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1. 超临界初值下 Euler-对齐系统的有限时间破裂
Finite-time breakdown of the Euler-alignment system for supercritical initial data
作者: Young-Pil Choi, Eitan Tadmor
类别: math.AP
发布日期: 2026-07-09
arXiv ID: 2607.08060v1
摘要翻译: 我们通过相关拉格朗日流的退化来研究 Euler-对齐系统经典解的有限时间破裂。该方法使我们能够将奇异性形成刻画为流的局部可逆性的丧失以及密度沿特征线的集中。对于常数通信核的情形，我们推导出流的显式公式，并在任意维数下获得一个精确的逐点破裂判据。在二维情形下，该判据可被重写为关于初始速度梯度的对称部分与初始涡量的闭式表达式。对于一般的非常数核，我们通过将主导压缩机制与非局部余项的扰动控制相结合，导出有限时间退化的充分条件。这些条件在任意维数下提供了定量的超临界破裂判据，从而补充了现有针对多维 Euler-对齐系统的亚临界整体正则性理论。
Comment: 无
链接: http://arxiv.org/abs/2607.08060v1
1. 研究对象和背景
本文研究如下 $d$ 维欧拉-对齐系统（Euler–alignment system）的古典解的有限时间破裂问题：
其中 $\rho = \rho(t,x) \geq 0$ 为密度，$u = u(t,x) \in \mathbb{R}^d$ 为速度场，$\kappa > 0$ 为对齐强度，$\phi : \mathbb{R}^d \to \mathbb{R}_{\geq 0}$ 为通信核（communication kernel），满足 $\phi \in W^{1,\infty}(\mathbb{R}^d)$。该系统源于粒子型集体动力学模型的连续极限，描述了通过非局部速度对齐机制相互作用的智能体群体行为；其数学结构介于压强为零的欧拉方程与非局部阻尼系统之间，兼具双曲性、非线性输运与积分型源项三重特征。
关键未知量为密度 $\rho$ 与速度 $u$ 的正则性演化，特别是 $C^1$ 类古典解能否全局存在。已有理论表明：当初始数据处于“次临界”（subcritical）区域时，解可全局延拓；而当初始数据落入“超临界”（supercritical）区域时，可能发生有限时间奇点。但现有次临界理论已较成熟（如 [42] 在任意维中给出基于 $\lambda_{\min}(\nabla^S u_0) \geq -\kappa \phi(d_x^\infty)$ 的全局正则性判据），而超临界破裂判据却长期缺失——尤其在 $d \geq 3$ 维，既无定量破裂条件，亦无几何机制刻画。一维情形虽有尖锐点态阈值 $u_0'(x) \geq - (\kappa \phi * \rho_0)(x)$（[10]），二维情形亦有基于负散度与梯度结构的破裂结果（[43], [26]），但这些方法依赖坐标系或低维恒等式（如 $\nabla \cdot u = \operatorname{tr}(\nabla u)$），难以推广至高维。
本文试图填补的核心空白在于：建立适用于任意维 $d \geq 2$ 的、基于拉格朗日流几何退化（Lagrangian degeneracy）的定量超临界破裂准则，并揭示其与初速度梯度对称部分 $\nabla^S u_0 := \frac{1}{2}(\nabla u_0 + (\nabla u_0)^\top)$ 的谱结构之间的本质联系。具体而言，作者摒弃传统欧拉观点下对 $\nabla \cdot u$、$\lambda_{\min}(\nabla^S u)$ 等量的微分不等式分析，转而考察拉格朗日流映射 $\eta(t,\cdot)$ 的雅可比行列式 $J(t,x) := \det \nabla \eta(t,x)$ 的零化行为。由连续性方程可得 $\rho(t,\eta(t,x)) = \rho_0(x)/J(t,x)$，故 $J(t^*,x^*) = 0$ 直接导致沿特征线的密度爆破（Dirac 质量形成），且蕴含 $\|\nabla u(\cdot)\|_{L^\infty}$ 在 $t^*$ 处不可积，从而违反 Beale–Kato–Majda 型延拓准则。因此，$J=0$ 不仅是几何退化标志，更是 $C^1$ 解破裂的充要判据（见 Proposition 2.1）。问题本质转化为：在何种初值条件下，$J(t,x)$ 必于有限时间内归零？这要求精确控制 $\nabla \eta(t,x)$ 的演化，而后者满足矩阵常微分方程 $\partial_t \nabla \eta = (\nabla u)(t,\eta) \nabla \eta$，其解的奇异行为完全由 $\nabla u$ 沿特征线的累积效应决定。本文正是在此拉格朗日框架下，首次实现了对任意维超临界初值的定量破裂刻画。
2. 主要定理或主要结果
本文核心结论分为两类：常核情形下的精确判据与非常核情形下的定量充分判据。
定理 3.1（常核 $\phi \equiv 1$ 下的精确破裂判据）：设 $\phi \equiv 1$，$\rho_0 \in C_0 \cap \mathcal{P}(\mathbb{R}^d)$，$u_0 \in C^1 \cap W^{1,\infty}(\mathbb{R}^d; \mathbb{R}^d)$。对任意 $x^* \in \operatorname{supp} \rho_0$，下列命题等价：
(i) 存在 $t^* > 0$ 使得 $\det \nabla \eta(t^*, x^*) = 0$；
(ii) 存在 $\alpha \in (0, 1/\kappa)$ 使得 $\det\big(I_d + \alpha \nabla u_0(x^*)\big) = 0$；
(iii) $\nabla u_0(x^*)$ 具有实特征值 $\lambda^* < -\kappa$。
进一步，若 $\lambda_{\min}(x^*) := \min\{ \lambda \in \sigma(\nabla u_0(x^*)) \cap \mathbb{R} : \lambda < -\kappa \}$ 存在，则首次破裂时间为
且此时 $\alpha(t^*) = -1/\lambda_{\min}(x^*)$，$\det \nabla \eta(t^*, x^*) = 0$。
定理 1.1（非常核下的定量超临界判据）：设 $d \geq 2$，$\rho_0 \in C_0 \cap \mathcal{P}(\mathbb{R}^d)$，$\phi \in W^{1,\infty}(\mathbb{R}^d; \mathbb{R}_{\geq 0})$，$u_0 \in C^1 \cap W^{1,\infty}(\mathbb{R}^d; \mathbb{R}^d)$。令 $\mu_1(x) \leq \cdots \leq \mu_d(x)$ 为 $\nabla^S u_0(x)$ 的特征值。若存在 $x^* \in \operatorname{supp} \rho_0$、$\delta \in (0,1]$ 及 $t_1 > 0$，使得
且扰动小性条件成立：
并满足非退化假设：
且 $r^* := \#\{j : 1 + \alpha_\infty(t_1) \mu_j(x^*) < 0\}$ 为奇数，则存在 $t^* = t^*(x^*) \in (0, t_1)$ 使得 $\det \nabla \eta(t^*, x^*) = 0$。
定理 4.3（二维情形的改进判据）：在 $d = 2$ 时，利用行列式结构 $\det(I + \alpha \nabla u_0) = 1 + \alpha \operatorname{tr}(\nabla u_0) + \alpha^2 \det(\nabla u_0)$，可将上述扰动小性条件替换为更精细的估计，避免使用算子范数，从而获得 sharper 的破裂阈值，其显式形式涉及 $\nabla^S u_0$ 的特征值与初始涡量 $\omega_0 = \nabla \times u_0$。
3. 研究方法、关键技术和核心工具
本文证明的核心障碍在于：对于非常核 $\phi \not\equiv \text{const}$，拉格朗日流梯度 $\nabla \eta(t,x)$ 无法显式求解，因而不能像常核情形那样直接分析 $\det(I + \alpha \nabla u_0)$ 的零点。作者提出的真正创新点是构造一种“有效时间分解”（effective-time decomposition）框架，将 $\nabla \eta$ 分解为一个主导压缩项与一个可控制的扰动余项。
首先，在第 4.1 节中，作者引入修正的“有效时间参数” $\alpha_\infty(t) := \frac{1 - e^{-\kappa \|\phi\|_{L^\infty} t}}{\kappa \|\phi\|_{L^\infty}}$，它源于常核情形下 $\alpha(t)$ 的自然推广，并满足 $\alpha_\infty(t) \in (0, 1/\kappa)$。接着，他们将 $\nabla \eta(t,x)$ 写为
其中余项 $R(t,x)$ 包含两部分：(i) 由反对称梯度 $\nabla^A u_0$ 引起的旋转效应，其大小受 $t \|\nabla^A u_0(x)\|$ 控制；(ii) 由核 $\phi$ 的空间变异性引起的非局部扰动，其大小由 $\beta t^2 e^{C_0 t}$ 控制（此处 $\beta$ 量化了 $\nabla \phi$ 与初速度振幅的耦合）。这一分解的关键在于：主导项 $I_d + \alpha_\infty(t) \nabla^S u_0(x)$ 完全由初值对称梯度的谱决定，而余项 $R$ 在 $t$ 较小时被指数级压制。
其次，作者利用行列式的连续性与代数性质，将 $\det \nabla \eta$ 的零点存在性归结为：当 $\alpha_\infty(t)$ 扫过区间 $[\alpha_\infty(t_1), t_1]$ 时，主导项的行列式 $\det(I_d + \alpha \nabla^S u_0(x^*))$ 从正值变为负值（由 $r^*$ 为奇数保证），而余项 $R$ 的范数在整个区间上严格小于主导项行列式的最小模（由扰动小性条件 $< \delta$ 保证）。这本质上是一个鲁棒的符号变化论证（robust sign-change argument），其技术核心是精细的 Gronwall 型估计与矩阵行列式扰动引理（Lemma 4.1）。
第三，在二维情形下（Theorem 4.3），作者 exploit 行列式的二次多项式结构：
由于 $\det(\nabla u_0) = \det(\nabla^S u_0) - \frac{1}{4} \omega_0^2$，该表达式可显式用 $\nabla^S u_0$ 的特征值 $\mu_1, \mu_2$ 和涡量 $\omega_0$ 表示为 $1 + \alpha(\mu_1 + \mu_2) + \alpha^2(\mu_1 \mu_2 - \frac{1}{4} \omega_0^2)$。这使得作者能直接分析该二次函数的判别式与根的位置，从而绕过高维算子范数估计，获得比 Theorem 1.1 更sharp 的扰动容限。
4. 与之前工作的比较
本文与既有工作的比较可从四个维度展开：
1. 与次临界理论（[42]）的互补性：文献 [42] 给出了任意维全局正则性的充分条件：$\lambda_{\min}(\nabla^S u_0) \geq -\kappa \phi(d_x^\infty)$ 且 $\operatorname{osc}\{u_0\} \leq \frac{\kappa \|\phi\|_{L^\infty}}{8 \|\nabla \phi\|_{L^\infty}}$。本文 Theorem 1.1 给出其“超临界对应物”：若在某点 $x^*$ 处 $\lambda_{\min}(\nabla^S u_0(x^*)) < -\Lambda_{\sup}(x^*)$，其中 $\Lambda_{\sup}(x^*)$ 显式依赖于 $\kappa \|\phi\|_{L^\infty}$、$\|\nabla^A u_0(x^*)\|$ 与非局部扰动尺度 $\beta$，则必发生破裂。二者共同刻画了临界阈值附近的完整相图，但存在间隙：$-\kappa \phi(d_x^\infty)$ 与 $-\Lambda_{\sup}(x^*)$ 并非精确互逆，这是扰动论证固有的非sharp性所致。
2. 与低维欧拉阈值理论（[10], [43], [26]）的推广性：一维尖锐阈值 $u_0' \geq -(\kappa \phi * \rho_0)$ 和二维基于散度的破裂条件均依赖于特定维数恒等式（如 $\nabla \cdot u = \operatorname{tr}(\nabla u)$ 或 $\det(\nabla u) = \frac{1}{2}((\nabla \cdot u)^2 - |\nabla u|^2) + \frac{1}{4} \omega^2$）。本文首次将破裂机制提升至任意维，其判据 $\mu_1(x^*) < -\kappa(1+\delta)\|\phi\|_{L^\infty}$ 直接关联到 $\nabla^S u_0$ 的最压缩方向，不依赖坐标系，具有本征几何意义。
3. 与拉格朗日方法（[3, 4, 22]）的深化：压力为零的欧拉方程中，$J=0$ 已被用于刻画质量集中（mass concentration），但其分析集中于均匀压力或特殊对称情形。本文将此几何视角系统应用于非局部对齐系统，不仅确认 $J=0$ 是破裂判据（Proposition 2.1），更发展出一套针对非局部源项的、可定量的拉格朗日退化分析框架，包括有效时间分解与扰动余项控制。
4. 与现有超临界结果（[43]）的适用范围扩展：文献 [43] 在二维中要求 $\nabla \cdot u_0 < 0$ 且 $\partial_{x_1} u_2, \partial_{x_2} u_1$ 满足额外结构假设，其破裂证明依赖于特定不等式技巧。本文 Theorem 4.3 的二维判据仅需 $\mu_1(x^*)$ 足够负且 $\omega_0$ 有界，无需对偏导数施加逐点符号限制，适用范围更广。然而，代价是本文所有破裂判据均为充分非必要条件，且对初值正则性要求较高（$u_0 \in C^1 \cap W^{1,\infty}$），而 [43] 允许更弱的初值。从提供内容中无法确认本文结果是否能覆盖 [43] 中的所有超临界情形，亦未显式展开与 [26] 的谱隙条件的数值比较。
了解领域论文（摘要翻译）
1. Korteweg-de Vries 方程中周期波在局部扰动下的非线性稳定性
Nonlinear stability of periodic waves in the Korteweg-de Vries equation under localized perturbations
作者: Emile Bukieda, Björn de Rijk
类别: math.AP
发布日期: 2026-07-09
arXiv ID: 2607.08401v1
链接: http://arxiv.org/abs/2607.08401v1
摘要翻译
中文标题: Korteweg-de Vries 方程中周期波在局部扰动下的非线性稳定性
摘要翻译: 我们研究 Korteweg-de Vries 方程中空间周期 cnoidal 波在局部扰动下的稳定性与渐近行为。标准的哈密顿系统稳定性论证在此情形下失效，因为局部扰动排除了将稳定周期波表征为某个适当能量泛函在有限个约束下的严格极小元的可能性。因此，周期波在局部扰动下的非线性稳定性问题在哈密顿系统中长期悬而未决；此前的结果仅处理了可通过变换至极坐标而约化为常值态的平面波。本文发展了一种新方法，通过结合变分论证、Floquet-Bloch 理论以及基于 Duhamel 公式的时空调制估计，克服了这一障碍。我们的框架适用于具有对称性的哈密顿系统中的一般周期波，并将非线性稳定性问题归约为验证某个适当守恒能量之二阶变分的扩散型谱稳定性条件。将该方法应用于 Korteweg-de Vries 方程中的 cnoidal 波，我们得到了首个关于哈密顿系统中不能约化为常值态的周期波在局部扰动下的非线性稳定性结果。
2. 具调制色散的非线性偏微分方程 III：乘性噪声
Nonlinear PDEs with modulated dispersion III: multiplicative noises
作者: Andreia Chapouto, Massimiliano Gubinelli, Guopeng Li, Jiawei Li, Tadahiro Oh
类别: math.AP, math.PR
发布日期: 2026-07-09
arXiv ID: 2607.08385v1
链接: http://arxiv.org/abs/2607.08385v1
摘要翻译
中文标题: 具调制色散的非线性偏微分方程 III：乘性噪声
摘要翻译: 我们研究圆周上带乘性噪声的随机调制 Korteweg–de Vries 方程（KdV）的路径式适定性，其中时间非齐次调制作用于线性色散项。(i) 在 Young 情形（即时间分数阶情形，Hurst 参数大于 $\frac 12$）下，我们在路径式意义下建立了随机卷积中一种新的噪声正则化现象：调制越不规则，空间正则性的增益（任意地）越大。进而证明：对任意 $s \in \mathbb R$ 及任意乘性 Young 噪声（无论其在空间上多么粗糙），只要调制足够不规则，该随机调制 KdV 方程在 $H^s(\mathbb T)$ 中路径式局部适定。(ii) 在粗糙情形（即时空白噪声情形）下，调制的不规则性不会在随机卷积上诱导任何平滑效应；事实上，空间正则性存在轻微损失。在此情形下，通过对乘性噪声项作轻微正则化，我们证明：对任意给定 $s \in \mathbb R$，只要噪声在空间上足够光滑，则该随机调制 KdV 方程在 $H^s(\mathbb T)$ 中路径式局部适定。我们通过结合以下两方面技术实现上述目标：(i) Chouk 与第二作者（2014）引入的非线性 Young 积分理论的 sewing 引理方法；(ii) 第一、第四与第五作者（2026）引入的借助随机张量估计与 sewing 引理将随机卷积路径式构造为 Young 积分或粗糙积分的方法。附录中，我们还给出一个具乘性 Young 噪声的随机调制 Schrödinger 方程的噪声正则化现象示例。
3. 通过非局部均匀化实现的非线性介质
Nonlinear Media via Nonlocal Homogenisation
作者: Andreas Buchinger, Nathanael Skrepek, Marcus Waurick
类别: math.AP, math.FA
发布日期: 2026-07-09
arXiv ID: 2607.08140v1
链接: http://arxiv.org/abs/2607.08140v1
摘要翻译
中文标题: 通过非局部均匀化实现的非线性介质
摘要翻译: 我们考虑一个描述具有非局部介电性的非线性静电介质的非线性偏微分方程。相应方程的存在性证明基于 Schauder 定理以及一个关于移动系数的新紧性定理（“Helga 定理”）。该技术利用了（算子论/拓扑）均匀化理论中的洞见。令人惊讶的是，即使未使用也无需单调性假设，基础定义域仅需为弱 Lipschitz 域，且对非线性项的导数无需任何假设。
4. 移动杂质下 Gross-Pitaevskii 模型中灰孤子的不稳定性
Instability of gray solitons in a Gross-Pitaevskii model with a moving impurity
作者: Paolo Antonelli, Martino Caliaro
类别: math.AP, math-ph
发布日期: 2026-07-09
arXiv ID: 2607.08190v1
链接: http://arxiv.org/abs/2607.08190v1
摘要翻译
中文标题: 移动杂质下 Gross-Pitaevskii 模型中灰孤子的不稳定性
摘要翻译: 本文借助具非零无穷远处边界条件的一维 Gross-Pitaevskii 模型（GP），研究移动杂质对稀薄玻色-爱因斯坦凝聚体的影响。杂质被建模为一个局域化的外部势，以恒定速度 $v \in \mathbf{R}$ 运动。在随动参考系中，我们研究了时不变解的存在性与稳定性；这些解具有物理意义，对应于凝聚体的超流行为。 对于亚音速区域中任意非零速度 $v$，我们证明存在一族时不变解，该族解从 GP 方程的一个（平移后的）灰孤子 $φ_{0,v}(x-s_0)$ 分岔而来，其中 $s_0 \in \mathbf{R}$；位置 $s_0$ 被确定为一个显式定义的有效势的极值点。此外，我们研究了这些态的谱稳定性。对于势强较小时的情形，我们证明：起源于该有效势极大值点的解族是谱不稳定的。该结论借助 Evans 函数方法获得。最后，我们形式地将该不稳定性结果应用于排斥型 delta 势情形。
5. 提升粗糙极大算子的最优弱型估计及其应用
Optimal Weak-Type Estimates and Their Applications of Lifted Rough Maximal Operators
作者: Dachun Yang, Wen Yuan, Yirui Zhao
类别: math.CA, math.AP, math.FA
发布日期: 2026-07-09
arXiv ID: 2607.08277v1
链接: http://arxiv.org/abs/2607.08277v1
摘要翻译
中文标题: 提升粗糙极大算子的最优弱型估计及其应用
摘要翻译: 设 $n\in\mathbb N\cap[2,\infty)$ 且 $Ω\in L^1(\mathbb S^{n-1})$ 满足 $Ω\not\equiv 0$。本文在上半平面引入一族新的提升粗糙极大算子 $\{\mathcal{M}_θ^Ω\}_{θ\in(0,\infty)}$，并建立其最优弱型估计。具体而言，我们证明：对任意 $p \in (1, \infty)$，如下估计（其中正的等价常数与 $f$ 无关）
对所有 $f\in L^p(\mathbb R^n)$ 成立当且仅当 $γ\in\mathbb R\setminus\{0\}$。对于端点情形 $p=1$ 且 $Ω\in L(\log L)(\mathbb{S}^{n-1})$，我们证明上述估计成立当且仅当 $γ\in (-\infty, -n) \cup (0, \infty)$。作为应用，我们在无需任何对数可积性假设的前提下，获得了广义泊松积分的弱型估计，从而正面回答了 Sjögren 与 Soria 在 [Israel J. Math. 95 (1996), 第 228 页] 中提出的问题。此外，尽管由 Calderón–Zygmund 旋转法导出的算子 $M^\ast_Ω$ 并非弱型 $(1,1)$，但我们发现其提升变体却是弱型 $(1,1)$ 的。进一步，我们建立了关于截断粗糙奇异积分的 Hardy 空间的新刻画。
6. 具Neumann边界条件的反应-扩散方程在一致扰动的非光滑区域上的全局吸引子的一致 $L^{ ty}$-有界性
Uniform $L^{\infty}$-Boundedness of Global Attractors for Reaction-Diffusion Equations with Neumann boundary condition in Uniformly Perturbed Non-Smooth Domains
作者: Antonio L. Pereira
类别: math.AP, math.DS
发布日期: 2026-07-09
arXiv ID: 2607.08061v1
链接: http://arxiv.org/abs/2607.08061v1
摘要翻译
中文标题: 具Neumann边界条件的反应-扩散方程在一致扰动的非光滑区域上的全局吸引子的一致 $L^{ ty}$-有界性
摘要翻译: 我们考虑一族定义在变化的非光滑区域 $\{Ω_μ\}_{μ\in Λ} \subset \mathbb{R}^n$ 上、具齐次Neumann边界条件的半线性抛物型方程。仅假设这些区域具有统一有界的体积、满足一致Jones条件，并具备一致椭圆性估计，我们在适当的分数阶Banach空间尺度上建立了该问题的适定性，并证明了全局吸引子的存在性。利用Moser-Alikakos迭代方法，结合一致Gronwall引理及Jones延拓算子的一致性质，我们证明该族吸引子在 $L^\infty(Ω_μ)$ 中具有一致有界性。最后，在区域体积收敛 $|Ω_μ\triangle Ω_0| \to 0$ 的假设下，我们构造了一套连接映射框架，证明该族吸引子在 $μ= 0$ 处关于强 $H^1$ 拓扑是上半连续的。
7. 拟解析电荷
The Pseudo-Analytic Charge
作者: Daniel Alayón-Solarz
类别: math.CV, math.AP
发布日期: 2026-07-08
arXiv ID: 2607.07910v1
链接: http://arxiv.org/abs/2607.07910v1
摘要翻译
中文标题: 拟解析电荷
摘要翻译: 带框架的Beltrami–Vekua方程 $Φ(w_{\bar z} - μw_z) + Ψ(\overline{w_z} - μ\overline{w_{\bar z}}) + \mathfrak{a}w + \mathfrak{b}\bar w = \mathfrak{f}$，其中 $|μ|<1$ 且 $|Φ|>|Ψ|$，携带一个分子场 $N = Φ\mathfrak{b} - Ψ\mathfrak{a} - W_L(Φ,Ψ)$，其加权模的积分给出拟解析质量。本文提取了该同一场所携带的整数。当 $N$ 的零点集紧含于某有界单连通区域中时，沿任意包围曲线的 $N$ 的环绕数——即拟解析电荷 $n \in \mathbb{Z}$——在未知函数的每一重组 $w = \varphi w' + ψ\bar w'$ 下、在方程的每一缩放下、以及在每一保向 $C^1$ 变量替换下均保持不变：重组将 $N$ 乘以正因子 $|\varphi|^2 - |ψ|^2$，故其不变性是精确的；而在多重连通区域上，其余两种作用仅在 $\mathbb{Z}/2\mathbb{Z}$ 中固定各分量电荷，而总电荷则精确不变。该电荷是一个Brouwer度：它局域化于 $N$ 的零点处，这些零点是涡旋，而该类作用既不产生也不消灭此类涡旋；孤立涡旋在数据扰动下持续存在当且仅当其局部电荷非零。该电荷仅通过框架的 $L$-Wronskian 涉及Beltrami系数，且在 $W_\partial(Φ,Ψ) \equiv 0$ 处与 $μ$ 无关——特别地，在平凡框架下，$N = \mathcal{B}$，此时电荷即为Beltrami–Vekua方程系数的规范不变环绕数。质量和电荷相互独立：$(0,\infty)\times\mathbb{Z}$ 中的每一对值均可实现。
不相关论文（基本信息）
1. 各向异性散射体附近的局部梯度增强
Localized gradient enhancement near anisotropic scatterers
作者: Weisheng Zhou, Huaian Diao, Hongyu Liu
类别: math.AP
发布日期: 2026-07-09
arXiv ID: 2607.08598v1
链接: http://arxiv.org/abs/2607.08598v1
摘要: This work investigates time-harmonic electromagnetic scattering governed by the Maxwell system in the presence of bounded anisotropic electromagnetic scatterers embedded in an intermediate anisotropic electromagnetic layer. We focus on the localized enhancement of the gradients of the total electric and magnetic fields in small boundary-attached neighborhoods of finitely many prescribed points on the outer interface of the surrounding layer. We show that, through a suitable construction of incident electromagnetic waves, the gradients of both the total electric field and the total magnetic field can be made arbitrarily large in these neighborhoods. Moreover, the localization radius may be chosen according to the prescribed gradient magnitude, thereby describing a localized high-gradient concentration mechanism for electromagnetic fields near anisotropic scatterers. The main strategy is based on the introduction of auxiliary boundary-attached electromagnetic neighborhoods and the associated electric and magnetic fields, which exhibit strong gradient variation near the prescribed points. Using the approximation property of Maxwell Herglotz wave functions, these auxiliary fields are then approximated by physically admissible incident waves in the neighborhood of the scatterers. Together with the well-posedness and continuous dependence of the anisotropic scattering problem, this implies that the corresponding scattered field can be controlled to be sufficiently weak in the relevant layer region. Consequently, the total field is dominated by the incident field near the prescribed points and inherits its large-gradient behavior.
2. 具有壁面附着与热力学生长抑制的生物膜反应器模型中平衡态的稳定性
Stability of Equilibria in a Biofilm Reactor Model with Wall Attachment and Thermodynamic Growth Inhibition
作者: Katerina Nik, Christoph Walker
类别: math.AP
发布日期: 2026-07-09
arXiv ID: 2607.08498v1
链接: http://arxiv.org/abs/2607.08498v1
摘要: The dynamics of a mathematical model for a chemostat-type reactor is investigated. The model describes the temporal evolution of suspended and wall-attached bacterial populations, with the latter represented as a one-dimensional biofilm, subject to a non-reproducing growth-limiting substrate and a reaction product formed through bacterial utilization of the substrate. In particular, it is shown that, in the regime where the trivial (washout) equilibrium is unstable, there exists a unique nontrivial equilibrium that is locally asymptotically stable. Under slightly stronger assumptions, uniform persistence and global asymptotic stability of the nontrivial equilibrium are established.
3. 压力与剪切率依赖粘性的Hele-Shaw流
Hele-Shaw Flow With Pressure and Shear Rate Dependent Viscosity
作者: Benedetta Calusi, Liviu Iulian Palade
类别: physics.flu-dyn, math.AP
发布日期: 2026-07-09
arXiv ID: 2607.08509v1
链接: http://arxiv.org/abs/2607.08509v1
摘要: This paper investigates the behaviour of a fluid characterized by a viscosity simultaneously depending on pressure and shear rate within a Hele-Shaw cell featuring a sharp corner geometry. The study extends previous analyses conducted on purely pressure-dependent (piezo-viscous) and yield-stress fluids, providing a new perspective on confined complex flows. Motivated by practical applications related to designing biomedical devices and flows of relevance to biomedicine area, thin film technologies, injection molding – to name only a few – the flow configuration considered here can highlight essential features of complex fluid behavior in narrow-gap geometries around a sharp edge. Starting from the governing equations for an incompressible generalized Newtonian fluid and employing an appropriate rheological model, we derive the modified flow equations adapted to the Hele-Shaw flow. A particular solution is obtained near the corner region. Numerical simulations complement the theoretical results, illustrating the influence of the rheological parameters on the flow behavior.
4. Alt-Phillips 问题极小元最优正则性的简短证明
A Short Proof of Optimal Regularity for minimizers of the Alt-Phillips Problem
作者: Kunyi, Ma
类别: math.AP
发布日期: 2026-07-09
arXiv ID: 2607.08685v1
链接: http://arxiv.org/abs/2607.08685v1
摘要: We give a self-contained short proof of optimal regularity for minimizers of the Alt-Phillips Free Boundary Problem for $γ\in (0, 1)$. We adopt a dichotomy argument that originates from [DS20].
5. 任意度量下 $S^3$ 中两个嵌入极小球面的存在性
Existence of two embedded minimal spheres in $S^3$ with an arbitrary metric
作者: Zhichao Wang, Xin Zhou
类别: math.DG, math.AP, math.GT
发布日期: 2026-07-09
arXiv ID: 2607.08631v1
链接: http://arxiv.org/abs/2607.08631v1
摘要: We prove that $S^3$ endowed with an arbitrary Riemannian metric $g$ admits at least two embedded minimal spheres. The proof is based on an iterative scheme of relative min-max constructions.
6. 关于子区域上特征值收敛至连续谱的一个注记
A note on the convergence of the eigenvalues in a subdomain to the continuous spectrum
作者: Miroslav Bulíček, Bjørn Fredrik Nielsen, Zdeněk Strakoš
类别: math.AP
发布日期: 2026-07-09
arXiv ID: 2607.08433v1
链接: http://arxiv.org/abs/2607.08433v1
摘要: The SIGEST paper Nielsen and Strakoš (2024) characterized the spectrum of the preconditioned operator $Δ^{-1}[\nabla \cdot (K\nabla u)]$ in a bounded open two-dimensional domain $Ω$, where $Δ$ denotes the Laplacian and $K(x,y)$ is a continuous symmetric matrix-valued function. An important part of the analysis states that for a diagonal tensor $K$ constant in an open subdomain $S \subset Ω$, the closed interval defined by its diagonal elements belongs to the spectrum of the preconditioned operator. This result is correct, but the proof in Nielsen and Strakoš (2024) must be refined. This paper presents a refined proof and extends the previous work. As shown in the cited papers, for any point $λ$ in the open interval defined by the elements of the diagonal tensor constant in $S$ and any point $(x_0,y_0)\in S$, a rectangular subdomain $Σ_l\subset S$ can be constructed such that the generalized eigenvalue problem associated with the preconditioned operator restricted to $Σ_l$, of arbitrarily small size, has the eigenvalue $λ$ and infinitely many eigenfunctions. These are given by solutions of a locally defined wave equation. However, such solutions of the locally restricted generalized eigenvalue problem cannot be extended to the whole domain $Ω$. Using instead rectangular subdomains whose size shrinks to zero, the present paper constructs a Weyl singular sequence of \emph{approximate} eigenfunctions associated with $λ$, proving that $λ$ belongs to the spectrum of the preconditioned operator. Since self-adjoint operators in a separable Hilbert space can have at most a countable set of eigenvalues, this shows that the eigenvalues of the locally defined operator converge to points of the continuous spectrum of the preconditioned operator on the entire domain.
7. 一种基于容度的Lyapunov型不等式方法：用于加权图上的椭圆问题
A capacitary approach to Lyapunov-type inequalities for elliptic problems on weighted graphs
作者: Mohamed Jleli, Bessem Samet
类别: math.AP
发布日期: 2026-07-09
arXiv ID: 2607.08405v1
链接: http://arxiv.org/abs/2607.08405v1
摘要: We initiate the study of Lyapunov-type inequalities for Dirichlet problems driven by the discrete p-Laplacian on weighted graphs. The approach is capacitary and is based on point p-capacities and the associated capacitary radii. First, we prove general Lyapunov-type inequalities on arbitrary connected locally finite weighted graphs. These inequalities provide intrinsic lower bounds, expressed in terms of the capacitary radii, for the positive part of the potential whenever the corresponding Dirichlet problem admits a nontrivial solution. Next, we estimate these capacitary radii in several geometric settings and prove the sharpness of the resulting Lyapunov-type inequalities. As an application, we derive lower bounds for the first weighted Dirichlet eigenvalue of the discrete p-Laplacian.
8. 具有临界局部与非局部相互作用的磁薛定谔-泊松系统的归一化半经典解
Normalized Semiclassical Solutions to Magnetic Schrödinger-Poisson Systems with Critical Local and Nonlocal Interactions
作者: Khaled Khachnaoui
类别: math.AP
发布日期: 2026-07-09
arXiv ID: 2607.08381v1
链接: http://arxiv.org/abs/2607.08381v1
摘要: We study the existence, multiplicity, and concentration of normalized semiclassical states for a magnetic Schrödinger–Poisson system in $\mathbb{R}^3$ featuring both the Sobolev-critical local nonlinearity $|u|^4u$ and a critical nonlocal Poisson interaction. The problem is considered under the prescribed mass constraint $\int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3,$ where $a>0$ denotes the prescribed mass and $\varepsilon>0$ is the semiclassical parameter. By combining constrained variational methods, a suitable penalization scheme, concentration–compactness arguments, and Ljusternik–Schnirelmann theory, we first prove the existence of a normalized semiclassical solution for sufficiently small $a$ and $\varepsilon$. We then establish a multiplicity result showing that, for every sufficiently small $\varepsilon>0$, the number of distinct normalized solutions is bounded from below by the Ljusternik–Schnirelmann category of the minimum set
Finally, we describe the semiclassical concentration phenomenon by showing that the maximum points of the resulting solutions approach $\mathcal M$ as $\varepsilon\to0$.
9. Kähler-Ricci 流的几何正则化
Geometric smoothing by the Kähler-Ricci Flow
作者: Eleonora Di Nezza, Vincent Guedj, Chinh Hoang Lu
类别: math.DG, math.AP, math.CV
发布日期: 2026-07-09
arXiv ID: 2607.08325v1
链接: http://arxiv.org/abs/2607.08325v1
摘要: We study the geometric regularization of a positive closed current by the (twisted) Kähler-Ricci flow on a compact Kähler manifold. We conjecture that the local Arnold multiplicities linearly decrease to zero, while the flow produces complete Kähler metrics in the Zariski open subset of points that have small Lelong numbers. We prove this conjecture in complex dimension 1 and provide several partial results in higher dimension.
10. 关于不同凸性概念下Wasserstein空间中邻近算子的稳定性
On the stability of proximal operators in Wasserstein spaces under different notions of convexity
作者: Simone Di Marino, Sara Farinelli, Emanuele Naldi
类别: math.OC, math.AP, math.FA
发布日期: 2026-07-09
arXiv ID: 2607.08209v1
链接: http://arxiv.org/abs/2607.08209v1
摘要: The proximal operator is a fundamental tool in variational analysis and optimization. In the setting of a Hilbert space, given a proper, lower semicontinuous convex functional, its proximal operator is non-expansive, that is, 1-Lipschitz continuous. In the Wasserstein setting, the contraction properties of this operator have been investigated from different perspectives by Carlen and Craig and Adve and Mészáros, among others, and are not completely understood. In this paper, we study the stability properties of proximal maps, with a particular focus on non-expansivity, under various notions of convexity of the functional that can be considered in the Wasserstein space.
11. 一种非解耦的时域直接采样方法用于弹性介质反散射问题
A Non-Decoupled Time-Domain Direct Sampling Method for Inverse Elastic Medium Scattering
作者: Lefu Cai, Hongjie Li, Xianchao Wang
类别: math.NA, math.AP
发布日期: 2026-07-09
arXiv ID: 2607.08067v1
链接: http://arxiv.org/abs/2607.08067v1
摘要: This work is concerned with an inverse medium problem for elastic waves, in which unknown inhomogeneities are reconstructed from time-resolved boundary measurements. We propose a novel time-domain direct sampling method for locating scatterers from a single incident source, without imposing specific assumptions on the temporal profile of the excitation. In particular, the imaging functional introduces a time-shifted correlation strategy that replaces the traditional $P$-$S$ wave decomposition with a travel-time alignment mechanism, thereby enabling direct imaging from the coupled elastic wave field. To analyze the proposed time-domain imaging functional, we employ Parseval’s identity for the Fourier–Laplace transform and reformulate the functional in the frequency domain. By exploiting properties of modified Bessel functions, we characterize the asymptotic behavior of the imaging functional and show that it attains its maximum at the target location, which enables reliable identification of the scatterer. Rigorous theoretical justifications are provided to substantiate the effectiveness of the proposed method. Numerical experiments are also presented to demonstrate its performance and applicability.
12. 带线性扰动的临界Hamilton系统存在无穷多个变号解
Infinitely many sign-changing solutions for critical Hamiltonian systems with linear perturbation
作者: Yuxia Guo, Congzheng Xuanyuan
类别: math.AP
发布日期: 2026-07-09
arXiv ID: 2607.08021v1
链接: http://arxiv.org/abs/2607.08021v1
摘要: In this paper, we study the following elliptic system \begin{equation}\label{main_1} \begin{cases} -Δu = |v|^{p-1} v + ε(αu + β_1 v), & \text{in } Ω, \ -Δv = |u|^{q-1} u + ε(β_2 u + αv), & \text{in } Ω, \ u = v = 0, & \text{on } \partial Ω, \end{cases} \tag{*} \end{equation} where $Ω$ is the unit ball in $\mathbb{R}^N$, $ε$ is a small parameter, $α$, $β_1$ and $β_2$ are real numbers, $(p, q)$ is a pair of positive numbers lying on the critical hyperbola \begin{equation} \frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2}{N}.\nonumber \end{equation} Under suitable assumptions and suitable restrictions on $(p,q)$ and $N$, we construct infinitely many sign-changing solutions to \eqref{main_1} which look like a positive radial solution to \eqref{main_1} crowned by $k$ negative bubbles arranged on a regular polygon of a suitable radius, whose energy can be arbitrarily large.
13. 依赖于未知函数的次黎曼拉格朗日量的弱KAM定理
Weak KAM theorems for subriemannian Lagrangians depending on the unknown function
作者: Renato Iturriaga, Héctor Sánchez Morgado
类别: math.AP
发布日期: 2026-07-08
arXiv ID: 2607.07966v1
链接: http://arxiv.org/abs/2607.07966v1
摘要: We extend some results of weak KAM theory to Lagrangians that are defined only on the horizontal distribution of a subriemannian manifold and depend on the unknown function