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1. 一维 Hou-Luo 模型与二维 Boussinesq 方程中具奇异剖面的新型自相似有限时间爆破：数值研究

Novel Self-similar Finite-time Blowups with Singular Profiles of the 1D Hou-Luo Model and the 2D Boussinesq Equations: A Numerical Investigation

作者: Bojin Chen, De Huang, Xiangyuan Li

类别: math.AP, math.NA

发布日期: 2026-04-02

arXiv ID: 2604.01868v1

摘要翻译: 针对一维 Hou-Luo 模型，我们给出了一类新型的自相似有限时间爆破情形。数值计算表明，初始满足特定导数退化条件的解可发展为渐近自相似的有限时间爆破，其自相似奇异剖面在某点处无界。此外，该爆破现象呈现两阶段特征：解首先在某一时刻 $\tilde{T}$ 发生局部 $L^{\infty}$ 爆破，随后在弱解意义下延拓至 $\tilde{T}$ 之后，并于之后的时刻 $T>\tilde{T}$（对某个 $p>0$）发展为局部 $L^p$ 爆破。进一步的数值研究表明，这两个阶段均具有渐近自相似性。最后，我们将数值研究推广至二维 Boussinesq 方程，发现了同样具有奇异剖面且呈现两阶段特征的类似自相似有限时间爆破。

Comment: 51 pages

链接: http://arxiv.org/abs/2604.01868v1

1. 研究对象和背景

本文的研究对象为流体力学中描述边界奇点形成机制的一维简化模型——Hou-Luo（HL）模型及其关联的二维Boussinesq方程。三维不可压Euler方程在无界空间$\ℝ^3$中光滑初值能否在有限时间爆破是数学流体力学的核心开放问题。Luo与Hou在固体边界附近数值观测到了自相似奇点形成现象，并由此提出了一维边界简化模型：

其中$(x,t)\in\mathbb{R}\times[0,T_{max})$，$\omega(x,t)$为拟涡量，$\theta(x,t)$为温度标量（或压力梯度代理），$\mathcal{H}$为实轴上的Hilbert变换。该模型保留了原始三维轴对称Euler方程在非对称轴上的主要非线性结构，是研究自相似爆破机制的极佳试验场。

自相似爆破的严格数学刻画依赖于形如

的剖面收敛性。将Ansatz代入原方程并令$t\to T$，由量纲平衡可得标度关系$\lambda=-1$，$\mu=\gamma-2$，其中$\gamma$与剖面远场衰减率紧密相关。若仅要求在$t\to T$时动态标度变量渐近收敛至稳态，则称为渐近自相似爆破。

既有理论（如Chen-Hou-Huang的计算机辅助证明与Huang-Qin-Wang-Wei的纯分析不动点方法）已完整刻画了“非退化”光滑奇对称初值（满足$\omega_{0x}(0)\neq 0, \theta_{0xx}(0)\neq 0$）下的爆破图景：解在原点对称点处发生单阶段$L^\infty$爆破，动态重标度后收敛至有界连续（正则）稳态剖面，且剖面在原点非退化。然而，对于更一般的“退化”初值情形（即$\omega_{0x}(0)=\theta_{0xx}(0)=0$），现有理论几乎空白。Liu的早期数值研究仅暗示退化情形可能导致不同于非退化情形的奇异行为，但缺乏对剖面正则性、多尺度结构或爆破时间精细分阶段的严格描述。退化条件破坏了标准能量估计所需的非零边界展开项，导致传统稳定性分析失效，且无法直接套用不动点框架证明正则剖面的存在性。本文正是瞄准这一理论空白，试图系统探究：在退化初值下，HL模型是否仍表现出自相似爆破？其极限剖面是否保持有界？爆破过程是否呈现多阶段动力学特征？该问题不仅关乎HL模型解的完整相空间结构，也为理解无界空间中低正则或特殊结构初值驱动的奇点形成机制提供关键的降维模型参照。

2. 主要定理或主要结果

在退化初值（光滑、奇对称且满足$\omega_{0x}(0)=\theta_{0xx}(0)=0$，或单边非负支集）假设下，作者通过高精度数值模拟与部分解析构造，揭示了HL模型的一种新型两阶段自相似有限时间爆破机制，并给出了具有显式表达式的奇异稳态剖面。主要结果可严格/数值表述如下：

1. 两阶段爆破动力学与时间尺度：数值实验表明，解首先在有限时间$\tilde{T}$于原点两侧对称点$x=\pm x_0(\tilde{T})\neq 0$处发生局部$L^\infty$爆破（第一阶段）。在$\tilde{T}$之后，解在弱意义下继续演化，奇点沿空间轨道向原点输运。在更晚的有限时间$T>\tilde{T}$，解在原点附近发展出局部$L^p$爆破（第二阶段），其中指数$p>0$由第二阶段渐近收敛的奇异自相似剖面决定。两阶段的动态重标度剖面均表现出渐近自相似性。

2. 渐近标度律与参数关系：引入动态标度变换

   $$ \omega(x,t)=C_\omega(\tau)^{-1}\Omega(C_l(\tau)x,\tau),\quad \theta(x,t)=C_\theta(\tau)^{-1}\Theta(C_l(\tau)x,\tau), $$

   其中标度因子满足$\frac{d}{d\tau}\ln C_\omega = c_\omega$, $\frac{d}{d\tau}\ln C_l = c_l$。数值拟合确认爆破标度指数满足$\lambda=-1$，$\mu=\gamma-2$，且对于退化情形观测到稳定的$\gamma=-\bar{c}_l/\bar{c}_\omega$收敛值，保证了自相似结构的鲁棒性。

3. 显式奇异稳态剖面（Theorem 2.3）：在弱解意义下，作者严格构造了单边奇异稳态解。对于原方程，稳态对为

   $$ (\bar{\omega}(x), \bar{\theta}(x)) = \left(\frac{\mathbf{1}_{\{x>0\}}}{\sqrt{x}}, \frac{\pi \mathbf{1}_{\{x>0\}}}{2}\right)。 $$

   对应到动态标度方程(2.5)的稳态，参数固定为$\bar{c}_l=2, \bar{c}_\omega=-1$，剖面为

   $$ (\bar{\Omega}(X), \bar{\Theta}(X)) = \left(\frac{\mathbf{1}_{\{X>1\}}}{\sqrt{X-1}}, \frac{\pi \mathbf{1}_{\{X>1\}}}{2}\right)。 $$

   该剖面在$X=1$（对应物理空间$x=(T-t)^\gamma$）处具有代数奇性$(X-1)^{-1/2}$，明确突破了非退化情形下有界连续剖面的限制。

4. 二维Boussinesq方程的推广：将退化初值设定与数值追踪框架应用于边界诱导的二维Boussinesq系统

   $$ \omega_t + u_1\omega_{x_1} + u_2\omega_{x_2} = \theta_{x_1},\quad \theta_t + u_1\theta_{x_1} + u_2\theta_{x_2} = 0, $$

   数值结果复现了完全相同的两阶段结构：先发生带正则剖面的$L^\infty$爆破，随后弱延拓至带奇异剖面的$L^p$爆破。从提供内容看，该推广主要依赖数值验证，显式稳态构造与严格弱解存在性证明的细节在后续章节，当前摘录仅给出定理陈述与数值对应关系。

3. 研究方法、关键技术和核心工具

本文面对的核心分析障碍在于：退化初值条件$\omega_{0x}(0)=\theta_{0xx}(0)=0$直接导致传统基于原点的Taylor展开非退化假设失效，使得[CHH22]中用于控制非线性项$u\omega_x$与$\theta_x$共振的加权$H^1$能量估计无法直接闭合；同时，Hilbert变换$\mathcal{H}(\omega)$在解趋近奇异剖面时对低正则性的敏感度急剧上升，常规主值积分的数值稳定性面临严峻挑战。为突破这些障碍，作者发展并整合了以下关键技术框架：

第一，全局动态标度重构与稳态收敛准则。作者采用时变坐标伸缩变换(2.2)-(2.3)，将有限时间爆破问题转化为动态标度变量$(\Omega,\Theta, c_l, c_\omega)$在$\tau\to\infty$时的稳态收敛问题。该重构的核心在于将爆破奇点“拉回”至固定计算域，并通过调节$c_\omega(\tau)$与$c_l(\tau)$的ODE演化捕获瞬时增长速率。引理2.2提供了从动态标度收敛到物理空间渐近自相似爆破的严格桥梁：只要满足积分衰减条件$\int_0^\infty |c_{l,\omega}(\tau)-\bar{c}_{l,\omega}|d\tau<\infty$，即可由极限定理严格导出$\lim_{t\to T}(T-t)\omega((T-t)^\gamma x/C_0,t)=-\bar{c}_\omega^{-1}\bar{\Omega}(x)$。该引理规避了直接追踪$T-t$奇异性带来的数值溢出，使得长时间渐近行为可计算。

第二，标度自由度固定与归一化条件。稳态方程(2.5)具有双参数缩放不变性$(\alpha\bar{\Omega}(\beta X), \alpha^2\bar{\Theta}(\beta X)/\beta, \alpha\bar{c}_l, \alpha\bar{c}_\omega)$。为消除数值求解时的漂移，作者在演化过程中动态施加约束条件（非退化文献中常用$\partial_\tau\Omega_X(0)=\partial_\tau\Theta_{XX}(0)\equiv 0$，退化情形则需调整锚定点以匹配奇点位置），从而保证标度因子$c_l, c_\omega$唯一确定，使数值轨迹能稳定逼近真实的吸引子稳态。

第三，显式弱解构造与Hilbert变换奇性处理。针对正则能量方法失效的问题，作者放弃强解正则性假设，转而在分布意义下直接验证候选奇异剖面。定理2.3通过分步积分与奇异核的主值对称性分析，严格证明了$\bar{\Omega}\sim (X-1)^{-1/2}$在加权空间下满足稳态方程。该构造避开了高阶导数估计，仅依赖$\mathcal{H}$对特定代数奇性的解析映射性质，为数值观测提供了坚实的解析锚点。

第四，弱延拓与两阶段数值追踪算法。面对第一阶段$L^\infty$爆破后的奇性，作者采用自适应网格细化与时间步长回溯策略，在$\tilde{T}$处切换至守恒律弱形式继续积分。通过监控局部$L^p$范数增长率与重标度残差，分离出$\tilde{T}$与$T$两个时间尺度，并利用对数导数拟合精确提取标度指数$\gamma$与$p$。该数值框架虽属计算分析范畴，但通过多初始条件测试与标度不变性检验，确保了现象的结构性而非偶然性。

4. 与之前工作的比较

本文在假设条件、剖面正则性、爆破阶段结构及适用范围上，与既有HL模型与Boussinesq爆破文献存在显著差异与实质性推进：

与非退化正则爆破理论[CHH22, HQWW25]的比较：既有工作严格限定于$\omega_{0x}(0)\neq 0$的奇对称非退化初值，证明解在原点发生单阶段直接爆破，动态标度剖面收敛至有界连续函数（$\bar{\Omega}\in L^\infty\cap H^p$）。本文彻底放宽该退化假设，允许$\omega_{0x}(0)=0$，揭示了完全不同的动力学路径：爆破位置初始偏离原点，且极限剖面必然包含代数奇点（局部无界）。代价在于，当前结果主要基于高精度数值验证与定理2.3的弱解存在性，缺乏非退化情形中完整的非线性轨道稳定性计算机辅助证明；同时，两阶段机制中$\tilde{T}$至$T$区间的弱解唯一性与适定性仍依赖物理延拓假设，尚未建立全空间的强解唯一性理论。

与早期退化数值探索[Liu17]的比较：Liu的学位论文首次暗示退化数据可能导致不同爆破模式并提出双尺度猜想，但未给出精细时间尺度划分与显式剖面构造。本文明确将猜想落地为“$L^\infty$爆破（正则剖面）$ o$ 弱延拓 $ o$ $L^p$爆破（奇异剖面）”的两阶段图景，并给出了显式弱稳态解$(\bar{\Omega},\bar{\Theta})$的闭合公式，同时验证了该机制在非对称（单边）初值下的鲁棒性，打破了以往研究对奇对称性的强制依赖，大幅拓展了适用初值类。

与自由空间三维Euler爆破[Elg21, CMZZ25]及二维Boussinesq边界爆破[WLGSB23, CH25]的关联与局限：Elgindi等人利用$C^{1,\alpha}$小Hölder指数初值构造了自由空间自相似爆破，但其正则性要求低于光滑函数；Córdoba等人则依赖无穷多涡量区的复杂拓扑结构。本文表明，即使在无限光滑但导数退化的初值下，降维模型依然能自发产生奇异自相似结构，这为无边界三维Euler方程的光滑爆破机制提供了新的启发方向。在二维Boussinesq方程方面，现有文献多聚焦于非退化边界爆破的严格证明。本文将退化奇异爆破成功迁移至该物理方程，数值上复现了两阶段特征。然而，从摘录内容看，Boussinesq情形的显式稳态构造与动态标度收敛的解析细节尚未完全展开，数值截断误差对弱延拓阶段$p$指数的敏感度仍需更高精度算法确认。总体而言，本文以牺牲部分解析严格性（相较于[CHH22]）为代价，换取了对退化相空间奇点形成机制的首次系统刻画与显式数学表达，填补了HL模型退化初值理论的关键空白。

了解领域论文（摘要翻译）

1. 紧型秩一黎曼对称空间上 Hölder 函数的 Cesàro 可和性与 Talbot 效应

Cesàro summability of Hölder functions and Talbot effect on rank one Riemannian symmetric spaces of compact type

作者: Utsav Dewan

类别: math.CA, math.AP

发布日期: 2026-04-02

arXiv ID: 2604.02305v1

链接: http://arxiv.org/abs/2604.02305v1

摘要翻译

中文标题: 紧型秩一黎曼对称空间上 Hölder 函数的 Cesàro 可和性与 Talbot 效应

摘要翻译: 在维数 $\ ge 2$ 的紧型秩一黎曼对称空间上，我们首先借助 Cesàro 平均给出了 Hölder 连续性的定量刻画。除若干逼近论应用外，我们还将其应用于研究源于衍射理论的著名物理现象“Talbot 效应”。更确切地说，对几乎每个固定时刻，我们根据初值的 Littlewood-Paley 投影的衰减性质，考察了 Schrödinger 演化的 Hölder 连续性及其分形轮廓。在此过程中，我们还分别建立了带型球函数在原点附近与割迹附近的一致振荡展开式，该结果可能具有独立的学术价值。

2. 无穷维环面上的 Kronecker 流

Kronecker Flow on the Infinite Torus

作者: Lorenzo Baroni

类别: math.DS, math.AP

发布日期: 2026-04-02

arXiv ID: 2604.02213v1

链接: http://arxiv.org/abs/2604.02213v1

摘要翻译

中文标题: 无穷维环面上的 Kronecker 流

摘要翻译: 本文研究无穷维环面上的 Kronecker 流。本研究的部分动机源于以下事实：许多哈密顿偏微分方程（PDE）与无穷格点系统允许存在不变环面（其维数可能为无穷），且在这些环面上的动力学是可线性化的。有限维 Kronecker 流的理论已十分完善：其动力学可归结为子环面上的非共振流，该性质等价于投影下的拓扑传递性、极小性以及唯一遍历性。我们证明，当环面维数为无穷时，上述性质依然成立的充要条件是频率的整（有限）线性组合构成一个自由阿贝尔群。进而，我们构造了一类轨道，其闭包局部同胚于一个球与 Cantor 集的直积，从而推广了 Sakbaev 与 Volovich 近期的结果。我们还证明了 Benjamin-Ono 方程容许此类解的存在。最后，我们证明了 Kronecker 流的分类问题与无挠可数阿贝尔群的分类问题之间的等价性。

3. 静息大爆炸奇点的局部形成

Localized formation of quiescent big bang singularities

作者: Andrés Franco-Grisales

类别: gr-qc, math.AP, math.DG

发布日期: 2026-04-02

arXiv ID: 2604.02104v1

链接: http://arxiv.org/abs/2604.02104v1

摘要翻译

中文标题: 静息大爆炸奇点的局部形成

摘要翻译: 我们证明了一项关于大爆炸局部形成的结果，该结果不要求初始数据邻近任何背景解。假设在开集 $U \\/subset \\mathbb{R}^3$ 上给定了爱因斯坦-非线性标量场方程组的初值。我们确立了初值所满足的一般性条件：若该条件在 $x \\in U$ 的充分大邻域内成立，则相应的最大整体双曲发展在 $x$ 的过去区域呈现局部静息大爆炸奇点，且伴随曲率爆破。我们通过引入一类新的类空超曲面叶层来实现局部化，该叶层由满足特定二阶微分方程的时间函数的水平集所刻画。此时间函数使得我们能够实现奇点的同步，并同时导出爱因斯坦方程的对称双曲表述。与以往关于大爆炸局部稳定性的结果相比，我们的新表述还具备两项关键优势。其一，该表述独立于具体的物质模型，因而有望用于证明标量场之外的其他物质模型中的大爆炸形成。其二，该表述使我们能够证明，我们的解在奇点处诱导出几何初值，从而完整刻画了趋于大爆炸的渐近行为。

4. 耗散边界条件下 Schwarzschild-AdS 中共形波动方程的有界性与衰减

Boundedness and decay for the conformal wave equation in Schwarzschild-AdS under dissipative boundary conditions

作者: Alex Tullini

类别: gr-qc, math.AP

发布日期: 2026-04-02

arXiv ID: 2604.02084v1

链接: http://arxiv.org/abs/2604.02084v1

摘要翻译

中文标题: 耗散边界条件下 Schwarzschild-AdS 中共形波动方程的有界性与衰减

摘要翻译: 我们在耗散边界条件下研究 4 维 Schwarzschild–Anti de Sitter 时空上的共形波动方程 $\square_g ψ+ \frac{2}{l^2} ψ= 0$。我们证明了 $ψ$ 的非退化能量的有界性及其以任意多项式速率 $(1+v)^{-n}$ 的衰减性，前提是控制（最高）$n$ 次的 $T$-交换能量。这一结果与 Dirichlet 边界条件下导出的逆对数衰减形成对比，并与耗散边界条件下纯 Anti-de Sitter 情形中所得结果一致。特别地，该衰减不受光子球处额外俘获效应的影响。

5. 某水文背景下的广义Riemann问题的半显式熵解

Semi-explicit entropic solution to a generalised Riemann problem in some hydrological context

作者: Brice Franke, Majid Lagnaoui, Catherine Rainer

类别: math.AP

发布日期: 2026-04-02

arXiv ID: 2604.01976v1

链接: http://arxiv.org/abs/2604.01976v1

摘要翻译

中文标题: 某水文背景下的广义Riemann问题的半显式熵解

摘要翻译: 本文研究具通量函数 $y\longmapsto G_{c,ρ}\left(y\right)=((1-ρ)c-y)\mathbb{1}_{\{y>c\}}-ρy\mathbb{1}_{\{y\leqslant c\}}$ 的一维标量守恒律在两种特定初值条件 $u(\cdot,0)=u_0$ 下的解。该方程为某一概念性水文模型的极限形式。当初值严格小于（分别大于）阈值水平 $c$ 时，方程退化为速度为 $p$（分别对应 $1$）的常速输运方程。我们的目标是精确刻画当初值跨越阈值 $c$ 时所发生的情形，该情形对应于Riemann问题的一种推广，并在此类情形下给出相应解的拟闭型表达式。

6. 视为Wasserstein梯度流的交叉扩散系统紧性方法综述

A review of compactness methods for cross-diffusion systems seen as Wasserstein gradient flows

作者: Mathias Dus, Ansgar Jüngel

类别: math.AP

发布日期: 2026-04-02

arXiv ID: 2604.01819v1

链接: http://arxiv.org/abs/2604.01819v1

摘要翻译

中文标题: 视为Wasserstein梯度流的交叉扩散系统紧性方法综述

摘要翻译: 本文提出了一种系统性方法，用于建立交叉扩散系统在适当能量泛函下的梯度流存在性。该方法基于利用 Jordan-Kinderlehrer-Otto 格式构造时间分段常数逼近。通过流交换技巧或五梯度不等式获得了逼近序列的紧性。上述方法在抛物型与双曲-抛物型 Busenberg-Travis 系统及其若干变体中得到了具体说明。本文综述了现有文献中的相关结果，并进一步探讨了模型的其他性质。

7. 散度方程至边界的解析性

Analyticity up to the boundary for the divergence equation

作者: Igor Kukavica, Qi Xu

类别: math.AP

发布日期: 2026-04-02

arXiv ID: 2604.01665v1

链接: http://arxiv.org/abs/2604.01665v1

摘要翻译

中文标题: 散度方程至边界的解析性

摘要翻译: 我们研究区域 $Ω$ 中散度方程 $\text{div}\, u = f$ 的解析正则性，其中在 $\partialΩ$ 上满足 $u=0$，$Ω$ 为任意有界解析区域且 $\int_Ω f\,dx=0$。若 $f$ 在 $Ω$ 的闭包上解析，则我们证明存在一个在 $Ω$ 的闭包上解析的解。

8. 全纯函数水平集的能量估计与 Calderón-Zygmund 理论的反例

Energy estimates for level sets of holomorphic functions and counterexamples to Calderón-Zygmund theory

作者: Yifei Pan, Guokuan Shao, Jianfei Wang, Jujie Wu

类别: math.CV, math.AP

发布日期: 2026-04-02

arXiv ID: 2604.01559v1

链接: http://arxiv.org/abs/2604.01559v1

摘要翻译

中文标题: 全纯函数水平集的能量估计与 Calderón-Zygmund 理论的反例

摘要翻译: 我们证明了 Calderón-Zygmund 理论中 $L^1$ 正则性的失效是一种普遍现象：$\ ext{C}^n$ 中的任意非常数全纯函数均构成泊松方程的反例。利用 Hironaka 奇点消解与 Łojasiewicz 梯度不等式，我们建立了尖锐的水平集估计，将调和分析与复结构的几何相联系，所得结果在代数几何中亦具有独立的研究价值。

9. 具有逆 Hölder 位势的薛定谔方程在 $L^p$ 及端点空间上的边值问题可解性

Solvability of boundary value problem for Schrödinger Equations with Reverse Hölder Potentials on $L^p$ and endpoint spaces

作者: Botian Xiao, Lin Tang

类别: math.AP

发布日期: 2026-04-02

arXiv ID: 2604.01544v1

链接: http://arxiv.org/abs/2604.01544v1

摘要翻译

中文标题: 具有逆 Hölder 位势的薛定谔方程在 $L^p$ 及端点空间上的边值问题可解性

摘要翻译: 本文利用层位势方法，研究椭圆型薛定谔型方程 $-\DIV(A(x)\nabla u(x,t))+V(x)u(x,t)=0$ 的 Neumann 边值问题与正则性边值问题的可解性。其中系数 $A(x)$ 有界可测、一致椭圆且与 $t$ 无关，位势 $V$ 属于逆 Hölder 类 $\mathcal{B}_q$，Neumann 边界条件为 $\partial_{ν_A}u(x,0)=f(x)\in H^p_{\mathcal{L}}(\rn)$，正则性边界条件为 $u(x,0)=g\in H^{1,p}_V(\rn)$。我们证明了当 $A$ 为满足 De Giorgi-Nash-Moser 估计的矩阵的微小 $L^\infty$ 扰动时，上述问题可解。此外，针对边界数据属于某类 Campanato 型空间的 Dirichlet 问题，我们还给出了相关双层位势的 Campanato 范数估计。

10. 关于具有 $L^p$ 熵产生的 Aviles-Giga 极限态

On Aviles-Giga limit states with $L^p$ entropy productions

作者: Xavier Lamy, Andrew Lorent, Guanying Peng

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.01439v1

链接: http://arxiv.org/abs/2604.01439v1

摘要翻译

中文标题: 关于具有 $L^p$ 熵产生的 Aviles-Giga 极限态

摘要翻译: Aviles-Giga 能量提供了映射序列，这些序列收敛于程函方程 \begin{align*} \mathrm{div}, m=0\text{ in }\mathcal D’(Ω),\quad |m|=1\text{ a.e. in }Ω,, \end{align*} 的弱解 $m\colonΩ\subset\mathbb R^2\to\mathbb R^2$，且其熵产生 $\mathrm{div}\,Φ(m)$ 为 $Ω$ 上的 Radon 测度，并受能量控制。此处，熵指所有满足如下条件的 $C^2$ 向量场 $Φ\colon\mathbb S^1\to\mathbb R^2$：对任意光滑解 $m_*$ 均有 $\mathrm{div}\,Φ(m_*)=0$。人们猜想，熵产生测度集中于 $m$ 的一维跳跃集上；若 $m$ 具有有界变差，该结论可由链式法则直接导出。特别地，若熵产生测度与 $L^p$ 函数一致，则该测度必为零：本文在 $p$ 不致过小且满足自然边界条件的条件下证明了这一结论。

11. 基于边界控制的ARZ交通模型中运动激波的指数镇定：显式增益与任意衰减率

Exponential Stabilization of Moving Shockwave in ARZ Traffic Model via Boundary Control: Explicit Gains and Arbitrary Decay Rate

作者: Mina Cao, Mamadou Diagne, Peipei Shang, Lei Yu

类别: math.OC, math.AP, math.DS

发布日期: 2026-04-01

arXiv ID: 2604.01382v1

链接: http://arxiv.org/abs/2604.01382v1

摘要翻译

中文标题: 基于边界控制的ARZ交通模型中运动激波的指数镇定：显式增益与任意衰减率

摘要翻译: 本文构造了边界反馈控制律，旨在 Aw-Rascle-Zhang (ARZ) 交通流模型中将交通拥堵稳定至预设的激波平衡态。我们通过基于激波位置的动坐标系，将对应的移动边界 $2 imes2$ 双曲系统（分别涵盖自由流与拥堵流态）转化为固定区域上的无激波 $4 imes4$ 增广系统。通过引入针对激波扰动的修正 Lyapunov 泛函，我们证明了：在所给反馈控制作用下，激波位置与系统状态的 $H^2$ 范数均可以实现镇定，且具有任意的指数衰减率。最后，数值模拟验证了上述镇定结果。

12. 对数薛定谔算子

Logarithmic Schrödinger operators

作者: Jorge J. Betancor, Estefanía Dalmasso, Juan C. Fariña, Pablo Quijano

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.01368v1

链接: http://arxiv.org/abs/2604.01368v1

摘要翻译

中文标题: 对数薛定谔算子

摘要翻译: 本文考虑 $\mathbb R^d$ 上具有非负位势 $V$ 且 $V\not\equiv 0$ 的薛定谔算子 $\mathcal L_V= -Δ+ V$。我们定义了对数薛定谔算子 $\log \mathcal L_V$，并证明了其主要性质。利用由 $\mathcal L_V$ 生成的算子半群 $\{T_t^V\}_{t>0}$，当 $V$ 满足指数 $q> \frac{d}{2}$ 的逆向 Hölder 不等式时，我们获得了 $\log \mathcal L_V$ 的点态表示。我们考虑适配于薛定谔情形的 Lipschitz 函数空间，借助与 $\mathcal L_V$ 相关联的分数阶积分求解初值问题

。

不相关论文（基本信息）

1. 低温费米气体的对易子估计

Commutator Estimates for Low-Temperature Fermi Gases

作者: Jacky J. Chong, Laurent Lafleche, Jinyeop Lee, Chiara Saffirio

类别: math-ph, math.AP, math.SP, quant-ph

发布日期: 2026-04-02

arXiv ID: 2604.02297v1

链接: http://arxiv.org/abs/2604.02297v1

摘要: We investigate the semiclassical regularity of thermal equilibria in the presence of a harmonic potential at low temperature; that is, we obtain the asymptotic behavior of the Schatten norms of commutators of the one-body operators associated with these equilibria and the position and momentum operators. We also obtain upper bounds in the magnetic field case for the Fock-Darwin Hamiltonian. Our estimates, in particular, allow us to observe several regimes depending on the joint behavior of the Planck constant, the temperature, and the strength of the magnetic field.

2. 攻防系统中大规模损耗的平均场博弈模型

A Mean-Field Game Model For Large-Scale Attrition in Attacker-Defender Systems

作者: Avetik Arakelyan, Tigran Bakaryan, Davit Alaverdyan, Naira Hovakimyan, Isaac Kaminer

类别: math.AP

发布日期: 2026-04-02

arXiv ID: 2604.02101v1

链接: http://arxiv.org/abs/2604.02101v1

摘要: This paper proposes a novel Mean-Field Game (MFG) framework for large-scale attacker-defender systems aimed at protecting one or multiple High-Value Units (HVUs). Motivated by classical agent-wise attrition models, we introduce a population-wise attrition mechanism formulated by statistical distance between populations, enabling a macroscopic description of weapon-based interactions between large populations. Leveraging this and Lions derivative on the space of probability measures, we derive the associated MFG system, which characterizes optimal strategies and the evolution of population distributions in attacker-defender interactions. We analyze the model by establishing upper and lower bounds on the defender density, ensuring physical consistency by preventing concentration and depletion. For numerical investigation, we develop a numerical scheme combining physics-informed neural networks with Sinkhorn method to solve attacker-defender MFG system. Simulations confirm the effectiveness of the framework and reveal key insights, including sensitivity to weapon strengths and population dispersion.

3. 反对称超薄铁磁双层中的最优斯格明子稳定性

Optimal skyrmion stability in antisymmetric ultrathin ferromagnetic bilayers

作者: Anne Bernand-Mantel, Valeriy V. Slastikov, Cyrill B. Muratov

类别: cond-mat.mes-hall, math-ph, math.AP, nlin.PS

发布日期: 2026-04-02

arXiv ID: 2604.02070v1

链接: http://arxiv.org/abs/2604.02070v1

摘要: We demonstrate the stray-field-mediated skyrmion stabilizing capabilities of ultrathin exchange-decoupled antisymmetric ferromagnetic bilayers based on conventional transition metal materials. Using an asymptotically exact micromagnetic model valid in the ultrathin film limit, we show that the antisymmetric tailoring of the bilayer allows the Dzyaloshinskii-Moriya interaction and the dipolar interaction to act synergistically to stabilize skyrmions, in contrast to the monolayer case, in which these energies compete. To obtain optimal stability of these skyrmions against collapse and bursting – the two fundamental processes determining skyrmion lifetime, we carry out an asymptotic analysis of the saddle point solution that separates the skyrmion from the demagnetized state. The result is an optimal stability line for compact skyrmions in the non-dimensional parameter space of the effective Dzyaloshinskii-Moriya interaction strength and the effective film thickness. Our predictions are confirmed by extensive micromagnetic simulations of antisymmetric bilayers, using magnetic parameters of the conventional Pt/Co/AlO$_x$ systems. Our results provide a new pathway for experimental observations of 10 nm radius zero-field skyrmions with lifetimes compatible with information technology applications.

4. 曲线缩短流讲义

Lectures on curve shortening flow

作者: Robert Haslhofer

类别: math.AP, math.DG

发布日期: 2026-04-02

arXiv ID: 2604.01981v1

链接: http://arxiv.org/abs/2604.01981v1

摘要: The curve shortening flow is a geometric heat equation for curves and provides an accessible setting to illustrate many important concepts from nonlinear partial differential equations, including maximum principle estimates, monotonicity formulas, Harnack inequalities and blowup analysis. All these techniques will be combined to give an exposition of Huisken’s proof of Grayson’s beautiful theorem that the curve shortening flow shrinks any closed embedded curve in the plane to a round point.

5. 等离子体物理中自由边界问题的尖锐谱估计

Sharp spectral estimates for free boundary problems arising in plasma physics

作者: Daniele Bartolucci, Aleks Jevnikar, Juncheng Wei, Ruijun Wu

类别: math.AP

发布日期: 2026-04-02

arXiv ID: 2604.01895v1

链接: http://arxiv.org/abs/2604.01895v1

摘要: We derive a sharp spectral estimate for a superlinear free boundary problem arising in plasma physics. The semilinear equation is coupled with a constraint, which forces the analysis of a non-local eigenvalue equation. Consequently the corresponding first eigenvalue, say $σ_1$, is not a standard one and it is shown that it cannot satisfy a general isoperimetric property of Faber-Krahn type. This motivates a careful analysis of the problem on balls in any dimension $N\geq 2$, where we prove that in fact $σ_1$ is always positive. The implications about the uniqueness problem for the Emden equation are also discussed.

6. 组织结构的多相交叉扩散模型：建模、分析与数值计算

Multiphase cross-diffusion models for tissue structures: modeling, analysis, numerics

作者: Ansgar Jüngel, Cordula Reisch, Sara Xhahysa

类别: math.AP

发布日期: 2026-04-02

arXiv ID: 2604.01827v1

链接: http://arxiv.org/abs/2604.01827v1

摘要: Volume-filling cross-diffusion equations for the components of a tissue structure are formally derived from mass conservation laws and force balances for the interphase pressures and viscous drag forces in a multiphase approach. The equations include Maxwell-Stefan, tumor-growth, thin-film solar cell models as well as novel volume-filling population systems. The Boltzmann and Rao entropy structures are explored. If the drag coefficients are all equal to one, the global-in-time existence of bounded weak solutions, their long-time behavior, and the weak-strong uniqueness of solutions to a regularized system are proved using entropy methods. In the general case, the resulting diffusion matrix is positively stable, ensuring local-in-time existence of solutions. Global-in-time existence of weak solutions is proved if the drag coefficients are sufficiently close to each other. This restriction is explained by the fact that the pressure forces are of degenerate type, while the drag forces are nondegenerate in the volume fractions. Numerical simulations are presented in one space dimension to illustrate the solution behavior beyond the entropy regime.

7. 小应变下营养物质存在时的形态弹性生长

Morphoelastic Growth in the Presence of Nutrients at Small Strains

作者: Helmut Abels, Julian Blawid, Georg Dolzmann

类别: math.AP

发布日期: 2026-04-02

arXiv ID: 2604.01812v1

链接: http://arxiv.org/abs/2604.01812v1

摘要: A model for morphoelastic growth, that is, growth influenced by elastic stress, driven by the absorption of nutrients is considered. The model features a multiplicative decomposition of the deformation gradient into an elastic contribution and a growth tensor. While the evolution of the system is governed by an ordinary differential equation for the growth tensor on a suitable Banach space, which depends on the elastic stresses and the concentration of a nutrient field, the total deformation is given by the solution of a quasi-static equilibrium equation arising from the formal Euler-Lagrange equations of a hyperelastic variational integral. The nutrient concentration is determined by a linear elliptic reaction-diffusion equation which is formulated in Lagrangian coordinates and whose coefficients depend on the growth tensor as well as the deformation gradient accounting for the change of material properties due to elastic deformation and growth. Existence and uniqueness of solutions of this fully coupled system of differential equations is proven via a fixed-point argument.

8. 局部有限图上对数p-拉普拉斯系统的基态解

Ground state solutions for logarithmic p-Laplacian systems on locally finite graphs

作者: Wenzheng Hu

类别: math.AP

发布日期: 2026-04-02

arXiv ID: 2604.01807v1

链接: http://arxiv.org/abs/2604.01807v1

摘要: In this paper,we study the discrete logarithmic p-Laplacian system

on locally finite graphs $G=(V,E)$, where $Δ_p u(x)=div(|\nabla u(x)|^{p-2}\nabla u(x))$ is the discrete p-Laplacian on graphs. Firstly, under certain assumptions on the potential $a(x)$ and $b(x)$, we establish two Sobolev compact embedding theorems in the case when different assumptions on $a(x)$ and $b(x)$, which leads to two different energy functionals, the one is not well-defined, while the other one is $C^1$ continuous. In the former case, we prove that the system admits a ground state solution by Nehari manifold method. In the latter case, we prove that the system admits a mountain-pass solution. Finally, we establish convergence results by analyzing the concentration behavior of ground state solutions.

9. 关于 Schur 拓扑的 Avscon

Avscon the Schur topology

作者: Andreas Buchinger, Marcus Waurick

类别: math.AP, math.FA

发布日期: 2026-04-02

arXiv ID: 2604.01803v1

链接: http://arxiv.org/abs/2604.01803v1

摘要: The aim of the course is to lead to an understanding of homogenisation processes in an operator-theoretic sense. In fact, using solely operator-theoretic means not referring to the particular form of the coefficients, we will identify an operator topology on the level of coefficients that will fully capture the convergence involved in the context of homogenisation. One upshot of this perspective will be that we will obtain homogenisation results for time-dependent partial differential equations (almost) for free.

10. 尺度临界曲线扩散流

Scale-critical curve diffusion flows

作者: Tatsuya Miura, Glen Wheeler

类别: math.AP, math.DG

发布日期: 2026-04-02

arXiv ID: 2604.01716v1

链接: http://arxiv.org/abs/2604.01716v1

摘要: We introduce and study a one-parameter family of curve diffusion flows with a scale-critical cubic curvature term for closed immersed planar curves. We first classify all closed stationary solutions, showing that they are precisely circles or a unique family of ``super-lemniscates’’. We then analyse the dynamical stability of homothetic circles. Under a sharp spectral condition, we establish, by purely variational methods, that any small perturbation of an $ω$-fold circle monotonically approaches the unit $ω$-circle after rescaling, translation, and reparametrisation. As a corollary, we determine the sharp ranges of the parameter for the stability of an embedded circle, and of all $ω$-circles. We also uncover a striking arithmetic structure in the stability landscape, where the stability of $ω$-circles depends non-monotonically on $ω$.

11. 含自旋扩散的Landau-Lifshitz-Bloch方程：整体强解与有限元逼近

The Landau–Lifshitz–Bloch equation with spin diffusion: Global strong solution and finite element approximation

作者: Agus L. Soenjaya

类别: math.NA, math.AP

发布日期: 2026-04-02

arXiv ID: 2604.01668v1

链接: http://arxiv.org/abs/2604.01668v1

摘要: The spin-diffusion Landau–Lifshitz–Bloch (SDLLB) system is a nonlinearly coupled system of quasilinear vector-valued PDEs which models the interaction between spin-polarised currents and magnetisation at high temperatures. The aim of this paper is twofold. Firstly, assuming the initial data is sufficiently small, we show the existence of a unique global strong solution to the SDLLB equation in a bounded domain $Ω\subset \mathbb{R}^d$, where $d\leq 3$, thus ensuring well-posedness of the model. Secondly, we propose a decoupled linearised fully-discrete finite element scheme to solve the problem. Despite the strong nonlinearity of the system, the proposed scheme only requires the solution of two completely decoupled linear systems per time-step. Assuming adequate regularity of the exact solution and a certain time-step constraint, we rigorously show that the numerical scheme converges at an optimal rate. Several numerical experiments corroborate our theoretical results.

12. 果蝇卵室细胞迁移边界运动：相场与趋化因子耦合模型

Cell Migration Boundary Motion in Drosophila Egg Chambers: A Combined Phase Field and Chemoattractant Model

作者: Naghmeh Akhavan, Alexander George, Michelle Starz-Gaiano, Bradford E. Peercy

类别: math.AP, q-bio.CB

发布日期: 2026-04-01

arXiv ID: 2604.01357v1

链接: http://arxiv.org/abs/2604.01357v1

摘要: In the Drosophila melanogaster egg chamber, the collective migration of border cells toward the oocyte is guided by spatial gradients of chemoattractants. While cellular responses to these cues are well characterized, the spatial distribution of chemoattractant within the tissue remains difficult to measure experimentally due to imaging limitations and extracellular complexity. In this study, we develop a spatially resolved mathematical framework to model local chemoattractant concentrations during border cell migration. We use a phase-field approach to represent the egg chamber geometry and define a diffusion-reaction system with spatially heterogeneous diffusivity that accounts for confinement by cellular domains. This framework allows chemoattractant diffusion to be restricted to extracellular space while remaining excluded from the interiors of nurse cells, the border cell cluster, and the oocyte, similar to what we observe in vivo. We simulate secretion from the oocyte and degradation throughout the domain, showing how geometry shapes the distribution of signaling molecules. We further couple this chemical field to a mechanical model of cluster migration that includes a tangential interface migration (TIM) force, allowing the cluster to respond to both chemoattractant gradients and cell-cell contact. Our results show that signal localization and tissue geometry jointly influence directional persistence and the speed of migration. Notably, geometric bottlenecks and intersections can flatten local gradients and slow migration, consistent with experimental observations. This modeling framework offers a tool to investigate how biophysical constraints shape signaling environments and guide collective cell movement in vivo.

13. 退化奇异抛物方程的脉冲控制有限时间镇定

Finite-time stabilization via impulse control of degenerate singular parabolic equations

作者: Walid Zouhair, Ghita El Guermai, Ilham Ouelddris

类别: math.AP, math.OC

发布日期: 2026-04-01

arXiv ID: 2604.01311v1

链接: http://arxiv.org/abs/2604.01311v1

摘要: This paper examines the impulse controllability of degenerate singular parabolic equations through a modern framework focused on finite-time stabilization. Furthermore, we provide an explicit estimate for the exponential decay of the solution. The proof of our main result combines a logarithmic convexity estimate with specific spectral properties. Finally, we establish the existence and uniqueness of the minimal norm impulse control associated with the system.