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1. 关于一维低粘性可压缩纳维-斯托克斯方程的全局正则性与精确衰减率研究

Global regularity and sharp decay rates to the 1D hypo-viscous compressible Navier-Stokes equations

作者: Chen Liang, Zhaonan Luo, Zhaoyang Yin

类别: math.AP

发布日期: 2026-03-16

arXiv ID: 2603.14914v1

摘要翻译: 本文研究一维等熵次粘性可压缩Navier-Stokes方程的全局正则性与最优衰减率问题。首先，我们证明了在稳定平衡态附近具有小初值情形的全局稳定性，特别地在Sobolev空间 $H^β$（其中 $\frac{1}{2}<β<1$）中建立了全局临界正则性。进而，通过bootstrap论证、傅里叶分片方法和能量方法，在附加低频率小性假设下得到了最优时间衰减率。我们发现 $L^2$ 能量具有自封闭性质，这一性质启发了我们获得高正则性初值下全局大解的存在性。通过纯能量方法，在 $\frac{1}{2}\leβ<\frac{3}{4}$ 范围内也推导出了最优时间衰减率。我们发现一个有趣现象：即使初值不具有 $L^2$ 小性， $\|(a,u)\|_{L^2}$ 仍会衰减。值得注意的是，在 $\frac{1}{2}\leβ<\frac{3}{4}$ 的情形中，低频率小性假设已被消除。

Comment: 无

链接: http://arxiv.org/abs/2603.14914v1

1. 研究对象和背景

本文研究一维（1D）亚粘性（hypo-viscous）可压缩Navier-Stokes方程（CNS equations）的全局正则性和最优时间衰减率。控制方程为：

$$ \begin{cases} \partial_t \rho + \partial_x (\rho u) = 0, \\ \partial_t (\rho u) + \mu (-\Delta)^\beta u + \partial_x (\rho u^2 + P(\rho)) = 0, \end{cases} $$

其中 $\rho$ 表示密度，$u$ 表示速度，压力 $P(\rho) = \rho^\gamma$（$\gamma > 1$），$\mu > 0$ 为粘性系数，$(-\Delta)^\beta$ 为分数阶拉普拉斯算子（$0 < \beta < 1$），其傅里叶符号为 $|\xi|^{2\beta}$。通过变换 $\rho = a + 1$，方程可改写为关于 $(a, u)$ 的变形式。经典CNS方程（$\beta=1$）在多维情形下已有丰富成果，如Matsumura和Nishida [31,32] 证明了小初值全局存在性并推导衰减率，Xin [46] 证明光滑解不存在性，Lions [29,30] 获得有限能量弱解。一维经典情况（$\beta=1$）也有研究，如Mellet-Vasseur [34] 在 $H^1$ 中证明全局强解，Li [23] 处理真空情形，Chen等 [3] 推广到粗糙初值，Chen等 [5] 利用谱分析和Fourier分裂得到最优衰减率。对于亚粘性情况（$0 < \beta < 1$），Li等 [27] 证明弱解无穷多，Wang-Zhang [44,45] 研究三维 $L^2$ 衰减但需高正则性初值。Liang-Luo-Yin [28] 近期处理二维情况在 $\frac{1}{2} \le \beta < 1$ 的全局正则性和衰减率。然而，一维亚粘性CNS方程在 $\frac{1}{2} \le \beta < 1$ 的全局正则性和最优衰减率尚未被研究，本文旨在填补此空白，并揭示一维情形的独特困难和方法创新。

2. 主要定理或主要结果

本文四个核心定理如下：

定理1.1（小解的全局正则性）：设 $\frac{1}{2} < \beta < 1$ 或 $\beta = 1$，若 $\beta < 1$ 则 $s \ge \beta$，若 $\beta = 1$ 则 $s > \frac{1}{2}$。初值 $(a_0, u_0) \in H^s$，存在小常数 $\delta > 0$，使得当 $\|(a_0, u_0)\|_{H^s} \le \delta$ 时，系统存在唯一全局强解 $(a, u) \in C([0, \infty), H^s)$。且能量泛函 $E_0(t) = \int_{\mathbb{R}} \left(2G(\rho) + \rho u^2\right) dx + \|(\sqrt{\gamma} a, u)\|_{\dot{H}^s}^2$ 与耗散 $D_0(t) = k\gamma \|\Lambda^\beta a\|_{\dot{H}^{s-2\beta+1}}^2 + \|\Lambda^\beta u\|_{\dot{H}^{s-2\beta+1}}^2$ 满足 $\frac{d}{dt}(E_0 + D_0) \le 0$，其中 $k > 0$ 为小常数。

定理1.2（小解的最优衰减率）：在定理1.1条件下，额外假设 $(a_0, u_0) \in \dot{B}^{-1/2}_{2,\infty}$ 且 $\|(a_0, u_0)\|_{H^s} + \|(a_0, u_0)\|_{\dot{B}^{-1/2}_{2,\infty}} \le \delta$。则存在 $C > 0$，对任意 $t > 0$ 及 $0 \le s_1 \le s$，有

$$ \|\Lambda^{s_1}(a, u)\|_{L^2} \le C (1+t)^{-1 + 2\beta s_1}. $$

特别地，若 $\|(a_0, u_0)\|_{L^1} \le \delta$（因 $L^1 \hookrightarrow \dot{B}^{-1/2}_{2,\infty}$），结果仍成立。

定理1.3（大解的全局正则性）：设 $\frac{1}{2} < \beta < \frac{3}{4}$ 或 $\beta = 1$，若 $\beta < 1$ 则 $s \ge 1+\beta$，若 $\beta = 1$ 则 $s > \frac{3}{2}$。初值 $(a_0, u_0) \in H^s$，存在常数 $M > 0$ 及小 $\delta(M) > 0$，使得当 $\|(a_0, u_0)\|_{L^2} \le M$ 且 $\|\partial_x (a_0, u_0)\|_{H^{s-1}} \le \delta(M)$ 时，系统存在唯一全局强解 $(a, u) \in C([0, \infty), H^s)$。且高阶能量 $E_1(t) = E_0(t) + E_s(t) + \langle u, 2k\Lambda^{-2+2\beta} a_x \rangle_{\dot{H}^{s-2\beta+1}}$ 与耗散 $D_1(t) = k\gamma \|\Lambda^{1+\beta} a\|_{\dot{H}^{s-2\beta+1}}^2 + \|\Lambda^{1+\beta} u\|_{\dot{H}^{s-2\beta+1}}^2$ 满足 $\frac{d}{dt}(E_1 + D_1) \le 0$。

定理1.4（大解的最优衰减率）：在定理1.3条件下，额外假设 $(a_0, u_0) \in \dot{B}^{-1/2}_{2,\infty}$。则存在 $C > 0$，对任意 $t > 0$ 及 $0 \le s_1 \le s$，有

$$ \|\Lambda^{s_1}(a, u)\|_{L^2} \le C (1+t)^{-1 + 2\beta s_1}. $$

该衰减率与定理1.2一致，且无需低频小性假设。注释指出，当 $\frac{1}{2} \le \beta < \frac{3}{4}$ 时，即使 $\|(a_0, u_0)\|_{L^2}$ 不具小性，$\|(a, u)\|_{L^2}$ 仍衰减。

3. 研究方法、关键技术和核心工具

研究方法与技术工具主要包括：

  • 能量方法：构造分阶能量泛函（如 $E_0, D_0$ 和 $E_1, D_1$），通过将方程乘以适当测试函数（如 $G'(\rho)$、$u$、$\Lambda^\beta a$、$\Lambda^\beta u$ 等）并积分，获得能量不等式。关键步骤包括分部积分和利用Sobolev嵌入控制非线性项。

  • Bootstrap 论证：在衰减率证明中，先假设衰减先验估计（如命题3.1中的 (3.10) 式），然后通过一致估计验证假设，迭代直至收敛。这用于处理一维情形下直接Fourier分裂法失效的问题（因衰减可能较慢）。

  • Fourier 分裂方法：结合Littlewood-Paley分解，将解频率局部化，分别估计高频和低频贡献。通过时间加权能量估计和迭代，导出最优衰减率。高频部分直接耗散，低频部分需精细分析。

  • 频率局部化与Besov空间理论：使用齐次Littlewood-Paley算子 $\Delta_j$ 分解，在Besov空间 $\dot{B}^s_{p,r}$（特别是临界空间 $H^\beta$ 和负指数 $\dot{B}^{-1/2}_{2,\infty}$）中刻画解的正则性。利用嵌入关系（如 $L^1 \hookrightarrow \dot{B}^{-1/2}_{2,\infty}$）放宽初值假设。

  • Sobolev 插值不等式：如引理2.2，用于估计中间范数，在能量估计中连接不同阶导数的 $L^p$ 范数。

  • 交换子与乘积估计：引理2.3至2.5提供分数阶交换子和乘积的 $L^p$ 估计，对处理非线性项（如 $au$、$u u_x$ 等）至关重要，确保能量估计的闭包。

  • 自封闭能量观察：对于大解（$\frac{1}{2} < \beta < \frac{3}{4}$），发现 $L^2$ 能量自封闭（方程(1.7)），即 $\frac{d}{dt} \int \left( \frac{1}{2} G(\rho) + \frac{1}{2} \rho u^2 \right) dx + \|\Lambda^\beta u\|_{L^2}^2 = 0$，这允许从 $L^2$ 控制高阶正则性，从而证明大数据全局解存在。

  • Lyapunov 型不等式与迭代论证：对大解，从能量估计导出Lyapunov不等式（如 $\frac{d}{dt} E_1 + C E_{1+\beta} \le 0$），结合Fourier分裂的迭代步骤，逐步改善衰减指数，最终达到最优率且移除低频小性假设。

核心工具还包括分数阶拉普拉斯算子的傅里叶定义、Gagliardo-Nirenberg不等式及各种嵌入定理。

4. 与之前工作的比较

本文与现有文献对比如下：

  • 对于经典CNS方程（$\beta=1$），多维结果成熟：Matsumura-Nishida [31,32] 在 $H^s \cap W^{s,1}$（$s > 2 + d/2$）中得小解全局存在和多项式衰减；Xin [46] 证明非平凡光滑解不存在；Lions [29,30] 得有限能量弱解。一维情形，Mellet-Vasseur [34] 在 $H^1$ 得全局强解；Li [23] 处理真空和热传导；Li-Xin [25,26] 研究熵有界解；Chen等 [3] 推广到粗糙初值。衰减率方面，Li-Liang [24] 得 $L^p$ 衰减但无显式率；Chen等 [5] 首次用谱分析和Fourier分裂得最优衰减率。相比，本文首次处理一维亚粘性（$\frac{1}{2} \le \beta < 1$），并揭示临界正则性空间为 $H^\beta$（$\beta \in (\frac{1}{2}, 1)$），而[5]需 $H^2$。

  • 对于亚粘性CNS方程（$0 < \beta < 1$），弱解存在由Li等 [27] 证明；Wang-Zhang [44] 研究三维 $L^2$ 衰减但初值需 $H^4 \cap L^1$；[45] 推广到Navier-Stokes-Poisson系统。Liang-Luo-Yin [28] 近期处理二维在 $\frac{1}{2} \le \beta < 1$ 的全局正则性和衰减率，方法依赖谱分析。本文创新在于：一维情形下，对于小解，通过bootstrap和能量方法避免复杂谱分析，得临界空间 $H^\beta$ 正则性；对于大解，利用 $L^2$ 能量自封闭性质，证明在 $\frac{1}{2} < \beta < \frac{3}{4}$ 时全局强解存在，且衰减率最优。特别地，在相同 $\beta$ 范围，移除之前工作（如[48]在多维Besov空间）依赖的低频小性假设，即使 $\|(a_0, u_0)\|_{L^2}$ 大仍得衰减。

  • 技术比较：经典衰减率证明常用Fourier分裂（如Ponce [36] 或Guo-Wang [12] 方法），但一维因衰减慢需额外技巧。本文小解衰减率证明采用bootstrap结合时间加权积分（步骤1），而大解通过Lyapunov不等式启动迭代（步骤2），最终统一衰减指数。此外，自封闭能量观察是处理大数据的关键，此前仅在高维或特殊系统见类似现象。

  • 未解决问题：与[28]类似，$\beta = 1$ 的临界正则性（$H^{1/2}$）仍开放，作者猜想对应Besov空间 $B^{1/2}_{2,1}$；大解在 $\frac{3}{4} \le \beta < 1$ 的最优衰减率未覆盖。这为未来工作提供方向。

2. 局域增强耗散：一种双协同性方法

Localized Enhanced Dissipation: A Hypocoercivity Approach

作者: Siming He

类别: math.AP

发布日期: 2026-03-15

arXiv ID: 2603.14657v1

摘要翻译: 本文研究含临界点剪切流中的被动标量解。通过构造精细的超压缩性泛函，我们建立了沿流向的增强耗散估计。

Comment: 无

链接: http://arxiv.org/abs/2603.14657v1

1. 研究对象和背景

本文研究二维环面$\mathbb{T} \times \mathbb{R}$上被动标量方程$\partial_t F + U(y)\partial_x F = \nu \Delta F$的解，其中$U(y)$是剪切流，$\nu \in (0,1]$是小扩散系数。核心现象是增强耗散：流动的剪切效应诱导产生小尺度，从而显著加速耗散，使某些模式的衰减快于纯热方程的时间尺度$O(\nu^{-1})$。对于严格单调剪切流（如Couette流$(y,0)$），已知$k \neq 0$的Fourier模式以速率$O(\nu^{1/3}|k|^3)$衰减。然而，当$U(y)$具有非退化临界点（如$U(y)=\sin y$）时，由于临界点附近速度梯度$U'$消失，剪切效应被抑制，已知最优全局增强耗散速率仅为$O(\nu^{1/2}|k|^2)$。本文关注更精细的问题：若初始数据远离临界点，是否能在局部（沿流线）获得更快的衰减率？即“沿着流线”的增强耗散估计，期望在单调区域恢复$O(\nu^{1/3})$速率，在临界层为$O(\nu^{1/2})$，两者之间存在平滑过渡。

2. 主要定理或主要结果

考虑方程$\partial_t f + iU f = \nu \partial_y^2 f$（通过关注特定Fourier模式$f(t,y) = \hat{F}_1(t,y)e^{\nu |k|^2 t}$将原问题约化至此）。定义修正剪切强度$B(y) = \max\{|U'(y)|, \nu^{1/4}\}$，空间-时间权重$W(t,y) = \exp\left(\varsigma \nu^{1/3} B^{2/3}(y) \max\{\nu^{-1/3}B^{-2/3}(y), \min\{t, \nu^{-1/2}\}\}\right)$（$\varsigma \in (0,1)$），以及参数函数$\varphi(t,y) = \min\{1, \nu^{1/3}B^{2/3}(y)t\}$。构造Hypocoercivity泛函

$$ \Phi[f] := \|f W\|_{L^2}^2 + \frac{\sqrt{\beta}}{4\nu^{1/3}} \|\varphi B^{-1/3} \partial_y f W\|_{L^2}^2 + \beta \nu^{1/3} \operatorname{Re}\langle \varphi^2 B^{-4/3} iU' f, \partial_y f W^2 \rangle + \frac{4\beta^3}{2} \|\varphi^{3/2} U' f B^{-1} W\|_{L^2}^2, $$

其中$\beta \in (0,1]$待选。主要定理断言：若$U$仅有有限个非退化临界点，则存在仅依赖于$U$的$\delta \in (0,1)$，使得

$$ \Phi[f(t)] \leq e^{3} \|f_0\|_{L^2}^2 e^{-\delta \nu^{1/2} t}, \quad \forall t \geq 0. $$

由于$W(t,y)$依赖于$y$，该估计隐含了局部衰减率：在$|U'(y)| \gg \nu^{1/4}$的区域（远离临界点），实际衰减率为$O\left(\nu^{1/3} \max\{|U'(y)|, \nu^{1/4}\}^{2/3}\right)$；特别地，在严格单调区域$|U'| \sim 1$，它恢复了单调流的最优速率$O(\nu^{1/3})$。

3. 研究方法、关键技术和核心工具

研究基于能量方法（Hypocoercivity），核心步骤与技术如下：

  1. 方程约化：通过关注单一Fourier模式$f$，将原输运-扩散方程转为hypoelliptic型，便于分析。
  2. 局部剪切度量：引入$B(y) = \max\{|U'(y)|, \nu^{1/4}\}$，在临界层$B \sim \nu^{1/4}$反映扩散主导的基准剪切，在单调区域$B \sim |U'| \sim 1$反映流动剪切主导。
  3. 空间-时间权重设计：权重$W(t,y)$的关键指数部分为$\varsigma \nu^{1/3} B^{2/3}(y) \max\{\nu^{-1/3}B^{-2/3}(y), \min\{t, \nu^{-1/2}\}\}$。其设计确保：
    • 在临界层（$B \sim \nu^{1/4}$）且$t \sim \nu^{-1/2}$时，$W \sim e^{\varsigma \nu^{1/2} t}$，给出$O(\nu^{1/2})$衰减。
    • 在单调区域（$B \sim 1$）且$t \sim \nu^{-1}$时，$W \sim e^{\varsigma \nu^{1/3} t}$，给出$O(\nu^{1/3})$衰减。
    • 在过渡区域，通过$\max\{...\}$实现两种行为的连续插值。
  4. Hypocoercivity泛函构造：$\Phi[f]$结合了$f$及其导数的加权$L^2$范数，并包含交叉项$\operatorname{Re}\langle iU' f, \partial_y f \rangle$以耦合不同方向。参数$\alpha = \frac{\sqrt{\beta}}{4\nu^{1/3}}$，$\gamma = \frac{4\beta^3}{2}$由$\beta$决定，$\varphi$用于区分时间尺度。
  5. 能量估计：计算$\frac{d}{dt}\Phi[f(t)]$，分解为四项$T_{L^2}, T_\alpha, T_\beta, T_\gamma$。对每项进行细致估计：
    • 使用Young不等式、Hölder不等式、分部积分。
    • 处理权重$B$、$\varphi$、$W$及其导数（如$B' \sim U''$）带来的复杂性。
    • 关键技巧：利用$B \geq \nu^{1/4}$调整权重幂次，将不同区域的项统一控制。
  6. 谱不等式：依赖Lemma A.1的谱不等式$\nu^{1/3} \|f B^{1/3} \varphi W\|_{L^2}^2 \leq \nu \|\partial_y f W\|_{L^2}^2 + C \nu^{1/3} \|U' B^{-2/3} f \varphi W\|_{L^2}^2$，用于在估计中置换不同范数。
  7. 参数选择：选取足够小的$\beta$（依赖于$\|U''\|_{L^\infty}$）和$\varseta = \beta^2$，使得合并所有项后$\frac{d}{dt}\Phi[f] \leq -\frac{\delta}{2} \nu^{1/2} \Phi[f]$对$t \geq \nu^{-1/2}$成立；对$t \leq \nu^{-1/2}$直接由泛函定义控制。从而得到全局估计。

4. 与之前工作的比较

关于有临界点剪切流的增强耗散，经典工作[3]首次将kinetic theory中的hypocoercivity技术引入，针对$U(y)=\sin y$等流，得到了全局衰减率$O(\nu^{1/2}|k|^2)$（即方程(1.4)）。后续研究[17,43,47]优化了常数，但速率未变。另一些方法包括resolvent分析[18,23,31,44,46]、Hörmander hypoelliptic方法[1]、随机方法[26]等。 本文的主要区别与贡献在于：

  1. 局部性视角：不满足于全局统一速率，而是精准刻画衰减率如何随空间位置$y$变化。明确指出在远离临界点的单调区域，衰减率可提升至$O(\nu^{1/3})$，与严格单调流的结果一致；在临界层保持$O(\nu^{1/2})$；中间过渡层实现平滑过渡。这回答了初始数据分布位置对衰减速率的影响。
  2. 权重构造创新：设计的权重$W(t,y)$显式依赖于局部剪切强度$B(y)$，是空间-时间权，而非仅时间权重。其指数形式$\nu^{1/3}B^{2/3} \max\{\nu^{-1/3}B^{-2/3}, \min\{t,\nu^{-1/2}\}\}$巧妙地将临界层与单调区域的不同行为结合。
  3. 泛函精细化：Hypocoercivity泛函中引入$\varphi$和$B$的分数幂（如$B^{-1/3}, B^{-4/3}$），以匹配不同$y$处的有效耗散率，这是对经典泛函（仅含常数系数）的显著推广。
  4. 恢复最优局部速率：严格证明了在$|U'(y)| \sim 1$的区域，估计中的局部衰减因子为$e^{-\delta \nu^{1/3} B^{2/3} t} \sim e^{-\delta \nu^{1/3} t}$，即恢复了单调流已知的最优速率$O(\nu^{1/3})$。论文还推测在过渡层该估计也是sharp的。
  5. 方法论意义：展示了hypocoercivity框架如何通过精心设计的空间依赖权重来处理具有局部几何变化（此处为临界点）的输运-扩散系统，为其他类似问题（如具有多个平衡点的系统）提供了可借鉴的技术路径。

3. 无限圆柱内二维不可压缩流的涡量限制

Vorticity confinement for 2D incompressible flows in an infinite cylinder

作者: Paolo Buttà, Guido Cavallaro

类别: math.AP, math-ph

发布日期: 2026-03-14

arXiv ID: 2603.13926v1

摘要翻译: 我们研究无限长圆柱中二维不可压缩流的涡旋约束问题。对于具有非负且紧支集初始涡度的纳维-斯托克斯解，我们导出了定量衰减估计：在距离初始支集增长速率分别如$\sqrt{t\log^αt}$（其中$α>1$）或如$t^β$（其中$β>1/2$）的区域之外，涡度质量分别以超多项式速率或拉伸指数速率衰减至任意小。分析结合了迭代格式与毕奥-萨伐尔核的反对称性质。对于欧拉情形，通过将此方法与文献[《数学物理通讯》, 367, 1077-1093, 2019]中的一阶矩估计相结合，我们恢复了该文献的约束界并稍作改进：涡度支集的直径增长至多如$(t\log t)^{1/3}$，而非$t^{1/3}\log^2 t$。

Comment: 无

链接: http://arxiv.org/abs/2603.13926v1

1. 研究对象和背景

本论文研究二维不可压缩流体在无限圆柱（等价于带周期性边界条件的无限长条 $S := \{(x_1,x_2) \in \mathbb{R}^2: x_1 \in \mathbb{R}, x_2 \in [0,2\pi]\}$）中的涡旋（vorticity）长期限制问题。作者考虑初始涡度 $\omega_0$ 非负、有界且紧支集的情形，重点考察涡度质量在空间中如何随时间扩散，特别是支撑集（$\Lambda(t) = \text{supp}\,\omega(\cdot,t)$）直径的增长速率。这一问题的背景可追溯至涡旋动力学的经典理论：在整平面上，已知支撑直径的增长受限于 $(t \log t)^{1/4}$（Córdoba等人，基于惯性矩守恒）；然而在无限圆柱几何中，由于周期性边界条件，惯性矩不再守恒，但Biot-Savart核具有关键的对偶性（反对称性），为获得非平凡限制提供了可能。此前，Buttà等人（[6]）针对Euler方程证明了支撑直径最多以 $t^{1/3} \log^2 t$ 增长，而Navier-Stokes情形（有粘性）在整平面上已有超多项式和拉伸指数衰减结果（如[11]）。本论文旨在将整平面上的衰减思想推广至圆柱几何，并改进Euler情况下的界。

2. 主要定理或主要结果

论文的主要结果分为Navier-Stokes方程和Euler方程两部分。对于Navier-Stokes方程（定理2.1）：设初始涡度 $\omega_0$ 非负、有界且紧支集于 $\Lambda_0 = \{y \in S: |y_1| \leq \bar{C}\}$。定义 $m_t(h) := \int_{|y_1|>h} \omega(y,t) \, dy$ 为距离初始支撑水平距离超过 $h$ 的涡度质量。则：(a) 对任意 $\ell>0$ 和 $\alpha>1$，存在 $T_{\alpha,\ell}>0$ 使得对所有 $t \geq T_{\alpha,\ell}$，有 $m_t(\sqrt{t \log^\alpha t}) \leq t^{-\ell}$；(b) 对任意 $\beta>1/2$ 和 $0<\delta<2\beta-1$，存在 $T_{\beta,\delta}>0$ 使得对所有 $t \geq T_{\beta,\delta}$，有 $m_t(t^\beta) \leq \exp(-t^\delta)$。这表明，当距离以 $\sqrt{t \log^\alpha t}$（超平方根速度）增长时，外部涡度以超多项式衰减；当以 $t^\beta$（超线性速度）增长时，则以拉伸指数衰减。

对于Euler方程（定理3.1）：设 $\omega(x,t)$ 为无粘性（$\nu=0$）解，且满足相同初始条件。定义 $d_\omega(t) := \sup\{|x_1 - y_1|: x,y \in \Lambda(t)\}$ 为支撑直径。则对任意 $\alpha>1$，有 $\lim_{t \to \infty} (t \log^\alpha t)^{-1/3} d_\omega(t) = 0$。即支撑直径的增长速率最多为 $(t \log t)^{1/3}$（对任意 $\alpha>1$，可取 $(\log^\alpha t)^{1/3}$ 作为上界），这改进了[6]中 $t^{1/3} \log^2 t$ 的界。

3. 研究方法、关键技术和核心工具

论文的核心技术是迭代方法与Biot-Savart核反对称性质的结合。首先，作者构造光滑截断函数 $W_{R,h}$（在 $|\xi|\leq R$ 处为1，$|\xi|\geq R+h$ 处为0），定义平滑化的涡度质量 $\mu_t(R,h) = \int_S [1 - W_{R,h}(x_1)] \omega(x,t) \, dx$，它介于 $m_t(R-h)$ 和 $m_t(R)$ 之间。利用涡度方程的弱形式（公式(1.7)）取测试函数 $f(x,t)=1-W_{R,h}(x_1)$，推导出 $\mu_t(R,h)$ 的时间导数。关键步骤是将速度 $u$ 表示为涡度通过圆柱Biot-Savart law（公式(1.3)）的卷积，并利用核 $\partial_{x_2} G(x,y)$ 的反对称性 $\partial_{x_2} G(x,y) = -\partial_{x_2} G(y,x)$，将时间导数重写为对称积分形式：

$$ \frac{d}{dt} \mu_t(R,h) = \iint_{|x_1|>R} H(x,y,t) \, dx dy - \iint_{|x_1|>R, |y_1|>R} H(x,y,t) \, dx dy - \nu \int_{|x_1|>R} W''_{R,h}(x_1) \omega(x,t) \, dx, $$

其中 $H(x,y,t)$ 对称且涉及 $[W'_{R,h}(x_1) - W'_{R,h}(y_1)]$ 和核导数。通过估计 $|\partial_{x_2} G(x,y)| \leq C \exp(-C'|x-y|)$（来自Green函数 $G$ 的显式表达式(1.4)），得到关键微分不等式：

$$ \frac{d}{dt} \mu_t(R,h) \leq \frac{C}{h^2} m_t(R) + \nu\text{-term}. $$

对于Navier-Stokes，忽略 $\nu$ 项，迭代该不等式 $n$ 次（$n \approx \log t$ 或 $t^\delta$，根据目标距离选择），利用初始时刻 $\mu_0(R,h)=0$（当 $R$ 足够大时）和总涡度质量 $M$ 有界，结合Stirling公式得到超多项式和拉伸指数衰减。迭代次数 $n$ 与 $R$ 和 $h$ 的关系通过设置 $R_0 = \sqrt{t \log^\alpha t}$ 或 $R_0 = t^\beta$，并选择 $h$ 使得 $R_n = 1$（或 $R/2$）来实现。对于Euler（$\nu=0$），上述迭代仅能得到 $m_t(R) \leq (t \log t)/R^2$ 之类的弱估计。为提高精度，作者引入了来自能量守恒的first moment估计（公式(1.8)）：$\sup_{t\geq 0} \int_S |x_1| \omega(x,t) \, dx < \infty$，这是[6]中的关键工具。在估计 $H$ 的积分时，将该moment估计与 $1/|y_1|$ 因子结合，得到更强的微分不等式 $\frac{d}{dt} \mu_t(R,h) \leq \frac{C}{R h^2} m_t(R)$。由此迭代得到 $m_t(R) \leq (t \log^2 t)/R^3$，令 $R = (t \log^\alpha t)^{1/3}$ 即得定理3.1的界。最后，通过比较支撑边界粒子的速度场（由远离中心的涡度贡献主导，该部分被Lemma 3.2和3.3控制），证明支撑直径不可能超过 $(t \log^\alpha t)^{1/3}$。

4. 与之前工作的比较

本论文的工作与之前文献有以下关联与改进：与整平面上Navier-Stokes结果的对比：[11]使用高阶矩方法得到类似定理2.1(b)的拉伸指数衰减。本文采用迭代+Biot-Savart反对称性的方法，统一处理了超多项式（a）和拉伸指数（b）衰减，且方法可推广至其他几何。整平面上还有[9]证明Euler支撑直径最多 $(t \log t)^{1/4}$，本论文Euler结果 $(t \log t)^{1/3}$ 虽弱于整平面界（因几何不同，缺乏惯性矩守恒），但优于[6]在圆柱中的先前结果。与[6]的直接对比：[6]针对Euler方程在无限圆柱中证明 $d_\omega(t) \leq C t^{1/3} \log^2 t$。本文改进为 $(t \log t)^{1/3}$，改进了对数因子。改进的关键在于利用first moment估计（[6]中已有）与迭代方法的更精细结合，特别是估计 $A_2$ 和 $A_3$ 项时通过乘以 $|y_1|/|y_1|$ 引入 $1/R$ 因子。与无限长条（strip）的对比：在带滑移或无滑移边界条件的无限长条中，Green函数无反对称性，且能量守恒不蕴含类似(1.8)的矩估计，仅能得到平凡 $O(t)$ 增长（见[5,13]）。本文圆柱几何因周期性边界条件保留了反对称性，故能获得更优的亚线性（Navier-Stokes）或次线性（Euler）增长界。方法论贡献：本文展示了迭代法结合核反对称性的通用性，可同时处理粘性与无粘情形。此外，Lemma 3.2将边界粒子速度分解为局部涡度贡献（由first moment控制）和远处涡度贡献（由核指数衰减控制），这种分解技巧对证明支撑直径的精细界至关重要。一致性：本文Navier-Stokes结果与整平面上的衰减率一致（外部质量衰减率仅取决于距离增长参数，而非几何细节），表明圆柱几何下涡旋限制行为在本质上与整平面类似，只要利用好核反对称性。

了解领域论文（摘要翻译）

1. Infinite-order multisoliton solutions to the Benjamin–Ono equation and soliton resolution

作者: Louise Gassot, Patrick Gérard

类别: math.AP

发布日期: 2026-03-16

arXiv ID: 2603.15419v1

链接: http://arxiv.org/abs/2603.15419v1

摘要翻译

中文标题: Benjamin-Ono方程的无穷阶多孤子解及其孤子分解

摘要翻译: 我们构造了一类Benjamin-Ono方程在直线上的无穷阶多孤子解，其初始数据具有缓慢的空间衰减性。我们证明，在长时间渐近行为下，此类解可分解为具有不同速度的无穷个独立孤子解的叠加，且不存在辐射项。

2. Lebesgue measure of distance sets with regular pins and multi-scale Mizohata-Takeuchi-type estimates

作者: Bochen Liu

类别: math.CA, math.AP, math.CO

发布日期: 2026-03-16

arXiv ID: 2603.15328v1

链接: http://arxiv.org/abs/2603.15328v1

摘要翻译

中文标题: 具有正则钉住的多尺度Mizohata-Takeuchi型估计的距离集勒贝格测度

摘要翻译: 假设 $E, F$ 是平面上的博雷尔集，满足 $\dim_{\mathcal{H}} E>1$，$\dim_{\mathcal{H}} E+\dim_{\mathcal{H}} F>2$，且 $F$ 的豪斯多夫维数与填充维数相等。我们证明存在 $y\in F$，使得钉住距离集

$$ Δ_y(E):=\{|x-y|:x\in E\} $$

具有正的勒贝格测度。特别地，该结果解决了平面距离集问题的正则情形。证明的主要组成部分包括多尺度好坏分解，以及带有任意小幂次损失的多尺度Mizohata-Takeuchi型估计。

3. Riesz transforms associated with the Grushin operator with drift

作者: Nishta Garg, Rahul Garg

类别: math.AP, math.CA, math.FA

发布日期: 2026-03-16

arXiv ID: 2603.15316v1

链接: http://arxiv.org/abs/2603.15316v1

摘要翻译

中文标题: 带漂移的Grushin算子的Riesz变换

摘要翻译: 我们考虑一类关于具有指数增长的测度对称的带漂移Grushin算子。对于相应的Riesz变换，我们研究了其在强型$(p, p)$（$1 < p < \infty$）及弱型$(1, 1)$意义下的有界性。

4. Numerical study of the sharp stratification limit towards bilayer models

作者: Théo Fradin

类别: physics.flu-dyn, math.AP

发布日期: 2026-03-16

arXiv ID: 2603.15287v1

链接: http://arxiv.org/abs/2603.15287v1

摘要翻译

中文标题: 陡峭密度分层向双层模型极限的数值研究

摘要翻译: 在地球物理尺度的海洋流动研究中，密度分层现象对系统动力学起着核心作用。描述密度分层作用的数学模型通常分为两类：一方面，连续分层模型——如本文所考虑的分层欧拉方程条带问题——能够准确刻画垂向效应，但其理论与数值复杂度较高；另一方面，双层模型通过分段常数分布近似密度分层。在后一种情况下，研究重点在于两层间自由界面的演化，由此得到 substantially simplified（显著简化）的模型。本文在线性化分层欧拉方程框架下，针对接近分段常数分布的密度剖面，比较了两种方法，并证明了向双层欧拉方程的收敛性。然而，存在剪切流时，双层模型的适用范围受开尔文-赫姆霍兹不稳定性限制。此时，我们采用适当的模态分解方法数值计算该线性化模型的色散关系，并提供数值证据表明：开尔文-赫姆霍兹不稳定性限制了两个广泛应用的双层模型——即双层欧拉方程与双层浅水方程——的适用性。

5. Large time behaviour of the fractional heat equation associated with the Dunkl Laplacian

作者: Suman Mukherjee

类别: math.AP

发布日期: 2026-03-16

arXiv ID: 2603.15275v1

链接: http://arxiv.org/abs/2603.15275v1

摘要翻译

中文标题: 与Dunkl拉普拉斯算子相关的分数阶热方程的大时间行为

摘要翻译: 本文考虑与Dunkl拉普拉斯算子相关的分数阶热方程，并证明：当初值是关于相应测度可积的函数时，该方程的弱解在时间趋于无穷时收敛于基本解。作为应用，本文亦证明了对应非线性方程的类似结果。

6. The one-dimensional compressible Navier-Stokes equations in critical regularity spaces

作者: Raphaël Danchin

类别: math.AP

发布日期: 2026-03-16

arXiv ID: 2603.15211v1

链接: http://arxiv.org/abs/2603.15211v1

摘要翻译

中文标题: 临界正则性空间中的一维可压缩Navier-Stokes方程

摘要翻译:
本文研究实数轴上的等熵可压缩Navier-Stokes方程。我们的主要目标是在初始数据为稳定常数状态的微小扰动情形下，于临界正则性框架内建立全局适定性。值得注意的是，尽管多维情形的相关结果已属经典，但据我们所知，一维情形至今尚未得到澄清。其原因是：在临界框架下，速度的正则性过低，导致某些非线性项无法控制。本文通过将方程置于质量拉格朗日坐标系中，克服了这一困难。在获得全局适定性结论后，我们进一步建立了最优时间衰减估计，并研究了高粘度极限问题，指出经时空缩放后，比容收敛于某常微分方程的解。

7. High viscosity limit for the multi-dimensional compressible Navier-Stokes equations

作者: Raphaël Danchin

类别: math.AP

发布日期: 2026-03-16

arXiv ID: 2603.15209v1

链接: http://arxiv.org/abs/2603.15209v1

摘要翻译

中文标题: 多维可压缩Navier-Stokes方程的高黏性极限

摘要翻译: 本文研究多方可压缩Navier-Stokes方程在初始数据为稳定常数解扰动情形下的高黏性极限（亦称惯性极限）。针对常黏性系数情形，我们证明经扩散尺度重整化后，密度将渐近满足一个具非线性阻尼的输运方程，且该方程即便对于大初值也是全局适定的。对于变黏性系数情形，我们建立了类似结论，此时密度极限方程中的阻尼项具有非局部性。

8. Curved Ingham inequalities and observability of the toroidal Schr{ö}dinger equation

作者: Bernhard H Haak, Philippe Jaming, Ming Wang, Yunlei Wang

类别: math.AP, math.CA

发布日期: 2026-03-16

arXiv ID: 2603.15193v1

链接: http://arxiv.org/abs/2603.15193v1

摘要翻译

中文标题: 曲率Ingham不等式与环面薛定谔方程的可观性

摘要翻译: 本文证明环面薛定谔方程的解可通过适当弯曲的时空轨迹（具有零勒贝格测度）实现观测。为此，我们仿照著名的Ingham不等式，建立了沿曲线的某些三角和的新上界与下界。在第二部分中，我们分别证明了低频与高频分量在任意短曲线上的可观性。对于低频分量，我们建立了所考察三角和的零点集所必须满足的强限制条件。

9. Singular limits for non-isentropic compressible rotating fluids

作者: Yajia Yu, Chenxi Su, Ming Lu

类别: math.AP

发布日期: 2026-03-16

arXiv ID: 2603.15064v1

链接: http://arxiv.org/abs/2603.15064v1

摘要翻译

中文标题: 非等熵可压缩旋转流体的奇异极限

摘要翻译: 在本文中，我们研究非等熵可压缩旋转流体的奇异极限。我们将毛细效应纳入α=1和α=0两种情况，并研究涉及低马赫数、低罗斯比数和高雷诺数项的Navier-Stokes-Korteweg方程。当α=1时，通过Rage定理推导出声波方程的色散估计。当α=0时，通过误差估计得到收敛结果。此外，我们还证明三维可压缩Navier-Stokes-Korteweg方程收敛到二维不可压缩Euler方程。

10. Two-phase free boundary problems for operators with nonstandard growth

作者: Fausto Ferrari, Monica Jacob, Claudia Lederman

类别: math.AP

发布日期: 2026-03-16

arXiv ID: 2603.15000v1

链接: http://arxiv.org/abs/2603.15000v1

摘要翻译

中文标题: 具有非标准增长算子的两相自由边界问题

摘要翻译: 本文聚焦于由具有非标准增长的偏微分方程所支配的自由边界问题，重点介绍部分最新成果。研究此类问题的兴趣源于其多样化的应用背景及所带来的数学挑战。

11. Smooth finite time singularity formation without quantization

作者: Istvan Kadar

类别: math.AP

发布日期: 2026-03-16

arXiv ID: 2603.14985v1

链接: http://arxiv.org/abs/2603.14985v1

摘要翻译

中文标题: 无量化情形下光滑有限时间奇点形成

摘要翻译: 我们遵循Hintz [Hintz23]的方法，从几何奇异解析的视角重新审视Krieger-Schlag-Tataru [KST09]关于聚焦型能量临界波方程在 $\mathbb{R}^{3+1}$ 中的有限时间奇点形成问题。我们构造了 $C^{\nu/2-}$ 正则的近似解，这些解收敛至以速率 $t^\nu$（$\nu>1$）收缩的多个孤子，并在不同测地线 $\{x=zt\}$ 上趋近原点。通过精细调节孤子的速度、尺寸与符号，我们能够构造出对任意 $\nu>8$ 均成立的光滑尝试解。利用鲁棒的能量估计，可将这些尝试解修正为精确解。

12. Optimal control of a tumor growth model with hyperbolic relaxation of the chemical potential

作者: Pierluigi Colli, Elisabetta Rocca, Jürgen Sprekels

类别: math.OC, math.AP

发布日期: 2026-03-16

arXiv ID: 2603.14960v1

链接: http://arxiv.org/abs/2603.14960v1

摘要翻译

中文标题: 具化学势双曲松弛的肿瘤生长模型最优控制

摘要翻译: 本文研究一类卡汉-希利亚德型肿瘤生长相场模型的最优控制问题，其中通常假设的化学势抛物松弛被替换为双曲松弛。分析涵盖了相演化所遵循的双阱势分别为正则型与对数型两种情况。我们证明了控制-状态算子在 suitable Banach spaces 中的弗雷歇可微性，并建立了关于伴随状态变量的变分不等式形式的必要最优性条件。进而利用这些必要条件推导出最优控制的稀疏性结果。

13. Local Existence and Finite-Time Singularity Formation in the Vlasov-Poisson-Isotropic Landau System

作者: Jin Woo Jang, Junsung Kim

类别: math.AP

发布日期: 2026-03-16

arXiv ID: 2603.14871v1

链接: http://arxiv.org/abs/2603.14871v1

摘要翻译

中文标题: Vlasov-Poisson-各向同性Landau系统的局部存在性与有限时间奇点形成

摘要翻译: 各向同性Landau（Coulomb）算子由Krieger与Strain（《偏微分方程通讯》，2012年）在动理论中提出。本研究考察空间非均匀的Vlasov-Poisson-各向同性Landau系统。我们首先建立了Cauchy问题的局部时间存在性理论：对于在适当范数下满足合适小性条件的初值，存在非负解于时间区间$[0,T]$上，其中寿命$T$依赖于初值的大小。

超越局部理论，我们探究了可能导致全局存在性失效的机制。在引力吸引情形下，若弱解满足特定的先验正则性与衰减假设，且初始引力场能量超过动能，同时产生的能量差主导碰撞算子的扩散效应，则有限时间奇点形成可能发生。由此，若进一步假设解在最大存在时间内属于适当的测度空间，则其在该时刻于物理空间中坍缩至单一位置。有限时间奇点形成的证明基于推导第二空间矩的上界，该上界在有限时间内变为负值。

14. Towards a Gagliardo-Type Theory of Fractional Sobolev Spaces on Arbitrary Time Scales

作者: Hafida Abbas, Abdelhalim Azzouz

类别: math.AP, math-ph

发布日期: 2026-03-16

arXiv ID: 2603.14715v1

链接: http://arxiv.org/abs/2603.14715v1

摘要翻译

中文标题: 论任意时间尺度上的Gagliardo型分数阶索博列夫空间理论

摘要翻译: 本文基于Lebesgue Δ-测度及定义在对角集外区域$Ω_T \subset T \times T$上的非局部相互作用能，提出了在任意时间尺度上构造分数阶索博列夫空间的Gagliardo型方法。对于常数阶分数$α∈(0,1)$及$1 ≤ p < ∞$，我们定义了相应的分数阶半范数与空间$W^{α,p}_Δ(T)$，并证明该构造在时间尺度的测度论框架下是良定的。特别地，我们确立了这些空间对于所有$1 ≤ p < ∞$均为巴拿赫空间、当$1 < p < ∞$时自反，且在$p=2$时为希尔伯特空间。随后，我们在自然的有界混合时间尺度类上证明了庞加莱型不等式，表明底层时间尺度的几何结构在强制性估计层面已影响理论构建。该构造在统一框架内同时还原了连续、离散与混合情形的理论。最后简要提示了向变阶非局部空间的可能推广，相关研究将另文开展。

15. On the stability of multi-dimensional rarefaction waves with vorticity

作者: Jin Jia, Tao Luo

类别: math.AP

发布日期: 2026-03-16

arXiv ID: 2603.14696v1

链接: http://arxiv.org/abs/2603.14696v1

摘要翻译

中文标题: 带涡旋度的多维稀疏波稳定性研究

摘要翻译: 本文研究二维等熵可压缩欧拉方程中多变量势流稀疏波的稳定性。证明依赖于在稀疏波区域内对声波与涡波同时进行能量估计，且过程中无导数损失。通过将欧拉方程重构为Luk-Speck在文献\cite{LukSpeck2D}中提出的波-输运系统实现上述估计，该方法消除了Luo-Yu在\cite{Luo-YuRare1, Luo-YuRare2}中无旋假设的限制。作为应用，我们在以下扰动区域内证明了分片连续解对黎曼问题的局部存在性：\emph{激波-稀疏波}扰动区与\emph{稀疏波-涡层-稀疏波}扰动区。

16. Quantitative Closure Analysis toward Ideal Fluids

作者: Gi-Chan Bae, Chanwoo Kim

类别: math.AP

发布日期: 2026-03-15

arXiv ID: 2603.14671v1

链接: http://arxiv.org/abs/2603.14671v1

摘要翻译

中文标题: 面向理想流体的定量封闭分析

摘要翻译: 我们针对一大类初始数据，在不借助任何渐近展开的前提下，建立了玻尔兹曼方程的不可压缩低马赫数/高雷诺数极限。通过利用局部麦克斯韦流形与宏观-微观分解，结合对声学分量的色散控制，我们获得了纯微观波动的定量估计，并实现了对熵波动与旋度分别由其初始数据支配的定量控制。据此，在二维空间中，重标度后的速度场与温度场将以DiPerna-Lions-Majda与Delort意义下的全局解形式，收敛至具有对流温度的不可压缩欧拉方程。

17. Nonlinear Wave Dynamics in Black Hole Spacetimes

作者: John Stogin

类别: math.AP

发布日期: 2026-03-15

arXiv ID: 2603.14638v1

链接: http://arxiv.org/abs/2603.14638v1

摘要翻译

中文标题: 黑洞时空中的非线性波动动力学

摘要翻译:
该论文系笔者2017年博士论文学位论文的略作重新编排版本。应导师Sergiu Klainerman之要求，本人将其发布于arXiv平台，因其指出该工作曾对部分学生有所助益。内容基本反映原论文的原始形态。

该论文详述了一种用于证明黑洞时空中非线性波动方程的整体有界性与衰减性结论的方法。该方法被应用于五个难度递增的示例问题中。首个问题涉及闵可夫斯基时空上的半线性波动方程，相对简明，适合尚处于偏微分方程研究入门阶段的读者。最后一个问题源于Ionescu与Klainerman在[IK14]中的提出，是证明慢旋转Kerr黑洞稳定性研究中的一步。其余中间问题包括：史瓦西时空上的半线性波动方程、在轴对称性附加假设下任意亚极端Kerr时空上的半线性波动方程，以及将最终问题限定于史瓦西情形的情形。

本论文所采用的方法基于若干特定进展，或可资其他相关问题借鉴。其中包括：一种构造Morawetz型估计的新方法（其稳健性体现于可成功应用于全部五个问题）；基于衰减层级的能量估计策略，用于类空超曲面（特别采用了弱衰减概念）；以及处理在对称轴上具有奇异因子项的技术。

18. Energy cascade for cross-shear length scales in free-shear three-dimensional incompressible viscous flows

作者: Ricardo M. S. Rosa

类别: math.AP, physics.flu-dyn

发布日期: 2026-03-15

arXiv ID: 2603.14626v1

链接: http://arxiv.org/abs/2603.14626v1

摘要翻译

中文标题: 自由剪切三维不可压缩粘性流动中关于交叉剪切长度尺度的能量级联研究

摘要翻译: 本文研究自由剪切流动中的能量级联现象，该流动由不可压缩牛顿流体方程描述，并受施加的平均剪切剖面驱动，采用混合周期与自由滑移边界条件。基于Foias与Prodi意义下关于平稳统计解的系综平均，我们证明了若干严格结果。通过将流动分解为水平高、低波数分量（对应垂直于平均剪切梯度的尺度），我们得到了包含耗散项、平均流动的剪切产生项以及脉动场能量通量项的能量收支关系。我们利用水平傅里叶模态的正交性对剪切产生项进行估计，并探讨脉动通量的能量级联。对于任意给定波数，我们定义其上方高频带对应的水平Taylor波数。我们证明了两个关于能量级联的结果：一个是关于能量通量本身，另一个是考虑了流向奇点可能导致能量损失的受限能量通量。这两个结论对满足以下条件的波数成立：其对应的水平Taylor波数高于粘性剪切波数，且其低频能量耗散率相对于总能量耗散率可忽略不计。从启发式角度看，这些严格条件恰好对应着经典Corrsin尺度至Kolmogorov耗散尺度之间的能量级联范围，这与传统湍流剪切流动理论的预期一致。

19. Mixing and enhanced dissipation in a time-translating shear flow

作者: Johannes Benthaus, Giuseppe Maria Coclite, Camilla Nobili

类别: math.AP, math-ph, physics.flu-dyn

发布日期: 2026-03-15

arXiv ID: 2603.14624v1

链接: http://arxiv.org/abs/2603.14624v1

摘要翻译

中文标题: 平移剪切流中的混合效应与增强耗散

摘要翻译:
本文部分受Vanneste与Byatt-Smith工作的启发，研究速度场为$\mathbf{u}(x,y,t)=(\sin(y-ct),0)$的平流-扩散方程中的混合效应与增强耗散问题。该剪切流以速度$c$作刚性平移，是时变临界点流动的典型范例。我们量化了扩散衰减性质如何依赖于平移速度$c$与扩散系数$ν$的关联。首先分析无粘输运问题，针对$t\lesssim c^{-1}$的时间尺度建立了时间平均的$H^{-1}$混合估计，其衰减速率快于定常流估计。基于这些估计，我们在平移速度满足$c=c_0ν^\ell$（$\ell\in(1/3,3/4)$）的中等速度 regime 中证明了增强耗散现象，并得到衰减速率$ν^{(1+2\ell)/5}$，该速率连续插值于具有简单临界点的定常剪切流精确速率$ν^{1/2}$与单调流的$ν^{1/3}$之间。这一结果量化了平移速度增加如何逐步削弱临界点的影响。通过直观比较无粘混合与增强耗散的时间尺度，可以解释下限端点$\ell=1/3$的形成机制。对于$c\gg 1$的情形，我们证明在固定时间区间内解始终保持与热方程解的接近性，因此快速平移平均化了平流效应并削弱了混合。混合估计依赖于利用临界点运动产生的抵消效应而精细化的平稳相位分析。增强耗散结果则需要将定常剪切流的Hypocoerc性框架适配于非自治情形：平移流动导致交换子层级无法以标准方式闭合，我们通过构造扩展能量泛函克服此困难。大$c$分析则利用了该 regime 中快速平移的平均效应。

20. Concentrated solutions to fractional Schrödinger-Poisson system with non-homogeneous potentials

作者: Lintao Liu, Haidong Yang

类别: math.AP

发布日期: 2026-03-15

arXiv ID: 2603.14490v1

链接: http://arxiv.org/abs/2603.14490v1

摘要翻译

中文标题: 具有非齐次势的分式薛定谔-泊松系统的集中解

摘要翻译: 本文主要研究分式薛定谔-泊松系统归一化解的几个极限性质，包括存在性、集中行为及局部唯一性。值得注意的是，我们在双重非局部情形下且无需假设势的齐次性，获得了关于归一化解存在性与渐近行为的结果，这在多个方面推广了文献\cite{GDCDS}的结论，并改进了我们先前在文献\cite{LIUYANG}中的工作。同时，通过引入新的技巧，我们还建立了解序列的若干精细性质，包括能量估计、衰减估计及一致正则性。

21. On the Limiting Behavior of $L^2$-Critical Pseudo-Relativistic Fermi Systems

作者: Bin Chen, Yinbin Deng, Yujin Guo, Chenyang Wang

类别: math.AP

发布日期: 2026-03-15

arXiv ID: 2603.14286v1

链接: http://arxiv.org/abs/2603.14286v1

摘要翻译

中文标题: 关于 L² 临界的伪相对论性费米子系统的极限行为

摘要翻译: 本文研究 L² 临界情形下伪相对论性费米子系统的基态。我们证明：该系统存在基态当且仅当吸引强度 $ a $ 满足 $ 0 < a < D_{4/3,2} $，其中 $ D_{4/3,2} \in (0, \infty) $ 是对偶分数阶 Lieb-Thirring 不等式的最优常数。进一步分析了当 $ a \nearrow D_{4/3,2} $ 时基态的极限行为。作为附带结论，本文也研究了该对偶分数阶 Lieb-Thirring 不等式优化子的定性性质。

22. Inverse boundary value problems of determining nonlinear coefficients for the JMGT equation

作者: Dong Qiu, Xiang Xu, Yeqiong Ye, Ting Zhou

类别: math.AP

发布日期: 2026-03-15

arXiv ID: 2603.14194v1

链接: http://arxiv.org/abs/2603.14194v1

摘要翻译

中文标题: Jordan-Moore-Gibson-Thompson (JMGT) 方程非线性系数的反边界值问题

摘要翻译: 本文研究非线性声学中具有Kuznetsov型与Westervelt型二次非线性的Jordan-Moore-Gibson-Thompson (JMGT) 方程的反边界值问题。证明了当观测时间大于最大边界到边界的测地线传播时间时，相应的边界Dirichlet-to-Neumann映射能唯一确定Westervelt型模型中的非线性系数β，以及Kuznetsov型模型中的系数对(β,κ)。该结论在欧氏空间及满足适当几何条件的紧黎曼流形上均成立。证明基于二阶线性化方法，结合几何光学与高斯射束解的构造，将唯一性反问题归结为相关测地线射线变换的单射性。

23. Sheets of Spectral Data of Stokes Waves in Weakly Nonlinear Models

作者: Benjamin Akers, Ryan P. Creedon

类别: math.AP

发布日期: 2026-03-14

arXiv ID: 2603.14090v1

链接: http://arxiv.org/abs/2603.14090v1

摘要翻译

中文标题: 弱非线性模型中斯托克斯波的谱数据片

摘要翻译: 我们研究一族形式为 $u_t + L u + (u^2)_x = 0$ 的弱非线性单向模型中，小振幅斯托克斯波的谱稳定性。我们引入一种摄动方法，在平直流态特征值简并附近，以碰撞模态的比值作为自由参数，将谱数据按波幅展开。这产生了谱数据片，其在固定波幅处的截面给出不稳定性的孤立区域。同一摄动框架可同时处理高频与本杰明-费尔不稳定性，并延伸至不连续色散关系（包括阿克斯-米莱夫斯基方程）。该工作首次为该模型提供了本杰明-费尔谱的解析近似，并在全部模型族中直接比较了高频与本杰明-费尔增长率。渐近预测通过弗洛凯-傅里叶-希尔方法和拟牛顿方法计算的数值谱得到了验证。

24. Global Well-Posedness for the Fourth-Order Nonlinear Schrödinger Equation with Potential in the Energy-Critical Case

作者: Hikaru Nakayama

类别: math.AP

发布日期: 2026-03-14

arXiv ID: 2603.13892v1

链接: http://arxiv.org/abs/2603.13892v1

摘要翻译

中文标题: 带势四阶非线性薛定谔方程在能量临界情况下的全局适定性

摘要翻译: 我们考虑在维度 $n \ge 5$ 下带势的非聚焦四阶非线性薛定谔方程

$$ i\partial_t u + Δ^2 u + Vu + λ|u|^{p-1}u = 0 \qquad (x \in \mathbb{R}^n,\ t \in \mathbb{R}), $$

。在能量临界情况 $p = \frac{n+4}{n-4}$ 下，在径向实值势 $V$ 的适当假设下，我们证明了径向初值在 $H^2(\mathbb{R}^n)$ 上的全局适定性。我们还表明，每个这样的解在 $H^2$ 中散射到双调和薛定谔方程的一个自由解。证明依赖于带势的四阶薛定谔算子的斯特里奇artz估计、与 $Δ^2+V$ 和 $Δ^2$ 相关的 Sobolev 范数的等价性、波算子的有界性、摄动稳定性理论，以及适应势存在的 Morawetz 型估计。这将早期无势情况的结果扩展到了一类径向势。

25. On singular $p$-Laplacian problems with discontinuous convection terms

作者: Umberto Guarnotta, Salvatore A. Marano

类别: math.AP

发布日期: 2026-03-14

arXiv ID: 2603.13811v1

链接: http://arxiv.org/abs/2603.13811v1

摘要翻译

中文标题: 具有不连续对流项的奇异 p-Laplacian 问题研究

摘要翻译: 本文建立了齐次 Dirichlet 边界条件下 p-Laplacian 问题的正强解存在性，其中反应项由在零点处具有奇性且高度不连续的非线性项与不连续对流项叠加构成。论证过程中采用了基于适当测度论结果的局部性论证方法（详见第 3 节）。

不相关论文（基本信息）

1. Boundary symmetry breaking via logistic damping in a chemotaxis-growth system

作者: Yiren Chen, Padi Fuster Aguilera, Vincent Martinez, Kun Zhao

类别: math.AP

发布日期: 2026-03-16

arXiv ID: 2603.15551v1

链接: http://arxiv.org/abs/2603.15551v1

摘要: We establish global stability for a chemotaxis-growth model with logarithmic sensitivity under dynamic Dirichlet boundary conditions on a 1D domain. We analyze both parabolic-parabolic and parabolic-hyperbolic systems. The key challenge is handling time-dependent boundary data for the unknown functions. We overcome this by introducing dynamic reference profiles which suitably interpolate boundary values. Using an expanded entropy functional measuring deviation from these profiles, we prove energy estimates the uniform boundedness of solutions and global asymptotic stability of perturbations.

2. Physics-informed fine-tuning of foundation models for partial differential equations

作者: Vlad Medvedev, Leon Armbruster, Christopher Straub, Georg Kruse, Andreas Rosskopf

类别: cs.LG, cs.AI, math.AP, math.NA

发布日期: 2026-03-16

arXiv ID: 2603.15431v1

链接: http://arxiv.org/abs/2603.15431v1

摘要: Foundation models for partial differential equations (PDEs) have emerged as powerful surrogates pre-trained on diverse physical systems, but adapting them to new downstream tasks remains challenging due to limited task-specific data and distribution shifts. While fine-tuning has proven transformative in natural language processing, best practices for adapting PDE foundation models remain underexplored. Although physics-informed training has successfully trained accurate solvers across a wide range of PDE problems, its potential for fine-tuning data-based foundation models has not been systematically studied. In this work, we introduce a physics-informed fine-tuning framework that adapts pre-trained PDE foundation models by incorporating physical constraints (PDE residuals and boundary conditions) directly into the fine-tuning objective. This enables effective adaptation in data-scarce regimes while promoting physical consistency. We evaluate our method on a downstream task composed of an unseen PDE class and compare it with data-driven finetuning counterparts. Our results demonstrate that physics-informed fine-tuning achieves competitive accuracy without requiring PDE solutions for training. Furthermore, a hybrid fine-tuning strategy yields superior generalization to out-of-distribution scenarios when only minimal training data is available. These findings establish physics-informed fine-tuning as a scalable and data-efficient paradigm, providing a physically interpretable pathway for adapting foundation models in scientific machine learning.

3. Pogorelov interior estimates for general sum-type Hessian equations

作者: Weisong Dong, Sirui Xu, Ruijia Zhang

类别: math.AP, math.DG

发布日期: 2026-03-16

arXiv ID: 2603.15345v1

链接: http://arxiv.org/abs/2603.15345v1

摘要: In this paper, we exploit the concavity of sums of Hessian operators to derive Pogorelov estimates for corresponding equations under the dynamic semi-convexity assumption, and we further obtain several Liouville-type results. Moreover, when k=n-1 and k=n we establish Pogorelov estimates in the admissible cone. As an application, we prove that any entire admissible solution in $\mathbb{R}^n$ with quadratic growth must be a quadratic polynomial.

4. Lipschitz solutions to mean field games with a major player and applications

作者: Charles Meynard

类别: math.AP, math.OC, math.PR

发布日期: 2026-03-16

arXiv ID: 2603.15204v1

链接: http://arxiv.org/abs/2603.15204v1

摘要: This paper introduces a notion of weak solution for the coupled system of master equations in mean field games with a major player. It extends the previously introduced notion of Lipschitz solutions in mean field games. By relying on a probabilistic representation of the system of master equations, we prove that there can exist at most one sufficiently smooth solution and that it is consistent with the associated Nash equilibrium. In this approach, coefficients are only required to be Lipschitz, in particular, no differentiability assumption with respect to probability measures is needed. In a second part, we apply this notion of solution to prove the existence and uniqueness of solutions to MFGs with a major player on intervals of arbitrary length. Our argument relies on assuming that the intensity of the Brownian common noise driving the state of the major player is sufficiently large, as well as a joint displacement monotonicity assumption between the coefficients of minor players and those of the major player. Most notably, this joint monotonicity allows us to prove that the threshold of volatility can be taken independently of the horizon of the game considered, without any long time decoupling assumption on the dependence between minor players and the major player. Finally, inspired by recent extragradient methods for mean field games, we present an algorithm that converges exponentially fast to the solution of the major-minor probabilistic system under this monotonicity assumption. Thanks to the generality of our approach, all results presented in this article hold for mean field games of controls with a major player.

5. Global stability in an age-structured SIRS malaria transmission model

作者: Seraphin Djaoue, Quentin Richard, Antoine Perasso, Irépran Damakoa

类别: math.AP

发布日期: 2026-03-16

arXiv ID: 2603.15201v1

链接: http://arxiv.org/abs/2603.15201v1

摘要: This paper proposes and analyzes a malaria transmission model structured by the chronological age of the human host population. The model couples an age-structured SIRS system for humans, incorporating waning immunity, with an SI system for mosquitoes under mass-action transmissions. Using integrated semigroup theory and spectral analysis, we establish the well-posedness of the model, derive the basic reproduction number, and prove the global asymptotic stability of the parasite-free equilibrium by using a Lyapunov functional, when $R_0\leq 1$, thereby excluding the possibility of backward bifurcation. Numerical simulations further suggest the global stability of the endemic equilibrium when $R_0>1$.

6. Length-explicit stability analysis of Helmholtz problems in leaky circular waveguides

作者: Leszek Demkowicz, Martin Halla, Jens Markus Melenk

类别: math.AP

发布日期: 2026-03-16

arXiv ID: 2603.15107v1

链接: http://arxiv.org/abs/2603.15107v1

摘要: Motivated by the study and simulation of long, coiled optical fibers we consider in this article a simplified model that is prevalent in the engineering community. Mathematically, the problem is specified as follows: Time-harmonic wave propagation is modeled by the Helmholtz equation; the waveguide is a bounded circular section with a transparent boundary condition on one end; the dissipation of energy is modeled by an impedance boundary condition on the outer hull of the waveguide. We show a stability estimate that is explicit in terms of the angular length of the waveguide. The analysis is based on a separation of variables ansatz and the study of the related (nonselfadjoint) modal eigenvalue problem. The key property there is to show that the modes form a Riesz basis in both $L^2$ and $H^1$ spaces. To this end we apply perturbation theory for selfadjoint operators and the concept of local subordination of perturbations [B. Mityagin and P. Siegl, JAM 139 (2019)]. Since the possibility of nontrivial Jordan chains cannot be ruled out, our whole methodology is conducted accordingly. In addition, in contrast to previous works, we include a bounded but heterogeneous part of the waveguide into our considered setting.

7. Lipschitz regularity of solutions to two-phase $p$-Laplacian free boundary problems with right hand side

作者: Fausto Ferrari, Claudia Lederman

类别: math.AP

发布日期: 2026-03-16

arXiv ID: 2603.15024v1

链接: http://arxiv.org/abs/2603.15024v1

摘要: We prove the local Lipschitz continuity of viscosity solutions for two-phase free boundary problems for the $p$-Laplacian with non-zero right hand side, where $p\in (1,\infty)$. This is the optimal regularity for the problem. We also obtain the local Hölder continuity for a larger class of problems. The results introduced here are new even in the homogeneous situation, that is, when the right hand side is zero. Our work applies to merely viscosity solutions, which allows a wide applicability.

8. Monotonicity for the fractional semi-linear problem in a half space

作者: Wenxiong Chen, Yahong Guo, Leyun Wu

类别: math.AP

发布日期: 2026-03-15

arXiv ID: 2603.14627v1

链接: http://arxiv.org/abs/2603.14627v1

摘要: In this paper, we study semilinear fractional equations

$$ (-Δ)^s u(x) = f(u(x)) $$

in a half-space and prove that all positive solutions are strictly increasing in the $x_n$-direction. Previous results typically require the solution $u$ to be globally bounded in $\mathbb{R}^n$. We substantially weaken this condition by assuming only that $u$ be bounded in each slab. Moreover, our analysis relies solely on the local Lipschitz continuity of the nonlinearity $f$, which is weaker than the conditions imposed in earlier works. As a crucial ingredient, we obtained a boundary H"{older} regularity estimate that requires only the boundedness of $u$ near the boundary. This represents a significant improvement over existing results, which often assumed global boundedness of $u$ throughout $\mathbb{R}^n$. The proof introduces a new idea that may be of independent interest. To derive the monotonicity, we employ the method of moving planes. We first obtain a narrow region principle in unbounded domains, which ensures that the moving plane procedure can be initiated from $x_n = 0$. We then establish two averaging effects for the solutions to fractional equations. These key ingredients guarantee that the planes can be moved continuously all the way to $x_n = \infty$. Previously, narrow region principle can only be applied to a single narrow region. It is for the first time that we establish a multiple narrow region principle that can be applied simultaneously to finitely many narrow regions. Compared with the traditional approaches, methods based on the {\em averaging effect} require substantially weaker regularity assumptions and can even accommodate unbounded solutions. We believe that these new ideas and techniques develop here will serve as powerful tools in studying qualitative properties of solutions to fractional equations.

9. Yamabe problems for formally self-adjoint, conformally covariant, polydifferential operators

作者: Jeffrey S. Case

类别: math.DG, math.AP

发布日期: 2026-03-15

arXiv ID: 2603.14340v1

链接: http://arxiv.org/abs/2603.14340v1

摘要: Formally self-adjoint, conformally covariant, polydifferential operators provide a general framework for studying variational problems, such as prescribing the scalar, $Q$-, or $σ_2$-curvatures, within a conformal class. We describe recent progress on Yamabe problems for such operators, including uniqueness results on the sphere and nonuniqueness results in general. We also highlight a number of open questions related to these operators, some of which constitute a possible blueprint for the general solution of the Yamabe problem for polydifferential operators.

10. On the $q$-analogue of Duhamel’s principle

作者: Mohammed Elamine Sebih, Serikbol Shaimardan, Irfan Ali

类别: math.AP

发布日期: 2026-03-15

arXiv ID: 2603.14274v1

链接: http://arxiv.org/abs/2603.14274v1

摘要: In this paper, we revisit the classical Duhamel’s principle and provide a self-contained proof of this fundamental tool for linear evolution equations and systems of coupled equations. Moreover, we establish a $q$-analogue of Duhamel’s principle for $q$-evolution equations of order $k\geq 1$ generated by Jackson’s $q$-difference operator.

11. Uniform boundedness for the two-dimensional Keller-Segel system with Gompertz growth

作者: Nohayla Alaoui, Mohamed Halloumi, Giuseppe Viglialoro

类别: math.AP

发布日期: 2026-03-15

arXiv ID: 2603.14261v1

链接: http://arxiv.org/abs/2603.14261v1

摘要: It is known that in two dimensions the classical Keller-Segel model can lead to cell aggregation. This behavior can be controlled by adding a logistic growth term with quadratic decay. Researchers have tried to find weaker damping mechanisms that can still stabilize the system. Previous work showed that, under suitable assumptions on the initial cell distribution, even weaker growth terms than the classical logistic one can prevent aggregation. In this paper, we study the effect of a Gompertz-type growth term in a minimal two-dimensional chemotaxis model. This term provides a weaker damping effect than those previously considered. We analyze how it influences the system and identify conditions that guarantee that solutions exist for all time and remain bounded.

12. Variational characterizations of weighted eigenvalue and basic reproduction rate for nonlocal dispersal systems and application

作者: Xiandong Lin, Jiazhuo Cheng, Qiru Wang

类别: math.AP

发布日期: 2026-03-15

arXiv ID: 2603.14256v1

链接: http://arxiv.org/abs/2603.14256v1

摘要: The basic reproduction rate is a crucial threshold parameter in infectious disease models. In nonlocal dispersal systems, its variational characterization is challenging due to the possible absence of a principal eigenvalue caused by non-compactness. In this paper, we aim to establish such a characterization even when the principal eigenvalue does not exist. To this end, we first study the spectral bound of a class of nonlocal dispersal operators, establishing a Collatz-Wielandt characterization as well as a Rayleigh-Ritz characterization when the operator is self-adjoint. Using this, we characterize the unique parameter value at which the spectral bound equals zero, covering both non-degenerate and partially degenerate cases, and subsequently obtain an explicit expression for the basic reproduction rate. To demonstrate the utility of our theoretical framework, we apply it to a nonlocal dispersal SIS epidemic model with saturated incidence rate. The analysis shows that, in the degenerate case of the saturation coefficient, the limiting behavior of the basic reproduction rate as the total population tends to zero is strikingly different from that in local diffusion case.

13. On the lower bound and upper bound of the Sum of Eigenvalues of the Fractional-Logarithmic Laplacian

作者: H. Hajaiej

类别: math.AP

发布日期: 2026-03-15

arXiv ID: 2603.14250v1

链接: http://arxiv.org/abs/2603.14250v1

摘要: We establish a lower bound and an upper bound to the sum of the Fractional-Logarithmic Laplacian. A main challenge in such a study comes from the fact that this operator has a Fourier symbol that is not globally monotone in its radial variable due to its low-frequency behavior.

14. Absence of the Lavrentiev phenomenon for degenerate parabolic double phase problems

作者: Bogi Kim, Youngchae Kim, Jehan Oh

类别: math.AP, math.FA

发布日期: 2026-03-15

arXiv ID: 2603.14235v1

链接: http://arxiv.org/abs/2603.14235v1

摘要: We establish the absence of the Lavrentiev phenomenon for degenerate parabolic double phase problems. Any finite-energy function in the natural parabolic class admits smooth approximations with convergence in the parabolic Sobolev space and convergence of the corresponding energy. We provide explicit gap bound conditions and derive improved bounds under additional assumptions such as boundedness or stronger time regularity.

15. A nonlocal transmission problem on a hybrid continuous-discrete domain

作者: Hafida Abbas, Abdelhalim Azzouz

类别: math.AP

发布日期: 2026-03-15

arXiv ID: 2603.14178v1

链接: http://arxiv.org/abs/2603.14178v1

摘要: We study a quadratic nonlocal variational problem on a hybrid domain formed by a compact interval and finitely many discrete points. The associated energy splits into continuous, discrete, and interface contributions. Our main estimate shows that the interface term yields a coercive coupling between the two phases and provides an equivalent hybrid norm. As a consequence, we prove existence and uniqueness of a minimizer for the corresponding variational problem and characterize it as the unique weak solution of the associated hybrid Euler–Lagrange system. The latter combines a nonlocal integral equation on the continuous component with a finite nonlocal algebraic system on the discrete nodes.

16. Existence of Solutions of the third term of the Connaughton-Newell Model with a source term

作者: Anh Viet Nguyen

类别: math.AP

发布日期: 2026-03-15

arXiv ID: 2603.14162v1

链接: http://arxiv.org/abs/2603.14162v1

摘要: The Connaughton-Newell equation is an approximation of three-wave kinetic equations using a fully non-linear coagulation-fragmentation model. This equation consists of three non-linear operators. In this paper, we proved that assuming a constant interaction kernel and a well-behaved source term, the third operator of the Connaughton-Newell equation has a solution.

17. On the Hausdorff dimension and singularities of the monopolist’s free boundary curve

作者: Robert J. McCann, Lucas D. O’Brien, Cale Rankin

类别: math.AP

发布日期: 2026-03-14

arXiv ID: 2603.14100v1

链接: http://arxiv.org/abs/2603.14100v1

摘要: The simplest genuinely multidimensional monopolist’s problem involves minimizing a linearly perturbed Dirichlet energy among nonnegative convex functions $u$ on an open domain $X \subset [0, \infty)^2$. The geometry of the region of strict convexity $Ω\subset X$ for the unique minimizer $u$ is of central interest. A relatively closed portion $X_1^0 \subset X$ of the domain is comprised of segments starting and ending on $\partial X$ along which $u$ is affine. For convex polygons and certain other domains $X \subset \mathbf{R}^2$, we build on results with Zhang to show that outside $X_1^0 \cup \{u=0\}$, the free boundary of $Ω$ is a continuous curve of Hausdorff dimension one, and that $Ω$ has density $1/2$ along it (and is $C^α_{\mathrm{loc}}$ for all $0<α<1$), except perhaps at a discrete set of singular points. We do this by showing that much of the free boundary solves an obstacle problem whose endogenous obstacle is $C^2$. For Rochet and Choné’s square $X=(a,a+1)^2$ with $a>0$, there is a point $x_a$ on the diagonal such that $X_1^0\cap \partial Ω\subset \{x_a\}$, and the discrete singularities mentioned above can only accumulate at $x_a$ or at the two ends of the analytic arc $\{u=0\} \cap \partial Ω$, plus the two limit points of $X \cap \partial Ω$ on $\partial X$. Where the regularity of the endogenous obstacle can be improved to $C^{2,\mathrm{Dini}}$, the free boundary becomes locally $C^\infty$ outside a closed set whose relative interior is empty.

18. Dimension reduction of fractional Sobolev seminorms on thin domains

作者: Andrea Braides, Andrea Pinamonti, Margherita Solci

类别: math.AP, math.FA

发布日期: 2026-03-14

arXiv ID: 2603.13968v1

链接: http://arxiv.org/abs/2603.13968v1

摘要: We study the asymptotic behaviour of Gagliardo seminorms in $H^s$ defined on thin films $Ω_\e=ω\times(0,\e)$. The first relevant order is $\e^{1-2s}$, at which the corresponding limit captures the vertical fractional oscillations through one-dimensional sections. The second relevant order produces dimension-reduction regimes that undergo a qualitative transition at the critical exponent $s=\tfrac12$. For $s<\tfrac12$, the dominant contribution is driven by interactions at finite planar distance, and the dimension-reduction scale is $\e^2$. In this regime, the limit is a lower-dimensional \emph{fractional} energy with an effective gain of $\tfrac12$ in the differentiability index. At the critical exponent $s=1/2$, the dimension-reduction scale is $\e^{2}|\log\e|$, and the limit is {\em local}, with dominant interactions at scales between $\e$ and $1$, producing a Dirichlet-type limit on $ω$. For $s>\tfrac12$, the dominant contribution is instead driven by interactions at distances of order $\varepsilon$, the dimension-reduction scale is $\e^{3-2s}$, and the second-order $Γ$-limit is still local. We also study the case $s=s_\e\to 1^-$, showing a Bourgain–Brezis–Mironescu-type result.

19. Channel transport: gating, geometry, and heterogeneous diffusion

作者: Sean D Lawley

类别: cond-mat.stat-mech, math.AP, math.PR

发布日期: 2026-03-14

arXiv ID: 2603.13918v1

链接: http://arxiv.org/abs/2603.13918v1

摘要: Channel-mediated transport is ubiquitous in biology. A series of works by different theoreticians have sought to determine how the diffusive flux through a channel depends on (a) stochastic gating, (b) channel geometry, and (c) heterogeneous diffusion. In this paper, we derive an explicit estimate for the diffusive flux through a channel that accounts for these three factors. We show that our estimate is exact in certain parameter regimes. We further use stochastic simulations to confirm that our estimate remains accurate across a very broad range of parameters. Our estimate differs from some results in the physics literature.

20. Nahm Poles and 0-Instantons

作者: Marco Usula

类别: math.DG, math-ph, math.AP

发布日期: 2026-03-14

arXiv ID: 2603.13805v1

链接: http://arxiv.org/abs/2603.13805v1

摘要: We study self-dual 0-connections, or 0-instantons, on asymptotically hyperbolic 4-manifolds. These connections develop a uniform singularity along the conformal infinity, and are asymptotic, at each point of the boundary, to a “Nahm pole” model solution on $H^4$. Examples include the Levi-Civita spin connections on $S^+$ over spin Poincaré-Einstein 4-manifolds. Inspired by the Fefferman-Graham expansion for Poincaré-Einstein metrics, we study the asymptotic expansion of these 0-instantons. We prove that the expansion is log-smooth, and that the coefficient of the first log term - which we call the 0-instanton obstruction tensor - is a conformal invariant related to the Weyl curvature of the ambient conformal metric. We then show that this invariant vanishes if and only if the 0-instanton is smooth modulo gauge. Finally, we study the renormalized Yang-Mills energy: we prove that, if the metric is asymptotically Poincaré-Einstein to third order, then this energy is a well-defined conformal invariant, and equals the negative Chern-Simons invariant of the conformal infinity.

21. Notes on an intuitive approach to elliptic homogenization

作者: Conor Rowan

类别: math.AP, math-ph

发布日期: 2026-03-14

arXiv ID: 2603.13661v1

链接: http://arxiv.org/abs/2603.13661v1

摘要: Elliptic homogenization is used to determine coarse-grained properties of materials with features on small scales for heat transfer and elasticity. When microstructural features of a material have rapid, periodic fluctuations, the solution corresponding to a “homogenized” coefficient field closely resembles the true solution based on the heterogeneous material. Most presentations of elliptic homogenization rely on methods from perturbation theory, which can make an intuitive, physical understanding of the homogenized coefficients elusive. In this set of notes, we derive the homogenized coefficients for one- and two-dimensional elliptic boundary value problems based on arguments which are physically motivated, and with no recourse to perturbation theory. Then, we discuss homogenization of the Laplace-Beltrami operator for heat conduction on thin surfaces with multiscale curvature, an example which has seen minimal treatment in existing literature.

22. Conformally invariant equations with negative critical exponents on the three dimensional hyperbolic space

作者: Debdip Ganguly, Jungang Li, Guozhen Lu, Jianxiong Wang

类别: math.AP

发布日期: 2026-03-13

arXiv ID: 2603.13601v1

链接: http://arxiv.org/abs/2603.13601v1

摘要: We establish a symmetry result for positive entire solutions with a prescribed growth rate to the following fourth order equation on the 3-dimensional hyperbolic space $\mathbb{H}^3$:

$$ P_2 u = - u^{-7}, $$

where $P_2$ denotes the fourth-order Paneitz operator. We prove that any positive solution $u$ on $\mathbb{H}^3$ exhibiting exponential growth at infinity must, up to hyperbolic isometries, be radial and strictly decreasing with respect to some point $P \in \mathbb{H}^3$. Fourth order equations with negative critical growth on 3-dimensional Euclidean space $\mathbb{R}^3$ has been studied by Choi and Xu in \cite{CX09 }, and subsequently by McKenna and Reichel \cite{MR03} and Xu \cite{Xu05}. Unlike the Euclidean case, the behavior of the Green’s function of $P_2$ is substantially different, which prevents us from using the moving plane (sphere) method directly.

23. Feedback Control and Local Convexification of Wasserstein Gradient Flows

作者: Dante Kalise, Lucas M. Moschen, Grigorios A. Pavliotis

类别: math.OC, math.AP, math.DS, math.FA

发布日期: 2026-03-13

arXiv ID: 2603.13588v1

链接: http://arxiv.org/abs/2603.13588v1

摘要: For free energies of the form

$$ F(μ) = E(μ) + σ\int_Ωμ\logμ\,dx, \quad σ> 0, $$

we study the Wasserstein gradient flow, a continuity equation also known as mean-field Langevin dynamics, around a stationary state $\barμ$ on the flat torus. Our first result identifies the Wasserstein Hessian of $F$ at $\barμ$ with a self-adjoint operator with compact resolvent on a Hilbert space of potential variables, and shows that, up to the natural Riesz isometry, this operator generates the linearized gradient flow. This spectral description allows us to design a finite-rank feedback law, via an algebraic Riccati equation, that shifts the closed-loop Hessian spectrum above any prescribed threshold $δ> 0$. As a consequence, the nonlinear closed-loop flow converges locally exponentially to $\barμ$ with rate $δ$. Under an additional second-order remainder assumption on the first variation, the corresponding closed-loop energy is also locally strongly convex in chart coordinates. We illustrate the framework on the flat torus and discuss extensions to multi-species systems, moment-constrained Fokker-Planck equations, and closed Riemannian manifolds.

24. Nonlinear parabolic problem with time fractional derivative

作者: Nikolai Kutev, Tsviatko Rangelov

类别: math.AP

发布日期: 2026-03-13

arXiv ID: 2603.13534v1

链接: http://arxiv.org/abs/2603.13534v1

摘要: Time fractional parabolic problem for p-Laplacian with double singular Hardy-type potential is considered. Comparison principle and appriory estimates for the weak solutions are proved. Existence of global weak solutions and finite-time blow-up are investigated depending on the optimal Hardy constant.