率空间依赖于 $\nu, \varepsilon$。作者应用 Skorokhod 定理在 Galerkin 近似族级别统一概率空间。为了获得 $L^2_\omega$ 收敛而非仅仅是概率收敛，作者利用了二维输运噪声在涡量水平上的特殊结构：在二维情况下，输运噪声在涡量方程中表现为纯输运项，在 $L^2$ 能量估计中完全抵消，从而提供了确定性的界：

   $$ \sup_{t \in [0,T]} \| w_t \|^2_{H^1} + \int_0^T \| w_s \|^2_{H^2} ds \leq C \| w_0 \|^2_{H^1} $$

   结合随机 Gronwall 不等式，最终得到 $L^2_\omega$ 收敛结果。

4. 噪声结构的精细缩放： 垂直噪声分量 $\xi^z_i$ 按 $\nu^\alpha$ 缩放，确保其在极限方程中消失而不留下额外阻尼。这一选择基于 Itô-Stratonovich 校正项中最高阶垂直导数 $(\xi^z_i)^2 \partial_3^2 u$ 的量级分析，使其与垂直拉普拉斯算子的贡献相匹配或更小，从而避免干扰埃克曼边界层的形成机制。

4. 与之前工作的比较

与确定性理论相比，本文首次将 Grenier-Masmoudi 的埃克曼边界层收敛结果推广到带有输运-拉伸噪声的随机情形。在确定性框架中，Grenier 和 Masmoudi [25] 通过构造边界修正项并仔细分析能量估计中的各项，证明了三维旋转 Navier-Stokes 方程在 $\nu/\varepsilon \to \beta$ 时收敛至二维阻尼方程。本文继承了这一思路，但面临额外的随机挑战：噪声项的乘性特征、Itô-Stratonovich 校正项的复杂结构以及概率空间的非一致性。

与最近的随机流体力学工作相比，本文有几个显著区别：

1. 噪声类型：工作 [47] 考虑的是加性噪声，且仅在水平方向作用。加性噪声不会改变方程的结构复杂度，而本文处理的输运-拉伸噪声是乘性的，且涉及水平和垂直方向的耦合。这使得能量估计更加复杂，需要处理噪声与解的非线性相互作用。
2. 投影算子的处理：在 [19] 中，第一作者研究了无旋转情况下的无黏极限，其中投影算子不包含零均值约束，因此可以利用 $P G_i P = P G_i$ 的弱交换性来简化 Itô-Stratonovich 校正项。而在本文中，为了消除 Coriolis 项的影响，必须使用包含零均值约束的投影算子 $P$，导致弱交换性失效。作者通过开发新的估计技术解决了这一问题。
3. 收敛强度：许多随机 PDE 的收敛结果仅给出概率收敛或几乎处处收敛。本文通过利用二维输运噪声在涡量水平上的特殊对称性，获得了更强的 $L^2_\omega L^\infty_t L^2_x$ 收敛结果。这是一个重要的技术突破，表明噪声的特定结构可以提供额外的正则性或确定性控制。
4. 初始数据正则性：工作 [47] 要求初始数据 $w_0 \in H^2$ 以获得高概率集上的均匀界，从而应用标准 Gronwall 论证。本文仅需 $w_0 \in H^1$，通过使用随机 Gronwall 不等式和更精细的能量估计，降低了初始数据的正则性要求。

局限性方面，本文假设初始数据是纯水平的。如果初始数据包含垂直分量，极限方程将变得更加复杂，可能涉及更多的维度约化机制。此外，当 $\beta = 0$ 时，虽然理论上允许 $\alpha = 1/2$，但证明中可能需要更细致的分析来处理临界尺度下的边界层行为。未来工作可以探索非水平初始数据的情形，以及更一般的噪声结构（例如依赖 $z$ 的相关函数）。

2. 粘性 Katz–Pavlović 二进模型的精确正则性阈值

Sharp Regularity Thresholds for Viscous Katz–Pavlović Dyadic Models

作者: Ziqian Zhang

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21312v1

摘要翻译: 我们研究两种壳层设置下具有非负光滑初始数据的无强迫粘性 Katz–Pavlović 二进模型。对于超指数尺度 $N_n=N_0^{b^n}$，其中 $N_0>1$ 且 $1<b<2$，我们证明 $α=1/(b+2)$ 是精确耗散阈值：在此值及以上，每个此类数据都产生全局光滑解；而低于此值时，适当紧支数据在有限时间内失去正则性。对于几何尺度 $N_n=N_0Λ^n$，当 $α\geq1/3$ 且 $4α-1>(\log2)/(2\logΛ)$ 时，全局正则性成立。特别地，对于 $Λ>2^{3/2}$，该结果达到 $α=1/3$，并与已知的经典爆破定理一起确定了经典壳层比中的精确阈值。

Comment: 38 pages

链接: http://arxiv.org/abs/2609.21312v1

1. 研究对象和背景

本文研究的是无外力驱动的粘性Katz–Pavlović对数模型，该模型是一组用于模拟不可压缩Navier-Stokes方程中能量级联现象的常微分方程系统。模型保留了相邻频率壳层之间的非线性相互作用，其形式为：

其中 $u_{-1}(t) \equiv 0$，$\nu > 0$ 是粘性系数，$N_n$ 表示第 $n$ 个频率壳层的尺度，$\alpha \geq 0$ 是耗散参数。

该模型在数学流体力学中具有重要意义，因为它提供了一个简化但保留核心物理机制（非线性能量转移与粘性耗散的竞争）的框架，有助于理解三维Navier-Stokes方程的全局正则性问题。已有理论表明，当耗散参数 $\alpha$ 足够大时，解保持全局光滑；而当 $\alpha$ 较小时，存在有限时间奇点。然而，对于不同的壳层尺度设置（如几何尺度 $N_n = N_0 \Lambda^n$ 或超指数尺度 $N_n = N_0^{b^n}$），具体的临界耗散阈值尚未完全确定。

本文旨在填补这一空白，针对两种壳层设置（超指数尺度和几何尺度）分别给出精确的正则性阈值。具体而言，对于超指数尺度 $N_n = N_0^{b^n}$（其中 $1 < b < 2$），作者证明了 $\alpha_c = 1/(b+2)$ 是全局正则性的精确阈值：当 $\alpha \geq \alpha_c$ 时，所有非负光滑初值都产生全局光滑解；而当 $\alpha < \alpha_c$ 时，存在适当的紧支集初值导致有限时间失去正则性。对于几何尺度 $N_n = N_0 \Lambda^n$，作者在条件 $\alpha \geq 1/3$ 且 $4\alpha - 1 > (\log 2)/(2 \log \Lambda)$ 下证明了全局正则性。特别地，当 $\Lambda > 2^{3/2}$ 时，结合已知的经典爆破定理，确定了 $\alpha = 1/3$ 为经典壳层比下的精确阈值。

2. 主要定理或主要结果

本文的主要结果分为两部分，分别对应于超指数尺度和几何尺度下的正则性阈值。

定理 1.1（超指数阈值）：设 $N_n = N_0^{b^n}$，其中 $N_0 > 1$，$1 < b < 2$，且 $\nu > 0$。 (1) 若 $\alpha \geq 1/(b+2)$，则每个非负初值 $h \in H_N^\infty$ 都生成唯一的全局非负有限能解，并且该解在全局范围内是光滑的。 (2) 若 $0 \leq \alpha < 1/(b+2)$，则存在非负的有限支撑初值，使得最大光滑存在时间为有限。 因此，$\alpha_c = 1/(b+2)$ 是全局正则性的精确阈值，且端点属于正则区域。

定理 4.1（更精确的爆破陈述）：对于每个 $s > 1/(b+2)$，存在适当的有限支撑初值，使得任何全局非负有限能解在有限时间内失去 $H_N^s$ 正则性。数据可以依赖于 $s$。

定理 1.2（几何壳层正则性）：设 $N_n = N_0 \Lambda^n$，其中 $N_0 > 0$，$\Lambda > 1$，且 $\nu > 0$。假设满足以下条件：

则每个非负初值 $h \in H_N^\infty$ 都生成一个全局光滑解，并且在具有相同初值的全局非负Leray-Hopf解中是唯一的。特别地，当 $\Lambda > 2^{3/2}$ 时，上述结论对所有 $\alpha \geq 1/3$ 成立。

通过结合Cheskidov关于 $0 < \alpha < 1/3$ 时的爆破结果 [2, Theorem 5.3]，对于任意固定的 $\Lambda > 2^{3/2}$，可以确定 $\alpha = 1/3$ 是所有非负光滑初值的全局正则性的精确阈值。

3. 研究方法、关键技术和核心工具

本文的核心技术障碍在于如何建立适用于不同壳层尺度设置的统一分析框架，并证明临界尾部的均匀衰减。作者的真正创新点在于引入了一种加权加速度估计和向后传播原理，从而能够处理超指数和几何壳层中的复杂非线性相互作用。

1. 加权加速度估计：通过对方程求导并利用权重 $2^{-n}$ 进行抵消，作者得到了关于 $A(t) := \sum_{n \geq 0} 2^{-n} |\dot{u}_n(t)|^2$ 的恒等式（Theorem 3.3）。由于非负性，剩余项是耗散的，因此有：

这一估计仅依赖于初值和固定参数，即使 $T_{\max} < \infty$ 也成立。权重 $2^{-n}$ 的选择是为了实现内部系数的抵消，这是构建加速度估计的关键。

2. 向后传播原理：对于超指数壳层，固定 $q = 1/(b+2)$ 并定义 $v_n = N_n^q u_n$。方程变为：

当 $\alpha \geq q$ 时，阻尼系数至少为 $\nu$。此外，由于 $2^{n/2} N_n^{-q} \to 0$，加速度估计 (1.3) 使得 $R_n$ 在高索引处一致小。作者的向后传播引理（Lemma 3.6）排除了存在固定的 $\epsilon > 0$ 以及序列 $n_j \to \infty$ 和 $t_j \in [0, T_{\max})$ 使得 $v_{n_j}(t_j) \geq \epsilon$ 的可能性。这些值会强制 $v_k(t_j) \to \infty$ 在某个固定的低壳层 $k$ 处，与能量界矛盾。由此得出临界尾部随时间一致趋于零。

3. 几何壳层的处理：对于几何壳层，权重为 $x_n = N_n^{1-2\alpha} u_n$，递推关系为：

其中 $A_\alpha = \Lambda^{1-4\alpha}$，$B_\alpha = \Lambda^{2\alpha-1}$。条件 $B_\alpha \geq A_\alpha$ 等价于 $\alpha \geq 1/3$，这是引理 3.6 所需的。在此几何结构中，$2^{n/2} = N_0^{-\rho} N_n^{\rho}$，其中 $\rho = (\log 2)/(2 \log \Lambda)$。因此，误差在第二个条件 (1.2) 下趋于零。再次使用向后传播得到临界尾部的均匀衰减，并通过 Proposition 2.5 中的经典判据得出光滑性和延拓性。

4. 超指数爆破构造：对于低于 $1/(b+2)$ 的情况，作者使用了一个独立的加权Lyapunov构造，类似于之前关于几何壳层的技术 [2]。所选的线性泛函在足够大的初值下满足Riccati下界，与其基于能量的上界相矛盾，从而完成了精确分类。

4. 与之前工作的比较

本文与既有工作的比较主要体现在以下几个方面：

1. 与Katz-Pavlović [5] 的比较：Katz和Pavlović引入了对数模型，并在Corollary 4.2.4中证明了当 $\alpha < 1/4$ 时的有限时间爆破。本文不仅扩展了这一范围，还针对超指数壳层给出了精确的阈值 $\alpha_c = 1/(b+2)$，这对于 $b \in (1, 2)$ 来说是一个新的结果。

2. 与Cheskidov [2] 的比较：Cheskidov建立了全局能量解、$\alpha \geq 1/2$ 时的全局正则性，以及当 $\alpha < 1/3$ 时足够大的非负初值的有限时间爆破。本文在几何壳层情况下将正则性范围扩展到 $\alpha \geq 1/3$，并在特定条件下（$\Lambda > 2^{3/2}$）确定了精确阈值 $\alpha = 1/3$。这与Cheskidov的结果相结合，完整地刻画了经典壳层比下的正则性行为。

3. 与Barbato, Morandin, Romito [1] 的比较：他们发展了一种不变区域论证来处理相邻分量，其Theorem A处理了 $2 < \beta \leq 5/2$ 范围内的非负有限能数据，并给出了正时间的平滑性。本文的参数与他们的参数通过 $\beta = 1/\alpha$ 和 $\lambda = \Lambda^\alpha$ 相关联。他们的选择 $\lambda = 2$ 对应于 $\Lambda = 2^\beta$。因此，Theorem 1.2 在该归一化下覆盖了 $5/2 < \beta \leq 3$，并将光滑初值的正则性范围扩展到阈值 $\beta = 3$。需要注意的是，初值类的区别：早期定理在其范围内从非负 $\ell^2$ 数据给出正时间平滑性，而本文的结果从光滑数据开始。

4. 与Palasek [7] 的比较：Palasek最近关于具有超指数分离壳层的Obukhov模型的工作为本文研究超指数Katz-Pavlović模型提供了动机。他的粘性爆破构造使用了光滑的外部力。尽管两个模型中的非线性转移机制不同，但他的构造表明，用超指数壳层替代几何壳层可以根本改变非线性转移和粘性之间的竞争。本文从相同的壳层几何结构出发，识别出相应的精确耗散阈值。

5. 与OpenAI报告的比较：最近，OpenAI报告了一个AI生成的三维不可压缩Navier-Stokes方程有限时间爆破的构造 [6]。截至撰写时，这一声明尚未经过数学界的独立审查；如果正确，它将构成一个重大突破。该构造使用了光滑的外部力。相比之下，本文研究的模型是无外力的，目的是隔离仅由非线性壳层间转移和粘性耗散引起的机制。

了解领域论文（摘要翻译）

1. 具有混合乘积型非线性的阻尼波动方程系统的临界阈值

Critical thresholds for the damped wave system with mixed product-type nonlinearities

作者: Trung Loc Tang, Tuan Anh Dao, The Anh Cung

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21975v1

链接: http://arxiv.org/abs/2609.21975v1

摘要翻译

中文标题: 具有混合乘积型非线性的阻尼波动方程系统的临界阈值

摘要翻译: 在本文中，我们研究了一类带有混合乘积型非线性项的耦合阻尼波动方程系统的 Cauchy 问题。我们已经成功确定了以非线性幂指数表示的临界曲线，该曲线界定了小初值温和解的全局（时间）存在性与全局温和解不存在性之间的界限。此外，全局存在性结果是通过 Schauder 不动点定理和 Banach 压缩原理建立的，而非存在性分析则依赖于一种适应于非线性乘积结构的新迭代方案。据我们所知，本工作首次提供了此类系统的临界阈值结果。

2. 二维环面上带势的散焦非线性薛定谔方程与随机初值

2D Torus defocusing NLSE with a potential and random initial data

作者: Nicolas Lindstrom

类别: math.AP, math-ph

发布日期: 2026-09-18

arXiv ID: 2609.21969v1

链接: http://arxiv.org/abs/2609.21969v1

摘要翻译

中文标题: 二维环面上带势的散焦非线性薛定谔方程与随机初值

摘要翻译: 我们考虑二维环面（$T^2$）上带有势 $V(x)\in H^{2^+}(T^2)$ 的非线性立方散焦薛定谔方程。初值 $u_0$ 由协方差为 $(-Δ+V)^{-1}$ 的中心高斯变量给出，因此几乎必然属于 $H^{0^-}(T^2)$。我们证明该方程在弱意义下对所有时间几乎必然适定，并且 $u-e^{it(Δ-V)}u_0$ 属于某个 $s>0$ 的 $H^{s}(T^2)$，且该解是截断方程解的极限。这一结果是 Bourgain 在其论文《Invariant Measures for the {2D}-Defocusing Nonlinear {Schr{ö}dinger} Equation》中首次给出的结果的推广。

3. 具有粗糙系数的退化抛物型 Riesz 变换的一个有界性结果

A boundedness result for degenerate parabolic Riesz transforms with rough coefficients

作者: Khalid Baadi

类别: math.CA, math.AP, math.FA

发布日期: 2026-09-18

arXiv ID: 2609.21949v1

链接: http://arxiv.org/abs/2609.21949v1

摘要翻译

中文标题: 具有粗糙系数的退化抛物型 Riesz 变换的一个有界性结果

摘要翻译: 在本文中，我们证明了与一类退化抛物型算子相关的抛物型 Riesz 变换在 $p \le 2$ 范围内的有界性。该退化抛物型算子的椭圆部分为散度形式，其复系数可测地依赖于空间和时间。退化性由 Muckenhoupt 类 $A_2(\mathbb{R}^n)$ 中的一个空间权函数决定。论证依赖于预解式族的抛物型梯度的非对角估计以及加权嵌入。我们在严格小于 $2$ 的一组指数范围内获得了有界性，且该范围由权函数的逆 Hölder 类显式量化。对于实系数情形，此量化是最佳的，并且可以向下延伸至 $1$。

4. 含电磁场 Wigner 方程的速度平均

Velocity Averaging for the Wigner Equation with Electromagnetic Fields

作者: François Golse, Jakob Möller

类别: math.AP, math-ph

发布日期: 2026-09-18

arXiv ID: 2609.21885v1

链接: http://arxiv.org/abs/2609.21885v1

摘要翻译

中文标题: 含电磁场 Wigner 方程的速度平均

摘要翻译: 我们讨论了基于 $L^2$ 的速度平均在 Wigner 动力学方程中的应用，该方程描述了受外部电磁场作用的粒子的密度算子的 Wigner 变换的量子演化。这将 [F. Golse, J. Möller: Commun. Math. Sci. (2026)] 中的基于 $L^2$ 的平均引理推广到了电磁场情形。我们采用了 Stratonovich [Sov. Phys. D., (1)414-418, 1956] 提出的磁或规范不变的 Wigner 变换，它与通常的定义相差一个涉及沿线段磁矢势环量的相位因子。磁 Wigner 变换满足广义相空间变量 $(x,v=ξ-A(t,x))$ 下的 Wigner 方程，这为速度平均的应用提供了更自然的设定。由于磁 Wigner 方程包含空间变量的一阶导数，我们在 $L^2$ 中获得了紧致性，但没有获得正则性的提升。

5. 剪切流附近的流体弹性波

Hydroelastic waves near shear flows

作者: Jian Li, Shaojie Yang

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21778v1

链接: http://arxiv.org/abs/2609.21778v1

摘要翻译

中文标题: 剪切流附近的流体弹性波

摘要翻译: 在本文中，我们证明，从非均匀剪切流分岔出来的、小振幅的对称双周期三维流体弹性波，在领头阶上必然是二维的。对二次阶可解性条件的细致分析表明，所有具有横向依赖性的傅里叶系数均恒为零。维数缩减完全由剪切流的垂直结构控制，并且在动态边界条件改变时是鲁棒的。

6. 深水中水弹性波的稳定化

Stabilization of hydroelastic waves in deep water

作者: Jian Li, Shaojie Yang

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21769v1

链接: http://arxiv.org/abs/2609.21769v1

摘要翻译

中文标题: 深水中水弹性波的稳定化

摘要翻译: 我们研究在无质量非线性 Cosserat 薄膜下方、深水中二维周期水弹性波通过局部压力反馈实现的稳定化问题。对于足够正则的解，在重力、弯曲刚度与局部化分布之间满足一个显式条件的情况下，我们证明了一个一致时空估计以及一个局部耗散可观测性不等式。作为推论，只要解保持在小的几何区域内，水弹性哈密顿量就会定量地衰减。证明结合了能量耗散恒等式、一种针对无限深度几何结构调整的 nonlinear multiplier method（非线性乘子法），以及对 Cosserat 弯曲能量的强制性分析。

7. Riesz 变换在常漂移下的无维弱 $(1,1)$ 估计

Dimension-free weak $(1,1)$ estimates for Riesz transforms with constant drift

作者: Suman Mukherjee

类别: math.FA, math.AP, math.CA

发布日期: 2026-09-18

arXiv ID: 2609.21458v1

链接: http://arxiv.org/abs/2609.21458v1

摘要翻译

中文标题: Riesz 变换在常漂移下的无维弱 $(1,1)$ 估计

摘要翻译: 设 $L_v=-Δ-2v\cdot\nabla$ 定义在 $L^2(\mathbb R^n,e^{2v\cdot x},dx)$ 上，其中 $v\in\mathbb R^n\setminus\{0\}$ 为常漂移向量。我们证明了完整的向量 Riesz 变换 $\nabla L_v^{-1/2}$ 对实值函数满足弱型 $(1,1)$ 估计，其常数至多为 $2$，且该常数关于维数和漂移向量一致有界。我们的证明将 Ouyang、Spector 和 Stockdale（https://arxiv.org/abs/2608.18068）的分数障碍方法推广到了指数加权情形。随后通过插值得到强 $L^p$ 界，适用于 $1<p\le 2$，且常数仅依赖于 $p$。

8. Weyl-Hörmander 演算中的核估计、弱 (1,1) 有界性及其他正则性性质

Kernel estimates for the Weyl-Hörmander calculus, Weak (1,1) boundedness and other regularity properties

作者: Duván Cardona, Julio Delgado, Manuel Martínez-Flores

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21692v1

链接: http://arxiv.org/abs/2609.21692v1

摘要翻译

中文标题: Weyl-Hörmander 演算中的核估计、弱 (1,1) 有界性及其他正则性性质

摘要翻译: 我们给出了 $S(m,g)$ Weyl-Hörmander 演算类中拟微分算子的核估计。这些新结果导出了适用于这些类的有界性定理，例如弱 (1,1) 型有界性，以及这些算子从 Hardy 空间到 Lebesgue 空间 $L^p$ 的有界性。我们证明，在权重 $m$ 满足适当的椭圆性条件下，可以得到 Fefferman 阶的弱 (1,1) 不等式，该不等式此前由第二作者类比于 Fefferman 的经典 $L^p$-有界性定理而建立。我们的方法将 Álvarez 和 Hounie、Nagase、Kumano-Go 等人先前对 Hörmander 类所证明的估计推广至 $S(m,g)$ 类，并在这一框架内引入了新的正则性性质，包括从 Hardy 空间 $H^p$ 到 $L^p$ 的映射以及在 Lorentz 空间上的有界性。我们发现，{f Uniform Hörmander Condition}（一致 Hörmander 条件）在某些有界性定理中起着基本作用。

9. Riesz型不等式在拼接下的稳定性

Stability of the Riesz-type inequalities under gluing

作者: Dangyang He

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21399v1

链接: http://arxiv.org/abs/2609.21399v1

摘要翻译

中文标题: Riesz型不等式在拼接下的稳定性

摘要翻译: 我们建立了紧致拼接下关于逆平方根不等式和 Riesz 变换的稳定性原理。在共同的 Sobolev 维数以下，逆估计从有限个模型流形传递到它们的连通和。一个与边界相容的 Poisson 参数列还证明了外部 Lipschitz 域上实对称一致椭圆散度型算子对每个 $1<p<\infty$ 满足 Neumann 逆不等式。对于正向变换，我们拼接了一个有界核模型和一个独立的整体空间端部模型，这两个模型在重叠区域上的系数一致。随后通过齐次 Fredholm 构造解决了取值于形式的误差。所得到的值域由两个模型的指数控制，而对于 Dirichlet 数据还有额外的限制 $p<n$。更广泛地，我们的方法阐明了有界内部问题与外部问题之间的相互作用，并为研究其他边值问题提供了一个灵活的框架。

10. 连续多孤子对 Calogero-Moser 粒子的时间一致逼近

Uniform-in-Time Approximation of Calogero-Moser Particles by Continuum Multisolitons

作者: Hedong Wang

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21314v1

链接: http://arxiv.org/abs/2609.21314v1

摘要翻译

中文标题: 连续多孤子对 Calogero-Moser 粒子的时间一致逼近

摘要翻译: 我们证明，经典有理 Calogero-Moser 系统的每个解在减去粒子位置共同的线性漂移后，均可被聚焦型连续 Calogero-Moser 方程的有理多孤子解在所有时间上一致地逼近。该构造将多孤子的逆谱矩阵识别为 Moser 时变 Hermitian 位置矩阵的秩一耗散扰动。随后，粒子间距的一致下界给出了多孤子极点的定量控制。这些极点的实部与实速度以 $\varepsilon^2$ 阶误差逼近平移后的粒子位置与速度，而极点高度为正且其和恰好等于 $\varepsilon$。利用多孤子密度的精确 Poisson 核表示，我们得到向原子粒子测度在 bounded-Lipschitz 距离下的收敛性，其收敛速率为 $O(\varepsilon(1+|\log\varepsilon|))$，且在时间上一致成立。我们还推导了固定 $\varepsilon$ 下极点高度的大时间分布：其中一个区分分支保持极限高度，而其余高度随时间二次衰减。显式的双粒子公式展示了该逼近，并表明实位置误差估计是锐利的（sharp）。

11. 非线性孔隙弹性力学的变分模型

A variational model of nonlinear poroelasticity

作者: James H. Adler, Xiaozhe Hu, Arkadz Kirshtein

类别: math.AP, math.NA, physics.flu-dyn

发布日期: 2026-09-18

arXiv ID: 2609.21294v1

链接: http://arxiv.org/abs/2609.21294v1

摘要翻译

中文标题: 非线性孔隙弹性力学的变分模型

摘要翻译: 我们推导了一个热力学一致的流体通过孔隙弹性介质流动的模型。从弹性与流体自由能密度、能量耗散率以及运动学约束出发，利用变分原理推导出力平衡方程；其中压力–密度本构关系作为变分结构的直接结果出现；同一运动学约束还提供了总通量输运结构。在理想气体极限下，该模型的线性化恢复经典的线性 Biot 方程。对于幂律流体能量，它给出等熵压力–密度关系。变分表述的一个关键优势在于，对更丰富物理现象（如热效应、化学反应或多组分流体）的扩展可以通过系统地扩充能量泛函和耗散泛函来实现，而无需重新设计力平衡或输运闭合条件。我们通过一个能量兼容的双场离散格式来支持该模型，并研究了三种侧边界处理方式和三种流体压缩性指数下的表面载荷固结问题。

12. Euler–Korteweg方程解的生命期下界

Lower bounds for the lifespan of solutions to the Euler–Korteweg equations

作者: Haruhiro Kurabe, Ryo Takada, Tomoaki Yoshizawa

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21244v1

链接: http://arxiv.org/abs/2609.21244v1

摘要翻译

中文标题: Euler–Korteweg方程解的生命期下界

摘要翻译: 我们考虑无旋流类中 Euler–Korteweg 方程的初值问题。在任意空间维数 $d \geqslant 2$ 下，我们建立了关于解的生命期的一个定量下界，该下界显式地展示了其对 Weber 数、Mach 数以及初始扰动的 Sobolev 范数的依赖关系。特别地，在低 Weber 数区域，若 Mach 数与 Weber 数之间存在适当的代数关系，则该下界可取任意大。

13. 二次无理环面上的最优 Strichartz 估计

Sharp Strichartz estimates on quadratic irrational tori

作者: Rui Liang, Shunlin Shen, Yuzhao Wang

类别: math.AP

发布日期: 2026-09-17

arXiv ID: 2609.21069v1

链接: http://arxiv.org/abs/2609.21069v1

摘要翻译

中文标题: 二次无理环面上的最优 Strichartz 估计

摘要翻译: 我们证明了二维周期 Schrödinger 流的最优 $L^{4}$ Strichartz 估计，其积分对角符号为 $ak_{1}^2+bk_2^2$，其中 $a,b \in \mathbb Z \setminus \{0\}$。这些估计展现出一种尖锐的算术二分性，取决于 $-ab$ 是否为平方数：对于由 $N$ 界定的频率，当 $-ab$ 不是平方数时，最优损失为 $(\log N)^{1/4}$；而当 $-ab$ 是平方数时，最优损失为 $N^{1/4}$。作为推论，我们表明矩形纵横比的任意小扰动可导致三次双曲型 NLS 在全局适定性与范数膨胀之间的转变。

14. 布尔立方体上的 $m^{2.943}$ Bohnenblust–Hille 界

An $m^{2.943}$ Bohnenblust–Hille Bound on the Boolean Cube

作者: Joseph Slote, Chun-Kai Tseng, Alexander Volberg

类别: cs.IT, math.AP, math.CA

发布日期: 2026-09-17

arXiv ID: 2609.21144v1

链接: http://arxiv.org/abs/2609.21144v1

摘要翻译

中文标题: 布尔立方体上的 $m^{2.943}$ Bohnenblust–Hille 界

摘要翻译: 令 $q_m=2m/(m+1)$，并设
β_0=\frac{3}{2}+\frac{1}{\log 2}=2.9426950408\ldots,
其中 $\\log$ 为自然对数。我们给出一个证明方案，表明：对任意 $\\varepsilon>0$，存在 $C_\\varepsilon<\\infty$，使得每个傅里叶次数至多为 $m$ 的复值函数 $f:\{-1,1\}^n\\to\\C$ 均满足
\left(\sum_{A\subseteq[n]}\abs{\wh f(A)}^{q_m}\right)^{1/q_m} \le C_\varepsilon m^{β_0+\varepsilon}\norm{f}_\infty.
相较于早期草稿中 $m^9$ 的估计，这一改进包含两个要素。前三个傅里叶层分别在尺度 $m$、$m^{3/2}$ 和 $m^{5/3}$ 上进行估计。这些损失被编码在权重 $M^{μ_r/r}r^B\\min\\{r,L_M\\}^8$ 中，其中 $L_M\\asymp_B \\log (M+1)$，且当 $r\\ge2$ 时 $μ_r=\\lceil3r/2\\rceil$。对于 $r<L_M$，采用奇偶兼容的中心窗口；而对于 $r\\ge L_M\\asymp_B\\log(M+1)$，则采用收缩的平衡窗口，在此区域内不使用奇偶性。主导的高次损失-增益因子为 $\\ee\\,2^{-B}$；通过仔细的统一界，可对任意 $B>1/\\log2$ 完成自举。$μ_r$ 的公式将在下文说明。

15. 具有变指数、齐次 Neumann 边界条件和广义 logistic 源的双非线性抛物问题

A doubly nonlinear parabolic problem with variable exponents, homogeneous Neumann boundary conditions and generalized logistic source

作者: Bogdan Maxim

类别: math.AP

发布日期: 2026-09-17

arXiv ID: 2609.20969v1

链接: http://arxiv.org/abs/2609.20969v1

摘要翻译

中文标题: 具有变指数、齐次 Neumann 边界条件和广义 logistic 源的双非线性抛物问题

摘要翻译: 本文旨在为具有变指数和齐次 Neumann 边界条件的双非线性抛物问题建立一套自包含的存在性与唯一性理论。我们的方法基于 Rothe 的时间离散化技术，并建立在作者先前论文中已确立的结果之上。此外，我们还研究了解的渐近行为。文中所有论证均以完整细节呈现，从而使该论文自包含。

不相关论文（基本信息）

1. $C^{1, Dini}$ 域中奇异势椭圆方程奇异集的界

Bounds of singular sets for elliptic equations in $C^{1, Dini}$ domains with singular potentials

作者: Zhiwei Wang, Jiuyi Zhu

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21988v1

链接: http://arxiv.org/abs/2609.21988v1

摘要: We study the quantitative codimension-two estimate for the singular set in the boundary neighborhoods of the solutions of \begin{equation*} Δu+V(x)u=0\qquad\text{in }Ω, \qquad u=0\qquad\text{on }\partialΩ, \end{equation*} where $Ω\subset\mathbb R^n$ is a bounded $C^{1,\mathrm{Dini}}$ domain and $V\in L^p(Ω)$ for some $p>n$. We first prove the explicit upper bound for the doubling index is given by $C(n,p,Ω)(1+\|V\|_{L^p(Ω)}^{\frac{2p}{3p-2n}})$.The analytic input is an interior volume estimate for solutions of second order elliptic equations with uniformly elliptic Dini leading coefficients and $V\in L^p$. Using the quantitative doubling index bound, boundary flattening, we show an explicit upper bound for singular sets in the neighborhood of the boundary of the $C^{1,\mathrm{Dini}}$ domain.

2. Bourgain-Brezis-Mironescu 极限中的均匀化与平均

Homogenization and averaging in the Bourgain-Brezis-Mironescu limit

作者: Andrea Braides, Stefano Mannella, Alec Jacopo Almo Schiavoni Piazza

类别: math.AP, math-ph

发布日期: 2026-09-18

arXiv ID: 2609.21933v1

链接: http://arxiv.org/abs/2609.21933v1

摘要: We study the simultaneous asymptotic behaviour of homogenization and the Bourgain–Brezis–Mironescu limit for quadratic fractional energies with periodic coefficients. The functionals depend on two small parameters: the period of the coefficients and the fractional exponent approaching the local limit. We prove that the $Γ$-limit is completely determined by the relative scaling of these parameters. In the two extreme regimes, one recovers the expected separation of scales, yielding either the homogenized Dirichlet energy or the energy with averaged coefficients. At the critical logarithmic scaling, however, the two mechanisms coexist, and the limit is a convex combination of the homogenized and averaged energies, with weights depending explicitly on the relative scaling of the two asymptotic processes. The proof combines the homogenization analysis of the short-range interactions with a blow-up argument showing that long-range interactions are simply averaged using a Riemann–Lebesgue-type argument.

3. 稳定阿贝尔希格斯场的积分曲率估计与测地线极限

Integral curvature estimates and geodesic limits for stable abelian Higgs fields

作者: Marco Badran, Marco A. M. Guaraco, Aria Halavati

类别: math.DG, math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21893v1

链接: http://arxiv.org/abs/2609.21893v1

摘要: We prove that, on closed three-manifolds, the limit interface of a sequence of stable abelian Higgs critical points with uniformly bounded energy is a finite union of smooth closed immersed geodesics, possibly with multiplicity and self-intersections. This follows from an integral estimate for the diffuse curvature, derived from an integral bound on the discrepancy.

4. 闭曲面上 Liouville 泛函的 Palais-Smale 序列的量子化

Quantization for Palais-Smale sequences of the Liouville functional on closed surfaces

作者: Andrea Malchiodi, Francesco Malizia, Luca Martinazzi

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21851v1

链接: http://arxiv.org/abs/2609.21851v1

摘要: In this paper we prove a quantization property for Palais-Smale sequences of functionals related to Liouville equations on compact surfaces. While these equations have been extensively studied over the past decades, such a quantization property had not been established so far, and has been often bypassed by the Struwe monotonicity trick. Our result, which relies on careful $L^p$ estimates to obtain gradient bounds and to perform neck analysis, allows to directly apply variational methods to obtain existence of solutions in non-resonant regimes. We also give an example of clustering of blow-up points and a stronger assumption that, on the contrary, guarantees that the blow-up points are isolated.

5. 第一旋度自由 Maxwell 特征值的形状优化

Shape optimisation of the first solenoidal Maxwell eigenvalue

作者: Antoine Henrot, Idriss Mazari-Fouquer, Yannick Privat, Dominik Stantejsky

类别: math.AP, math.OC

发布日期: 2026-09-18

arXiv ID: 2609.21794v1

链接: http://arxiv.org/abs/2609.21794v1

摘要: We study the minimisation of the first solenoidal Maxwell eigenvalue $λ_1^{\mathrm{Max}}(Ω)$ in perfectly conducting cavities through the scale-invariant functional $J_{\mathrm{Per}}(Ω):=\mathrm{Per}(Ω)λ_1^{\mathrm{Max}}(Ω)$ in $\mathbb R^3$. While the corresponding unconstrained problem is known to be ill-posed, we focus on bounded convex cavities and prove that $J_{\mathrm{Per}}$ has a positive infimum in this class. The proof relies on the study of the three main possible behaviours for sequences of convex domains with bounded perimeter: convergence to a bounded convex domain with non-empty interior, collapse to a planar domain or collapse to a one-dimensional domain. The case of a planar collapse is handled through projection arguments, yielding a lower bound, while that of a one-dimensional collapse is controlled by a different argument, showing that in that case $J_{\mathrm{Per}}$ must diverge to $+\infty$. Although the question of existence of an optimal shape (that is, ruling out planar collapse) remains open, we derive several negative results and obstructions to existence; typically, no minimiser with a $\mathscr C^{3,+}$ boundary exists. This relies on a fine analysis of first-order optimality conditions.

6. 加权 Kolmogorov 方程：转移估计、局部有界性与 Harnack 不等式

Weighted Kolmogorov Equations: Transfer Estimates, Local Boundedness, and Harnack Inequalities

作者: Kaj Nyström

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21737v1

链接: http://arxiv.org/abs/2609.21737v1

摘要: Let $m\geq1$, $k\geq0$, $n=m+k$, and write $x=(v,z)\in\mathbb R^m\times \mathbb R^k$, where $v$ is the active velocity variable and $z$ is passive with respect to $Y=v\cdot\nabla_y+\partial_t$. We consider

where $w=w(x)\in A_2(\mathbb R^n)$ and the measurable matrix $A$ has ellipticity and size controlled by $w$. For arbitrary such weights, including weights depending on the active variables, we prove a weighted hypoelliptic transfer estimate, a kinetic Sobolev inequality, local boundedness, the weak Harnack and Harnack inequalities, local Hölder continuity, and the strong minimum principle. The Harnack argument uses a full-variable weighted measure-data compactness theorem. An $\mathrm L^p$ velocity-averaging estimate gives compactness of smooth moments of $wu$, and weighted Poincaré inequalities in all diffusive variables reconstruct $u$. At a two-phase limit, testing with non-negative profiles produces transport directions given by weighted mean velocities. A $\mathrm{BV}$ argument applied to the limiting subsolution inequality yields the no-jump principle needed for expansion of positivity and a weighted kinetic ink-spots lemma. The theory includes the weight $|λ|^{1-2s}$ in the extension of Garofalo and Tralli for fractional powers of the Kolmogorov operator.

7. $\mathbb{R}^3$ 中 Allen–Cahn 方程的稳定 De Giorgi 猜想

Stable De Giorgi conjecture of the Allen–Cahn equation in $\mathbb{R}^3$

作者: Yong Liu, Tianci Luo, Kelei Wang, Juncheng Wei, Yong Wei, Ke Wu

类别: math.AP, math.DG

发布日期: 2026-09-18

arXiv ID: 2609.21680v1

链接: http://arxiv.org/abs/2609.21680v1

摘要: We prove that every bounded stable entire solution $v$ of the Allen–Cahn equation in $\R^3$ is one-dimensional. As a consequence, the full De Giorgi conjecture in $\mathbb{R}^4$ is true. We also obtain local curvature estimates for stable solutions. The proof strategy is inspired by the recent breakthrough work of Chan, Fernández-Real, Figalli and Serra [J. Amer. Math. Soc. 2026], by reducing the stabilty condition for the Allen-Cahn equation to a weak stability condition on a surface (the zero set) and then utilizing Gauss-Bonnet formula. For this purpose, we first use the stability condition to get a sublinear bound for a weighted integral that controls the zeros where the solution is far from planar. If such zeros exist, we isolate a bounded set of them and join $1-v^2$ near this set to derivatives of one-dimensional transitions farther away. By controlling the interaction between these transitions, we derive the weak stability condition on the zero set, which is then used to bound a weighted integral of the squared curvature on the regular part of the zero set by a cutoff gradient integral and a controlled error. We use this inequality to bound the intrinsic area and construct logarithmic cutoffs. The resulting compactly supported test function has a negative contribution near this set that exceeds all joining and cutoff errors, contradicting stability.

8. 多面体的中心仿射谱几何

Centro-affine spectral geometry of polytopes

作者: Yingxiang Hu, Mohammad N. Ivaki

类别: math.MG, math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21670v1

链接: http://arxiv.org/abs/2609.21670v1

摘要: In this paper, we develop a polyhedral version of Milman’s centro-affine spectral framework for full-dimensional origin-symmetric polytopes. A weighted graph Laplacian on the facets plays the role of the smooth centro-affine Laplacian. For $n\geq 3$, we prove that its first nonconstant even eigenvalue satisfies

whenever $P$ is a local minimizer of $λ_{1,e}$ among origin-symmetric polytopes with the same facet normals, but not necessarily the same normal fan.

9. 五阶扰动消失时向动力学 $Φ^4_3$ 模型的收敛 I: Paracontrolled 方法

Convergence to the Dynamical $Φ^4_3$ Model under a Vanishing Quintic Perturbation I: A Paracontrolled Approach

作者: Zikai Chen, Seiichiro Kusuoka

类别: math.PR, math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21552v1

链接: http://arxiv.org/abs/2609.21552v1

摘要: We study local convergence to the dynamical $Φ^4_3$ model under a vanishing quintic perturbation. More precisely, on the three-dimensional torus we consider

for $α\in(\frac56,1)$, where $ξ_\varepsilon$ is a spatial mollification of space-time white noise. Although the quintic coefficient vanishes, its contractions generate divergent linear and cubic contributions. We identify suitable mass and cubic counterterms that compensate these divergences. Using paracontrolled calculus, we construct the required enhanced stochastic data and prove that their renormalized higher-order components vanish, while the remaining coordinates converge to the enhanced data of the dynamical $Φ^4_3(λ)$ model. We further establish local well-posedness and stability of the associated deterministic solution map. Consequently, for well-prepared initial conditions, $u_\varepsilon$ converges in probability, as a random local solution germ, to the renormalized dynamical $Φ^4_3(λ)$ solution. In particular, the cubic counterterm allows convergence strictly below the threshold $α=1$ arising when only linear renormalization is allowed.

10. 具有正则或排斥相互作用的聚集-扩散方程：高斯与自相似渐近

Aggregation–Diffusion Equations with Regular or Repulsive Interactions: Gaussian and Self-Similar Asymptotics

作者: Grzegorz Karch, Yuri Soga

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21456v1

链接: http://arxiv.org/abs/2609.21456v1

摘要: We study the large-time behavior of solutions to the aggregation–diffusion equation

Our main results identify a transition, governed by the regularity and singularity of the interaction kernel, between Gaussian and nonlinear self-similar asymptotics. We prove that for regular interaction kernels, as well as for sufficiently mild singular repulsive kernels, the nonlinear drift is asymptotically negligible and solutions exhibit the same Gaussian large-time behavior as solutions of the linear heat equation. In contrast, for the critical logarithmic kernel $ K(x)=-\log |x|, $ the interaction persists at the diffusive scale and leads to genuinely nonlinear asymptotics. More precisely, for every mass $M>0$, we prove the existence and uniqueness of a self-similar solution of mass $M$ and show that every solution with mass $M$ and with finite second moment converges in all $L^p$-norms to this self-similar solution as $t\to\infty$.

11. 驻定调和映射的指定奇点集

Prescribed singular sets for stationary harmonic maps

作者: Wenshuai Jiang, Chang Li, Wenyou Yu

类别: math.AP, math.DG

发布日期: 2026-09-18

arXiv ID: 2609.21443v1

链接: http://arxiv.org/abs/2609.21443v1

摘要: A central conjecture in the regularity theory of harmonic maps asserts that the singular set of a stationary harmonic map has codimension at least three. In this paper, we construct counterexamples to this folklore conjecture. We construct stationary harmonic maps whose singular sets $S$ satisfies $\dim S=n-2$ and $\mathcal{H}^{n-2}(S)=0$ . The target is a smooth compact manifold diffeomorphic to $\mathbb{S}^2 \times \mathbb{T}^3$, with metric arbitrarily close in $C^\infty$ to the standard product metric.

12. $GSBD^p$ 中薄域上的 Korn 不等式与 Poincaré-Korn 不等式

Korn and Poincaré-Korn inequalities for thin domains in $GSBD^p$

作者: Davit Harutyunyan

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21261v1

链接: http://arxiv.org/abs/2609.21261v1

摘要: We prove Korn’s first inequalities with and without boundary conditions in $GSBD^p$ ($1<p<\infty$) in $C^{1,1}$-regular thin domains with Lipschitz thickness boundaries with a constant $C h^{-p},$ that is optimal if the domain mid-surface contains a flat region, e.g., if the domain contains a piece of plate. For the one with boundary condition, we impose zero Dirichlet boundary conditions on the thin lateral faces of the shell, where we prove that the same estimate holds with no subtracted rigid motion like in the classical case for Sobolev vector fields.

13. 变指数对数源强阻尼粘弹性波动方程在任意能量水平上的爆破

Blow-up at arbitrary energy levels for a strongly damped viscoelastic wave equation with a variable-exponent logarithmic source

作者: Menglan Liao, Mingxue Zhang

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21256v1

链接: http://arxiv.org/abs/2609.21256v1

摘要: We study a strongly damped viscoelastic wave equation with a logarithmic source whose measurable exponent satisfies $2<p_-\le p(x) \le p_+<2^*$. An optimal pointwise correction to the source–potential inequality defines a \emph{shifted energy}. Under suitable kernel conditions, we prove finite-time blow-up for negative shifted energy and for a class of nonnegative shifted energies with no fixed upper energy threshold. Explicit upper lifespan bounds follow from an estimate for reaching negative shifted energy and a concavity argument. We also construct smooth blow-up data at every prescribed real energy level and show that our lifespan estimate covers additional data in the constant-exponent case. Explicit examples are given to illustrate the blow-up results.

14. 临界奇异各向异性 Moser-Trudinger 不等式的极值与阈值

Extremals and Thresholds for Critical Singular Anisotropic Moser-Trudinger Inequalities

作者: Kaiwen Guo, Yanjun Liu

类别: math.AP, math.FA

发布日期: 2026-09-18

arXiv ID: 2609.21250v1

链接: http://arxiv.org/abs/2609.21250v1

摘要: Let $N\ge2$, $q>1$, and $0<β<N$, we study extremals for the critical singular anisotropic Moser–Trudinger functional under

For $b<N$, the supremum is attained. At $b=N$, put

For $q<q_-$, the supremum is attained for every $a>0$. For $q_-\le q<q_+$, there exists $a_c\in(N,\infty]$ such that the supremum is attained for $0<a<a_c$ and is not attained for $a>a_c$ if $a_c<\infty$. The argument does not determine attainment at $a=a_c$. Radial blow-up analysis gives a concentration bound in terms of a nonlinear Green function. The first nonzero Taylor term determines the strict comparison, an exact Euclidean reduction shows that the threshold is independent of $F$, a separate supplement gives an alternative proof for $q=N$ using the classical Green function and compact Taylor terms.

15. 关于具有移动边界的非局域登革热模型：干湿季与脉冲干预的影响

On a nonlocal dengue model with moving boundaries: The impact of dry-wet season and impulsive intervention

作者: Han Zhang, Zhigui Lin, Huaiping Zhu

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21235v1

链接: http://arxiv.org/abs/2609.21235v1

摘要: Seasonal succession and human intervention profoundly influence dengue fever transmission, and both humans and mosquito vectors can disperse over long distances. This paper proposes a nonlocal reaction-diffusion dengue model with moving boundaries that incorporates dry-wet seasonality and impulsive intervention. We first present the existence and uniqueness of the global solution to the model, then introduce the definition of the generalized principal eigenvalue, prove its uniqueness when the kernel functions coincide via the method of separation of variables, and further discuss its associated properties. Using this eigenvalue as an index, we obtain a spreading-vanishing dichotomy and derive two criteria that characterize the spreading and vanishing states, respectively. Our results extend both the nonlocal diffusion model without impulsive intervention and the model with impulsive intervention but without seasonal succession. Finally, we validate our theoretical findings and investigate how various factors affect dengue transmission through numerical simulations. The numerical results indicate that a shorter dry season makes disease containment more difficult, whereas more intensive impulsive intervention enhances control efficacy.

16. 周期变化环境中脉冲干预的登革热模型

Dengue fever model with impulsive intervention in a periodically varying environment

作者: Han Zhang, Inkyung Ahn, Zhigui Lin

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21205v1

链接: http://arxiv.org/abs/2609.21205v1

摘要: Pulse interventions represent a highly effective measure for infection control, as they influence disease transmission through short-term actions. In addition, the habitat ranges of dengue vectors and hosts exhibit periodic variations driven by environmental and climatic factors. To investigate the effects of impulsive interventions and domain evolution on disease transmission, we propose a dengue fever reaction-diffusion model that incorporates impulsive perturbations in a periodically varying domain. By applying the Poincar$\acute{e}$ map and the Krein-Rutman theorem, we establish the existence of the principal eigenvalue for the periodic eigenvalue problem with impulses, thereby extending previous studies on reaction-diffusion equations in fixed domains without impulsive effects. Sufficient conditions governing the long-term dynamics of periodic solutions are derived using the comparison principle and monotone iteration theory. Numerical simulations corroborate the theoretical findings and elucidate the effects of impulsive-intervention intensity and periodic domain evolution on disease transmission patterns. Our results indicate that increasing the intensity of pulse interventions suppresses disease transmission, whereas a larger magnitude of domain variation impedes disease control.

17. 具有收获项的扩散Nicholson方程的行波存在性

Existence of Traveling Waves for a Diffusive Nicholson Equation with Harvesting

作者: Adrián Gómez, Helí Elorreaga, Cesar Guayasamin

类别: math.AP

发布日期: 2026-09-18

arXiv ID: 2609.21166v1

链接: http://arxiv.org/abs/2609.21166v1

摘要: We prove the existence of traveling wave solutions for a diffusive Nicholson model with a delayed linear harvesting term and two delayed-response channels. Our approach combines monotone heteroclinic connections of the associated scalar delay equation with the large-speed traveling-wave theorem of Faria, Huang and Wu. For $0<σ<r$, we use a previous upper- and lower-solution construction under an explicit parameter condition. For $0<r\leqσ$, we establish a new monotone heteroclinic connection. To fit the abstract framework, we embed the scalar equation into an auxiliary two-dimensional reaction–diffusion system and prove that every relevant traveling wave lies on the invariant diagonal. We then determine an explicit parameter region in which the required spectral hypotheses hold. Consequently, the original diffusive Nicholson equation admits traveling waves connecting the trivial and positive equilibria for all sufficiently large wave speeds.

18. 强磁场中的双阱：无穷多个耦合常数取值下隧穿效应的缺失

Double-wells in a strong magnetic field: absence of tunneling for infinitely many values of the coupling constant

作者: Kevin Buck, Charles L. Fefferman, Javier Gómez-Serrano, Amaury Hayat, Jacob Shapiro, Michael I. Weinstein

类别: math.AP

发布日期: 2026-09-17

arXiv ID: 2609.21078v1

链接: http://arxiv.org/abs/2609.21078v1

摘要: We prove the fixed-potential conjecture of Fefferman, Shapiro, and Weinstein for inversion-symmetric magnetic double wells. We construct a number $L_0>0$ and one smooth, compactly supported, nonpositive, nonradial single-well potential $v$, independent of both the coupling parameter $λ$ and the well displacement, such that for every fixed $L\ge L_0$ the associated magnetic double-well Hamiltonian has infinitely many exact degeneracies of its two lowest eigenvalues as $λ\to\infty$. The magnetic hopping coefficient also vanishes along an infinite sequence tending to infinity. The potential consists of a radial core and two small, reflection-related perturbations supported on one-sided quadratic cusps. Their log-flat profiles isolate two dominant hopping contributions, which we evaluate by steepest descent after inserting the radial ground-state asymptotics. The resulting asymptotic cosine expression, together with error bounds for the even–odd splitting, yields infinitely many exact eigenvalue crossings and changes of ground-state parity.

19. Ornstein-Uhlenbeck 演化方程与障碍问题的变分方法

A variational approach to Ornstein-Uhlenbeck evolution equations and obstacle problems

作者: Erica Ipocoana, Annalaura Rebucci

类别: math.AP

发布日期: 2026-09-17

arXiv ID: 2609.21003v1

链接: http://arxiv.org/abs/2609.21003v1

摘要: We develop a variational formulation of the Ornstein-Uhlenbeck evolution equation based on a Gaussian divergence structure, in which the drift term is absorbed into the first variation of a weighted Dirichlet energy rather than treated as a lower-order perturbation. For the linear equation, the distributional, energy weak, and variational notions of solution coincide and determine the same unique solution. The same framework yields a variational formulation of the associated parabolic obstacle problem, for which we prove existence and then uniqueness under a standard additional regularity assumption on the obstacle. The same mechanism applies to nonlinear Gaussian gradient flows associated with convex integrands of standard $p$-growth. Finally, we extend the Gaussian-divergence formulation to the force-free kinetic Fokker-Planck equation with periodic position variables, establishing the same equivalence for the linear equation.

20. 具有反向 Hölder 势的 Schrödinger 方程在无穷远处的唯一延拓性

Unique continuation at infinity for Schrödinger equations with Reverse Hölder Potentials

作者: Blair Davey

类别: math.AP

发布日期: 2026-09-17

arXiv ID: 2609.21048v1

链接: http://arxiv.org/abs/2609.21048v1

摘要: In this article, we study unique continuation properties at infinity for solutions to generalized Schrödinger equations with potential functions that belong to the reverse Hölder class. For equations of the form $-div (A \nabla u) + V u = 0$ in $\mathbb{R}^n$, where $A$ is bounded and elliptic and $V \in RH_p$ for some $p \in [\frac n 2, \infty]$, we prove that if a solution doesn’t grow too quickly at infinity, then it must be trivial. We use $d_V$, the Agmon distance function associated to $V$, to quantify the threshold growth rate. More precisely, there exists a constant $γ_0 > 0$ so that if $|u(x)| \lesssim \exp(γd_V(x, 0))$ for some $γ< γ_0$ and every $x \in \mathbb{R}^n$, then $u$ must be trivial. The result may be interpreted as a Liouville-type theorem and is related to Landis’ conjecture. Our proof techniques are inspired by Z. Shen’s exponential decay estimates for fundamental solutions of Schrödinger operators and involve the application of a Fefferman-Phong inequality.

21. 度量空间中的耗散演化

Dissipative Evolutions in Metric Spaces

作者: Lauren Conger, Franca Hoffmann, Giuseppe Savaré

类别: math.AP, math.DS

发布日期: 2026-09-17

arXiv ID: 2609.20998v1

链接: http://arxiv.org/abs/2609.20998v1

摘要: We study a novel class of evolution variational inequalities (EVI) driven by bifunctions $\mathsf{b}:\mathrm{D} \times \mathrm{D} \to \mathbb{R}$ defined on a subset $\mathrm{D}$ of a metric space $(\mathrm{X},\mathrm{d})$ and satisfying a natural dissipativity condition

We provide general conditions for existence, stability, regularity, approximation and asymptotic behavior of solutions, by proposing a metric framework that generalizes the classical structure of evolutions driven by monotone operators in Hilbert spaces. A motivating application is the setting of minimax and multispecies coupled gradient flows, in which each of $N$ species evolves in the direction of steepest descent of its own energy functional and the joint system is not a gradient flow of any single energy. To construct solutions, we introduce a variational movement scheme (VMS), a time discretization scheme in which each update is a saddle point of a metrically penalized bifunction, generalizing the classical minimizing movement/ JKO scheme for single species gradient descent. In the coupled multispecies case, the VMS reduces to a Nash equilibrium problem. We prove existence of discrete solutions to the VMS via a general approach which combines dissipativity of $\mathsf{b}$ with a new notion of convexity along suitable $\textit{barycentric interpolations}$ (which is inspired by convexity along generalized geodesics, a crucial notion for gradient flows). Together, these conditions provide a zeroth-order notion of game-theoretic monotonicity applicable to general metric spaces without linear structure.

22. 变传播速度 Helmholtz 方程的笛卡尔完美匹配层：适定性与指数收敛

Cartesian perfectly matched layers for the Helmholtz equation with variable propagation speed: well-posedness and exponential convergence

作者: César Ortiz-Saavedra, Mauricio A. Londoño-Arboleda

类别: math.AP

发布日期: 2026-09-17

arXiv ID: 2609.20992v1

链接: http://arxiv.org/abs/2609.20992v1

摘要: We analyse the Cartesian perfectly matched layer (PML) approximation of the two-dimensional Helmholtz equation $-c^2Δu-ω^2 u=f$, in which the propagation speed $c$ is variable inside the region of interest and equal to a constant $c_\infty$ outside a disc $B_R$. Existing analyses of the Cartesian PML assume a homogeneous, or piecewise homogeneous, background. We assume of $c$ only a two-sided bound, so that discontinuous speeds – layered media and compact inclusions – are covered. This is possible because the speed enters the PML sesquilinear form only through its zeroth order coefficient, the principal part being built from the stretching profiles alone. Well-posedness of the physical problem at this regularity follows from a Lippmann-Schwinger equation, uniqueness from Rellich’s lemma and unique continuation. A Morawetz multiplier yields a sharp non-trapping condition on $c$, namely $\nabla c(x)\cdot(x-x_0)<c(x)$ for some $x_0$, which reduces in the radial case to the Herglotz condition $\frac{d}{dρ}(ρ/c(ρ))>0$; its failure produces a trapped bicharacteristic. The variable speed is invisible to the essential spectrum: after factoring out $c^2$, the two operators differ by a compactly supported multiplication, so the spectral analysis of the homogeneous Cartesian layer applies verbatim. This yields well-posedness of the truncated problem for every $ω>0$, with a stability constant independent of the layer width, and convergence to the physical solution in $H^1$ at a rate exponential in $ωγ_Mδ_M$, where $γ_M$ and $δ_M$ are the absorption strength and the layer thickness. Numerical experiments with a low-velocity inclusion confirm the exponential decay in both $δ$ and $ω$.