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1. Growth estimates for axisymmetric Euler equations without swirl

无旋轴对称欧拉方程的增长估计

作者: Khakim Egamberganov, Yao Yao

类别: math.AP

发布日期: 2025-12-15

arXiv ID: 2512.13456v1

链接: http://arxiv.org/abs/2512.13456v1

详细分析

中文标题: 无旋轴对称欧拉方程的增长估计

研究对象和背景

本文研究的是三维空间 $\mathbb{R}^3$ 中无旋轴对称欧拉方程（axisymmetric Euler equations without swirl）的解在时间演化过程中的增长性质。这类方程描述了不可压缩、无粘性流体在三维空间中的运动，其核心是vorticity form（涡量形式）下的欧拉方程：

其中 $u(\cdot,t)$ 表示流体的速度场，$\omega(\cdot,t) := \nabla \times u(\cdot,t)$ 是对应的涡量场。第二式称为Biot–Savart 定律，它将速度 $u$ 从涡量 $\omega$ 中恢复出来。右侧项 $\omega \cdot \nabla u$ 被称为涡旋拉伸项（vortex stretching），这是三维欧拉方程中产生潜在奇异性与无限增长的关键机制。

然而，在本文考虑的特殊情形下——即“无旋轴对称”（without swirl）设置中，这种拉伸效应被显著简化。具体来说，假设速度场在柱坐标 $(r,\theta,z)$ 下具有如下形式：

即没有切向分量（$u_\theta = 0$），这被称为“无旋”或“无周向速度”。在这种设定下，涡量也仅有一个非零分量：角向分量 $\omega^\theta$，且满足：

此时，第一个方程退化为一个标量输运方程：

这一方程揭示了一个关键守恒律：量 $\omega^\theta / r$ 沿着粒子轨迹保持不变。由此可得，对于所有 $1 \leq p \leq \infty$，范数 $\|\omega^\theta / r\|_{L^p(\mathbb{R}^3)}$ 在时间上是守恒的。这个守恒律极大地限制了涡量的增长方式：由于 $\omega^\theta = r \cdot (\omega^\theta / r)$，若 $\omega^\theta / r$ 有界，则 $\omega^\theta$ 的增长只能通过 $r$ 增大来实现，即流体颗粒向外移动至更大的半径时，涡量可能随之放大。

因此，研究的重点转向了两个方面：

1. 径向矩（radial moment）的增长，例如 $P_k(t) := \int_{\mathbb{R}^3} r^k |\omega^\theta| dx$；
2. 涡量 $L^p$-范数的增长行为，尤其是 $\|\omega(\cdot,t)\|_{L^p}$ 是否随时间趋于无穷。

本文的研究区域是整个三维空间 $\mathbb{R}^3$，未涉及边界条件，属于全空间问题。函数空间方面，作者假设初始数据满足：

并进一步要求某些矩有限（如 $P_2(0) < \infty$ 或 $Z(0) < \infty$）。这些条件保证了解的全局存在性（由 Danchin [12] 等人建立），同时允许使用能量方法和插值不等式进行分析。

特别地，当初始涡量满足关于 $z$ 轴奇对称且在上半空间非负的条件：

则系统呈现出类似于“反平行涡环”（anti-parallel vortex rings）的动力学行为。这种构型下，两个符号相反的涡结构相互吸引并向外扩张，导致中心区域的径向速度增强，从而推动涡量向更大的 $r$ 扩散，引发 $\omega^\theta$ 的增长。

历史上，Childress [6] 曾提出一个著名猜想：此类构型可能导致最优的涡量增长速率，并推测 $\|\omega\|_{L^\infty}$ 可能达到 $t^{4/3}$ 的增长率。数值模拟支持这一猜测，但严格的数学证明仍属开放问题。此前的工作多集中在上界估计或特定类型初值下的下界构造，而对一般光滑紧支初值能否产生无界的 $L^p$-范数增长尚无定论。

综上所述，本文的研究对象是一个物理意义明确、数学结构特殊的三维欧拉子系统——无旋轴对称流；其背景在于理解在无奇异性形成的前提下，涡量是否仍可在无限时间内持续增长，以及这种增长的速度极限何在。该问题不仅关乎流体力学模型的长期行为，也为探索三维欧拉方程是否存在有限时间爆破提供间接线索。

主要定理或主要结果

本文的主要成果分为上界估计和下界估计两部分，分别对应不同的初值条件。

上界定理（Theorem 1.1）

设初始速度 $u_0$ 和涡量 $\omega_0$ 满足：

则对于任意 $k \geq 2$，若 $P_k(0) < \infty$，有：

此外，还有以下速度与涡量的上界：

这些上界与 Childress 猜想中的渐近行为一致，表明它们可能是最优的。

下界定理（Theorem 1.2）

在上述条件基础上，若进一步假设初值满足奇对称与符号条件：

并且 $P_2(0) < \infty$, $Z(0) < \infty$，其中 $Z(t) := \int z \omega drdz$ 为垂直矩，则存在常数 $c > 0$，使得：

此结果改进了先前文献中 $P_2(t) \gtrsim t^{4/3-\varepsilon}$ 的下界。

$L^p$-范数增长定理（Theorem 1.4）

在相同对称与符号条件下，若 $Z(0) < \infty$，则存在常数 $c(\omega_0,u_0) > 0$，使得对所有 $1 \leq p \leq \infty$，

这是首个对光滑、紧支初值在 $\mathbb{R}^3$ 全空间中建立的 $L^p$-范数幂律增长结果。

涡量逃逸定理（Theorem 1.5）

在相同条件下，定义在区域 $\Pi_+ \cap \{r \leq R\}$ 上的“涡量质量”为：

则存在依赖于初值的常数 $C, R_0$，使得对所有 $R \geq R_0$，

这意味着：几乎所有的涡量最终都会以时间积分的方式逃逸到 $r \to \infty$。

研究方法、关键技术和核心工具

a) 核心分析工具

本文综合运用了多种经典与现代分析技术：

• 能量方法：利用动能 $E(t)$ 的守恒性作为基本控制量。文中定义 $E_0 := \frac{1}{\pi} \iint r(u_r^2 + u_z^2) drdz$ 为常数，用于控制速度场的大小。

• 插值不等式：关键一步是引用 Lim–Jeong [24] 中的一个非标准插值不等式：

  $$ \|u_r\|_{L^\infty(\Pi)} \lesssim \| \sqrt{r} u \|_{L^2(\Pi)}^{1/3} \left\| \frac{\omega}{r} \right\|_{L^\infty(\Pi)}^{1/2} \| r^2 \omega \|_{L^1(\Pi)}^{1/6}, $$

  将径向速度的 $L^\infty$ 范数用能量、守恒量和矩来控制。这是连接矩增长与速度、涡量增长的核心桥梁。

• 微分不等式求解：通过对矩 $P_k(t)$ 建立微分不等式（如 $P_2'(t) \lesssim \sqrt{P_2(t)}$），然后积分得到多项式增长上界。

• Cauchy–Schwarz 不等式：广泛用于处理积分表达式，特别是在下界估计中结合速度积分进行放缩。

• 正则化技巧：为了处理含绝对值的矩 $P_k(t) = \iint r^k |\omega| drdz$ 的导数，作者引入光滑逼近 $f_\epsilon(s) = \sqrt{s^2 + \epsilon^2} - \epsilon$ 来绕过 $|s|$ 不可微的问题，并通过控制余项 $g_\epsilon(s)$ 实现极限过渡。

b) 关键技术突破

1. 新表达式的发现：传统方法中，$P_2'(t)$ 和 $Z'(t)$ 被表示为复杂的双重积分（核函数形式）。本文的一大创新是推导出基于速度场的新表达式：

   $$ P_2'(t) = \int_0^\infty r u_r(r,0,t)^2 dr, $$
   $$ -Z'(t) = \int_0^\infty z \frac{u_z(0,z,t)^2}{r} dz + \iint_\Pi r \frac{u_r(r,z,t)^2}{r} drdz. $$

   这些表达式更为简洁且易于操作，避免了繁琐的核估计。

2. 下界估计策略：对于 $P_2'(t)$，利用 Cauchy–Schwarz 不等式将其下界与某个区间上的平均径向速度联系起来：

   $$ P_2'(t) \geq \left( \int_a^b u_r(r,0,t) dr \right)^2 \left( \int_a^b r^{-1} dr \right)^{-1}. $$

   选择 $a(t) \sim (1+t)^{-1/3}, b(t) \sim 1+t$，并结合速度衰减估计，得出 $P_2'(t) \gtrsim (\log t)^{-1}$，进而积分得 $P_2(t) \gtrsim t / \log t$。

3. 利用 $-Z'(t)$ 的定量下界：文章证明，若在某时刻有足够多的涡量留在 $r \leq R$ 区域内，则 $-Z'(t)$ 必须大于某个正量 $c m^4 R^{-4}$。由于 $Z(t)$ 单调递减且有下界，故 $-Z'(t)$ 在时间上可积，从而迫使 $m_R(t)$ 不能长期维持在一个正值附近，必须整体趋于零（在 $L^4_t$ 意义下）。

4. 独立于矩的 $L^p$ 增长证明：不同于以往依赖 $P_2(t)$ 增长的方法，本文通过分析 $-Z'(t)$ 的行为，直接控制涡量分布的空间扩展速度，从而获得对所有 $p$ 成立的 $\|\omega\|_{L^p} \gtrsim t^{1/4}$ 的 limsup 增长。

与之前工作的比较

a) 突破与改进

1. 首次建立光滑初值的 $L^p$ 幂律增长：尽管 Choi–Jeong [10] 和 Gustafson–Miller–Tsai [20] 对 patch-type 初值得到了 $\|\omega\|_{L^\infty} \gtrsim t^\alpha$ 类型的结果，但对于光滑、紧支初值，此前没有任何工作能证明 $\|\omega\|_{L^p}$ 会趋向无穷。本文 Theorem 1.4 首次实现了这一点，填补了重要空白。

2. 放宽正则性假设：Lim–Jeong [24] 得到 $\|\omega\|_{L^\infty} \lesssim t^{4/3}$ 依赖于初值紧支的条件；而本文 Theorem 1.1 在更弱的 $P_2(0) < \infty$ 条件下就得到了相同上界，适用范围更广。

3. 新视角下的下界分析：以往对 $P_2'(t)$ 的估计依赖复杂核函数的符号分析；本文采用基于速度的新公式，使估计更透明、更具几何直观性。

4. 揭示涡量逃逸机制：Theorem 1.5 提供了新的物理图像：并非所有涡量都参与高速扩张的“头部”，而是大部分以“尾部”形式缓慢向外迁移，最终在时间积分意义上全部逃逸至无穷远。这深化了对涡量输运机制的理解。

综上，本文不仅在技术上有所创新，更在概念层面推进了对三维欧拉方程长期动力学的认知。

2. Lifespan of the Non-resistive Hall-MHD System with Small Magnetic Gradient

具有小磁场梯度的无电阻Hall-MHD系统的寿命

作者: Linbin Yang, Taoran Zhou

类别: math.AP

发布日期: 2025-12-14

arXiv ID: 2512.12524v1

链接: http://arxiv.org/abs/2512.12524v1

详细分析

中文标题: 具有小磁场梯度的无电阻Hall-MHD系统的寿命

研究对象和背景

本文研究的是无电阻轴对称Hall-MHD系统（Non-resistive Hall-MHD system）在初始磁场梯度较小时，其强解的存在时间下界估计。该系统描述了导电流体与磁场之间的相互作用，特别引入了Hall项以刻画电子与离子运动脱耦的物理效应，这在高密度、低碰撞频率的等离子体中（如恒星形成、磁约束聚变装置）起着关键作用。

所考虑的三维非电阻Hall-MHD系统如下：

其中 $(u,h): \mathbb{R}^+ \times \mathbb{R}^3 \to \mathbb{R}^3 \times \mathbb{R}^3$ 分别表示速度场和磁场，$p: \mathbb{R}^3 \to \mathbb{R}$ 是压力函数，$\mu$ 是常数粘性系数，$\mu_0$ 是真空磁导率，$\nu_0$ 是Hall效应的比例常数。初始条件为：

由于 $\mu_0, \nu_0$ 在证明中不起本质作用，作者不失一般性地设 $\mu_0 = \nu_0 = 1$。此外，本文聚焦于轴对称解（axially symmetric solutions），即在柱坐标 $(r,\theta,z)$ 下，速度 $u$ 和磁场 $h$ 具有特定结构：

其中基向量定义为：

这一设定意味着速度无方位角分量（$u_\theta = 0$），而磁场仅有方位角分量（$h_r = h_z = 0$）。根据已知的局部适定性结果（如Li [17]），若初始速度 $u_0 \cdot e_\theta = 0$，则此性质在演化过程中保持不变。

在此轴对称假设下，原系统可简化为以下形式：

对应的涡度 $w = \nabla \times u$ 也具有轴对称结构：$w = w_\theta e_\theta$，其中 $w_\theta = \partial_z u_r - \partial_r u_z$。进一步推导得 $w_\theta$ 满足：

为了更有效地分析系统行为，作者引入两个关键的重构变量：

这两个量在轴对称流体力学中具有重要意义：$\Omega$ 与涡量的“广义”分布相关，而 $H$ 被Li和Yang [18]指出满足Beale-Kato-Majda（BKM）型正则性准则，即解的爆破与 $\|\nabla H\|_{L^\infty}$ 的增长密切相关。

利用上述变换，$(\Omega, H)$-系统满足如下方程组：

本文的核心关注点是解的寿命（lifespan），即局部光滑解能持续存在的最大时间 $T_* > 0$。以往研究表明，非电阻Hall-MHD系统即使在高正则初值下也可能不全局适定（Chae-Weng [3]），因此定量估计其存在时间成为重要课题。特别是当初始磁场梯度 $\|\nabla h_0\|_{L^\infty}$ 或 $\|\nabla H_0\|_{L^\infty}$ 很小时，人们期望解能在较长的时间内保持正则。

文中所用函数空间为标准的Sobolev空间 $H^k(\mathbb{R}^3)$，初始数据假设属于 $H^3(\mathbb{R}^3)$，并满足轴对称结构及散度自由条件。边界条件为全空间情形，无边界影响。研究区域为 $\mathbb{R}^+ \times \mathbb{R}^3$，即时间从零开始到某个有限或无限时刻。

主要定理或主要结果

本文的主要成果分为粘性和无粘两种情况，分别给出了解保持在 $H^3$ 正则性的寿命下界估计。

定理 1.1（粘性情形）

设 $(u,h)$ 是系统 (HMHD) 的局部光滑轴对称解，初始数据 $(u_0, h_0, H_0) \in H^3(\mathbb{R}^3)$，且满足 $\nabla \cdot u = u_0 \cdot e_\theta = h_r = h_z \equiv 0$。若初始重构磁场梯度满足：

则存在常数 $C_*$，使得当时间 $t \leq T_*$ 时，解 $(u,h)$ 保持在 $H^3$ 中，其中 $T_*$ 满足：

这里 $E_0 = \|u_0\|_{H^3} + \|h_0\|_{H^3}$ 表示初始能量大小。

说明：该定理表明，只要初始的 $\nabla H_0$ 足够小（$\varepsilon \to 0$），即使磁场本身可以很大（见Remark 1.5），解的存在时间就可以任意长，呈双重对数增长趋势。

定理 1.3（无粘情形）

考虑无粘情形（$\mu = 0$），其余设定同上。若初始条件满足更强的小梯度条件：

则存在常数 $C_*$，使得当 $t \leq T_*$ 时，解 $(u,h)(t,\cdot)$ 保持在 $H^3$ 中，且：

说明：无粘情形下的寿命下界包含四重对数结构，虽然仍趋于无穷（当 $\varepsilon \to 0$），但收敛速度比粘性情形更慢。这是由于缺乏耗散项导致控制难度增加所致。

此外，作者在 Remark 1.4 中给出了一个具体的初值构造示例，验证了条件 (1.7) 的可行性：令 $h_{\theta,0} = M \phi(2r/a - 3)$，其中 $\phi$ 是标准光滑截断函数，则通过计算可得 $\|\nabla h_0\|_{L^\infty} \sim M/a$，$\|\nabla H_0\|_{L^\infty} \lesssim M/a^2$，故选择 $a \sim \varepsilon^{-1}$ 即可满足小梯度要求。

研究方法、关键技术和核心工具

a) 核心分析学工具

论文采用了一系列现代偏微分方程中的经典与前沿分析工具，主要包括：

1. 能量估计方法（Energy Estimates）
   这是最基本也是最重要的技术之一。通过对方程进行高阶导数作用（如 $\nabla^3$），并在 $L^2$ 内积意义下测试，得到关于 $\|(u,h,H)\|_{H^3}$ 的微分不等式。例如，在引理 2.6 中，作者建立：

   $$ \frac{d}{dt} \|(u,h,H)(t,\cdot)\|_{H^3}^2 \leq C \|\nabla(u,h,H)(t,\cdot)\|_{L^\infty} \|(u,h,H)(t,\cdot)\|_{H^3}^2, $$

   再结合Grönwall不等式即可获得高阶范数的增长率控制。

2. Gagliardo-Nirenberg 插值不等式
   用于连接不同Sobolev范数之间的关系。例如在估计 $\|u_r/r\|_{L^\infty}$ 时，使用：

   $$ \left\| \frac{u_r}{r} \right\|_{L^\infty} \leq C \left\| \nabla \frac{u_r}{r} \right\|_{L^2}^{1/2} \left\| \nabla \frac{u_r}{r} \right\|_{L^6}^{1/2} \leq C \|\Omega\|_{L^2}^{1/2} \|\nabla \Omega\|_{L^2}^{1/2}, $$

   利用了 $\nabla(u_r/r)$ 与 $\Omega$ 的控制关系（引理 2.5）。

3. 最大正则性理论（Maximal Regularity for Heat Flow）
   在粘性情形中，作者利用热方程的最大 $L^q L^p$ 正则性来控制涡度梯度 $\nabla w$。具体而言，由引理 3.2 可知算子：

   $$ A f := \int_0^t \nabla^2 e^{(t-s)\Delta} f(s,\cdot) ds $$

   在 $L^q(0,T; L^p(\mathbb{R}^d))$ 上有界（$1 < p,q < \infty$）。应用于涡度方程：

   $$ \partial_t w - \Delta w = -\nabla \times (w \times u) - \partial_z(H h), $$

   可得：

   $$ \|\nabla w\|_{L^2_T L^4} \lesssim \|\nabla \times (w \times u)\|_{L^2_T L^4} + \|\partial_z(H h)\|_{L^2_T L^4}, $$

   从而将高阶导数估计转化为非线性项的时空积分估计。

4. Commutator 估计（Kato-Ponce 不等式）
   处理高阶导数与乘积项的交换误差。例如在引理 2.2 中，作者使用 Kato-Ponce 型估计：

   $$ \|[ \nabla^m, f \cdot \nabla ] g \|_{L^2} \leq C (\|\nabla f\|_{L^\infty} \|\nabla^m g\|_{L^2} + \|\nabla g\|_{L^\infty} \|\nabla^m f\|_{L^2}), $$

   用于处理能量估计中出现的非线性项，如 $\nabla^3(u \cdot \nabla u)$ 的展开。

5. Grönwall 不等式
   贯穿全文的基本工具，用于从微分不等式推出积分型增长估计。例如在引理 2.6 中，通过Grönwall引理得出：

   $$ \|\nabla h(t,\cdot)\|_{L^\infty} \leq \|\nabla h_0\|_{L^\infty} \exp\left( C \int_0^t (\|\nabla u\|_{L^\infty} + \|\partial_z H\|_{L^\infty}) ds \right), $$

   将梯度增长归结为速度梯度和 $H$ 垂直导数的累积效应。

b) 技术性难题的解决技巧

1. 控制 $u_r / r$ 的 $L^\infty$ 范数
   在轴对称流体中，$u_r / r$ 在 $r=0$ 处可能出现奇性。作者通过引入 $\Omega = w_\theta / r$ 并利用其满足的方程 (3.2) 来间接控制。通过对方程做 $L^2$ 能量估计，结合 $I_2 \leq 0$（源于边界项非正）和 $I_3$ 的Young不等式处理，最终得到：

   $$ \|\Omega(t,\cdot)\|_{L^2}^2 + \int_0^t \|\nabla \Omega(s,\cdot)\|_{L^2}^2 ds \leq \|\Omega_0\|_{L^2}^2 + \int_0^t \|H(s,\cdot)\|_{L^4}^4 ds, $$

   再结合插值不等式得到 $\|u_r / r\|_{L^\infty}$ 在时间上的积分估计 (3.1)，这是后续所有估计的基础。

2. 利用 $H$ 的 $L^p$ 守恒律
   由 $(\Omega,H)$-系统中 $H$ 的输运-拉伸方程可知，$H$ 满足守恒律：

   $$ \|H(t,\cdot)\|_{L^p} = \|H_0\|_{L^p}, \quad \forall p \in [2,\infty], $$

   这是一个强有力的先验估计，极大简化了非线性项的处理。例如在估计 $\|h_\theta\|_{L^p}$ 时，只需乘以 $r$ 并利用 $H$ 的守恒即可。

3. 迭代闭合估计策略
   整个证明采用“bootstrap-type”思路：先假设在某个时间区间 $[0,T_*)$ 内 $\|\nabla H\|_{L^\infty} \leq 1$ 成立（见 (3.9)），然后通过一系列嵌套估计（如 $\|u_r/r\|_{L^\infty}$ 控制 $\|\nabla u\|$，进而控制 $\|\nabla H\|$）最终验证该假设在 $T_*$ 内成立，并反推出 $T_*$ 的具体下界。

4. 分层对数尺度分析
   由于目标是获得多重对数型寿命估计，作者必须精细追踪各项增长率。例如在粘性情形中，通过：

   $$ \int_0^t \|\nabla u\|_{L^\infty} ds \lesssim t^{4/7} \|\nabla w\|_{L^2_t L^4}^{6/7} \ll \log(\log(\varepsilon^{-1})), $$

   实现了足够缓慢的增长速率，从而保证 $\|\nabla H(t)\|_{L^\infty}$ 在长时间内仍远小于1。

与之前工作的比较

a) 突破与改进

本文相较于已有文献做出了若干实质性推进，主要体现在以下几个方面：

1. 首次给出精确寿命下界估计
   尽管已有工作如 Chae-Degond-Liu [2]、Dai [5] 建立了局部适定性，Chae-Weng [3] 指出非全局适定性，Li-Yang [18] 给出了单分量正则性准则，但均未对“解能活多久”提供定量估计。本文首次针对轴对称非电阻Hall-MHD系统，给出了明确的寿命下界表达式，填补了该领域的空白。

2. 突破传统小初值限制，仅需小梯度
   经典PDE理论中，长时存在性通常依赖于整体小初值（small data）。而本文仅要求 $\|\nabla H_0\|_{L^\infty} \ll 1$，允许磁场本身幅度很大（如Remark 1.5所示，$M \gg 1$ 也可接受）。这是一种显著放宽的条件，更具物理现实意义——实际等离子体中磁场强度可能很高，但若其空间变化平缓（梯度小），系统仍可能稳定演化较久。

3. 揭示 $H = h_\theta / r$ 的核心角色
   作者延续并深化了 Li-Yang [18] 的思想，系统性地使用 $H$ 和 $\Omega$ 作为基本变量，不仅简化了方程结构，更重要的是突出了 $\nabla H$ 作为“爆破指示器”的地位。这种重构方法为分析Hall-MHD提供了新的视角。

4. 区分粘性与无粘机制，展现耗散效应
   论文分别处理 $\mu > 0$ 与 $\mu = 0$ 情形，发现粘性有助于延长寿命（三重对数 vs 四重对数）。这定量揭示了粘性耗散在抑制非线性增长中的积极作用，体现了数学模型对物理机制的敏感性捕捉。

5. 结合现代PDE工具链完成精细估计
   相比早期仅用能量法的工作，本文综合运用了最大正则性、Commutator估计、插值不等式等多种高级工具，实现了对非线性项的紧致控制。尤其是利用热流最大正则性估计 $\|\nabla w\|_{L^2 L^4}$，避免了直接处理四阶导数，提高了估计效率。

综上所述，本文不仅在技术深度上有所突破，更在问题导向上提出了新方向：不再局限于“是否存在”，而是深入探究“存在多久”，并将答案精确表达为初始梯度的函数，具有重要的理论与应用价值。

3. Robust series linearization of nonlinear advection-diffusion equations

非线性对流-扩散方程的鲁棒级数线性化

作者: T. Forrest Kieffer, Jakob Cupp, John S. Van Dyke, Paraj Titum, Michael L. Wall

类别: math.AP, math-ph, physics.flu-dyn

发布日期: 2025-12-12

arXiv ID: 2512.12019v1

链接: http://arxiv.org/abs/2512.12019v1

详细分析

中文标题: 非线性对流-扩散方程的鲁棒级数线性化

研究对象和背景

本文研究一类非线性偏微分方程（PDE），其形式为对流-扩散过程，并引入一个辅助参数 $ \delta $ 来连续地“变形”从线性到非线性的系统。核心思想是通过构造关于 $ \delta $ 的泰勒级数展开，将原非线性 PDE 转化为一个由无穷多个线性、受迫 PDE 构成的层次系统，从而实现对非线性问题的“线性化解法”。

文章主要关注两类经典非线性模型：

1. 非线性对流（以 Burgers 方程为代表）
2. 非线性扩散（以 p-Laplacian 算子为代表）

一般对流-扩散方程

在引言部分，作者首先给出了对流-扩散方程的一般形式：

其中 $ u: (0, \infty) \times \mathbb{R}^d \to \mathbb{R} $ 是待求解的密度函数，$ j $ 是通量（flux）。初始条件为 $ u(0, x) = g(x) $。

当通量 $ j $ 是 $ u $ 及其梯度的线性函数时，方程退化为线性对流-扩散方程。例如，在一维情况下，若取 $ j_a = vu $（对流通量）和 $ j_d = -\nu \nabla u $（Fick 扩散通量），则总通量 $ j = j_a + j_d $ 导出经典的线性对流-扩散方程：

非线性对流：Burgers 方程的变形

当对流速度与浓度本身相关时，会出现非线性对流。典型例子是Burgers 方程：

文中考虑了一种更一般的“同伦变形”（homotopy deformation），定义如下形式的线性同伦 Burgers 方程（在第5页公式(13)中给出）：

其中 $ \delta \in [0,1] $ 是变形参数。当 $ \delta = 0 $ 时，方程退化为线性对流-扩散；当 $ \delta = 1 $ 时，则得到标准的 Burgers 方程。外部力项 $ f $ 和初值 $ g $ 根据具体问题设定。

作者特别强调了两种边界条件（BCs）下的分析：

• 在无限域 $ I = \mathbb{R} $ 上，要求 $ u(t,x;\delta) \to 0 $ 当 $ |x| \to \infty $。
• 在周期域（如 $ I = [0,1] $）上，采用周期边界条件 $ u(t,0;\delta) = u(t,1;\delta) $。

函数空间方面，初值 $ g $ 通常假设属于 $ L^1(\mathbb{R}) $ 或具有更高正则性（如 $ C^1(S^1) $ 在周期情形下）。Sobolev 空间 $ W^{m,p}(\Omega) $ 也在文中被提及，用于描述解的正则性。

非线性扩散：p-Laplacian 模型

另一类重要模型涉及非线性扩散，其核心算子是 p-Laplacian：

当 $ p=2 $ 时，$ \Delta_2 $ 即为标准的 Laplace 算子。对应的演化方程称为 p-Laplacian 演化方程：

此外，还考虑了稳态的推广 Poisson 方程。

为了应用级数线性化方法，作者同样引入了一个依赖于 $ \delta $ 的变形，定义同伦 p-Laplacian 演化方程（第10页公式(40)）：

其中同伦函数 $ h: [0,\delta_p] \to \mathbb{R} $ 满足 $ h(0)=0 $（对应 $ p=2 $）和 $ h(\delta_p) = p-2 $（对应目标 $ p $ 值）。

该研究的物理背景广泛存在于流体动力学（如 Navier-Stokes 方程的简化模型 Burgers 方程、大涡模拟 LES 中的亚格子尺度模型）、图像处理、最优传输等问题中。传统直接数值模拟（DNS）在处理湍流等多尺度现象时面临巨大计算挑战。本文提出的级数线性化方法旨在提供一种新的分析框架，并为未来利用量子计算等新范式求解非线性 PDE 铺平道路，因为量子算法天然适用于线性系统。

主要定理或主要结果

论文的主要理论成果体现在对 Burgers 方程变形的严格数学分析上。

定理（解析性与无穷收敛半径）

对于线性同伦 Burgers 方程 (HPE) 在无限域 $ \mathbb{R} $ 上且无外力 $ f=0 $ 的情况，设初值 $ g \in L^1(\mathbb{R}) $ 且为实值函数。则存在一个与 Cole-Hopf 变换相关的函数 $ w(t,x;\delta) $，使得原方程的解 $ u(t,x;\delta) $ 可表示为：

其中 $ \mathcal{L}_\delta = (\delta-1)v\partial_x + \frac{1}{R_e} \partial_x^2 $ 是一个线性算子，$ \mathcal{T}g(x) = \int_{-\infty}^x g(y)dy $，$ R_e $ 为雷诺数。

主要结论：映射 $ \delta \mapsto u(t,x;\delta) $ 在 $ \delta = 0 $ 处是全纯的（holomorphic），并且其泰勒级数展开的收敛半径为无穷大，即对任意 $ t > 0 $ 和几乎处处的 $ x \in \mathbb{R} $，有：

这意味着，解 $ u(t,x;\delta) $ 关于 $ \delta $ 的泰勒级数：

在整个复平面 $ \mathbb{C} $ 上收敛，特别地，在 $ \delta = 1 $ 处也收敛，从而可以精确逼近 Burgers 方程的解。

推论（级数解的有效性）

作为上述定理的直接推论，由递推关系生成的函数序列 $ \{u_n\}_{n\geq 0} $ 满足一个线性 PDE 层次系统（见下文），其部分和 $ S_N(t,x;1) = \sum_{n=0}^N u_n(t,x) $ 在 $ N \to \infty $ 时逐点收敛到真实解 $ u(t,x;1) $。

此外，文中通过数值实验验证了以下重要现象：

• 对于Dirac-delta 初值（基本解），级数收敛速度依赖于参数 $ v $ 的选择，选取 $ v = \nu / \ell_0 $（即 $ v = 1/R_e $ 在无量纲形式下）比 $ v = u_0 $ 具有显著更好的收敛性。
• 提出了再馈送（refeeding）技术：在短时间步长内计算部分和，然后将其作为下一时间段的初值重新启动级数展开。该技术极大地改善了长时间模拟的数值稳定性与精度。
• 对于周期域上的受迫 Burgers 方程，使用该级数方法能够准确捕捉到湍流状态下的能量谱标度律 $ E(k) \sim k^{-2} $，与理论预期一致。

对于 p-Laplacian 模型，虽然没有给出类似的严格证明，但通过数值证据表明，所构造的级数展开在 $ p $ 远离 2 的区域（即 $ |p-2| > 1 $）仍然收敛，说明该方法具有非微扰性（non-perturbative），超越了传统的 $ p \approx 2 $ 的小参数近似范围。

研究方法、关键技术和核心工具

a) 核心分析学工具

1. Cole-Hopf 变换：这是处理 Burgers 方程的核心工具。通过变换 $ u = -\frac{2\nu}{\delta} \partial_x \ln w $，将非线性 Burgers 方程转化为关于 $ w $ 的线性热方程。这一变换使得作者能够获得解 $ u(t,x;\delta) $ 的显式表达式，从而可以直接分析其关于参数 $ \delta $ 的解析性质。这是证明无穷收敛半径的关键。

2. 复分析（全纯函数理论）：作者将参数 $ \delta $ 视为复变量 $ \delta \in \mathbb{C} $。通过分析 Cole-Hopf 解表达式中的分子和分母（均为 $ \delta $ 的复合函数），证明它们是 $ \delta $ 的全纯函数，且分母在 $ \delta = 0 $ 附近不为零。这保证了解 $ u(t,x;\delta) $ 在 $ \delta = 0 $ 处是局部全纯的。进一步论证分母在整个复平面无零点，从而得出收敛半径为无穷大的结论。

3. Fourier 分析与谱方法：在处理周期边界条件的问题时，作者大量使用了 Fourier 变换和快速 Fourier 变换（FFT）。通过将 PDE 层次系统转换到频域，每个 $ u_n $ 的方程变为一个带有源项的常微分方程组。这使得可以高效地进行数值求解。Parseval 恒等式也被用来计算动能和能量谱。

4. Duhamel 原理（常数变易法）：对于线性受迫 PDE 层次系统中的高阶项 $ u_n (n \geq 1) $，其解可以用 Duhamel 公式表示为积分形式。这不仅提供了理论上的表示，也为数值积分求解奠定了基础。

5. Sobolev 空间与函数空间理论：文中隐含地使用了 Sobolev 空间 $ W^{m,p}(\Omega) $ 来讨论解的存在性、唯一性和正则性。例如，要求初值 $ g \in L^1(\mathbb{R}) $ 或 $ g \in C^1(S^1) $，是为了确保 Cole-Hopf 变换后的初值足够光滑，从而使后续的线性演化有意义。

b) 技术性难题的解决技巧

1. 非线性项的处理——层次化线性化：这是本文最核心的技术。面对非线性项 $ \delta u \partial_x u $，作者不是直接求解，而是将整个解 $ u(t,x;\delta) $ 展开为 $ \delta $ 的幂级数。通过将这个级数代入原方程并按 $ \delta $ 的幂次匹配系数，巧妙地将一个非线性 PDE 分解为一个线性 PDE 的无穷层次系统。每一层的方程都是线性的，其非齐次项（源项）仅依赖于前几层的解。这使得复杂的非线性问题被转化为一系列可管理的线性问题。

2. 收敛性控制——再馈送（Refeeding）：一个关键的数值观察是，即使理论级数全局收敛，其部分和 $ S_N $ 在固定 $ N $ 下随时间增长而误差增大。为克服此困难，作者提出“再馈送”技巧：不在一个长区间上计算高阶展开，而是在很短的时间步 $ \Delta t $ 内计算低阶（如 $ N=8 $）的部分和 $ S_N(\Delta t, x; 1) $，然后将此结果作为下一个时间步的“新初值”，重新开始级数展开。这种“重置”初值的方法有效抑制了误差累积，实现了长时间高精度模拟。

3. 参数优化：作者发现，辅助参数 $ v $（线性对流速度）的选择对级数收敛速度有显著影响。通过量纲分析，指出 $ v = \nu / \ell_0 $（即 $ v = 1/R_e $）是一个“自然”的选择，并通过数值实验验证了它比 $ v = u_0 $ 能带来更快的收敛。

4. 同伦路径的设计：对于 p-Laplacian 模型，作者考虑了“普通”和“对偶”两种不同的变形路径 $ h(\delta) $。通过比较不同路径下的收敛行为，揭示了变形方式的选择直接影响级数的收敛速率和收敛半径。这为实际应用中如何设计最优变形提供了指导。

与之前工作的比较

a) 突破与改进

本文在非线性 PDE 的级数求解方法上做出了显著的突破，主要体现在以下几个方面：

1. 从微扰到非微扰：以往许多基于级数展开的方法（如摄动理论）本质上是微扰的（perturbative），要求非线性项非常小（即 $ \delta \ll 1 $）。而本文通过对 Burgers 方程的严格证明，首次确立了所构造的级数具有无穷收敛半径。这意味着该方法是非微扰的（non-perturbative），可以在 $ \delta = 1 $（完全非线性）处精确使用，极大地扩展了级数方法的适用范围。

2. 严格的数学基础：尽管类似的思想（如 Homotopy 分析法）早已存在，但本文为特定变形下的 Burgers 方程提供了严格的数学证明，证明了级数的全局收敛性。这为该类方法的应用提供了坚实的理论保障，使其从一种启发式技巧提升为一种可靠的数学工具。

3. 放宽对初值的限制：与许多需要高正则性初值（如 $ C^\infty $）的分析方法不同，本文的证明仅要求初值 $ g \in L^1(\mathbb{R}) $，这是一个相对宽松的条件。这使得该方法可以应用于更广泛的物理场景，包括具有奇异性（如 Dirac delta）的初值。

4. 超越 Burgers 方程的普适性：本文的工作不仅限于 Burgers 方程。它将同样的级数线性化框架成功应用于更复杂、更少被研究的非线性扩散模型（p-Laplacian）。通过数值探索，展示了该方法在 $ p $ 值远离 2 时依然有效，这为研究强非线性扩散现象提供了新途径。

5. 面向未来的计算范式：本文明确指出了该方法在量子计算领域的潜在应用价值。由于每一步都是求解线性 PDE，可以通过离散化转化为线性方程组，进而可能利用 HHL 等量子算法进行加速。这为解决传统计算难以处理的大规模非线性 PDE 问题开辟了全新的可能性。

综上所述，本文通过严谨的数学分析和创新的数值技术，发展了一种强大而鲁棒的级数线性化方法，不仅在理论上证明了其对经典非线性方程的有效性，而且展示了其在更广泛模型和未来计算技术中的巨大潜力。

了解领域论文（摘要翻译）

1. Well-posedness of multidimensional nonlocal conservation laws with nonlinear mobility and bounded force

具有非线性迁移率和有界外力的多维非局部守恒律的适定性

作者: Antonin Chodron de Courcel

类别: math.AP, math-ph

发布日期: 2025-12-15

arXiv ID: 2512.13535v1

链接: http://arxiv.org/abs/2512.13535v1

摘要翻译

中文标题: 具有非线性迁移率和有界外力的多维非局部守恒律的适定性

摘要翻译: 本文在较弱的核函数假设下，建立了具有非线性迁移率的非局部守恒律在多个空间维度上的局部时间存在性和唯一性结果，其中核函数被假定为有界且具有有限的全变差。与线性迁移率情形不同，即使核函数光滑，解仍可能在有限时间内产生激波。我们通过粘性消失法构造了熵解，并为此逼近格式提供了收敛速率估计。

2. Quasi invariant Gaussian measures for the nonlinear Schrödinger equation on $\mathbb T^2$

二维环面上非线性薛定谔方程的拟不变高斯测度

作者: Leonardo Tolomeo, Nicola Visciglia

类别: math.AP

发布日期: 2025-12-15

arXiv ID: 2512.13113v1

链接: http://arxiv.org/abs/2512.13113v1

摘要翻译

中文标题: 二维环面上非线性薛定谔方程的拟不变高斯测度

摘要翻译: 本文研究在定义于 $\mathbb{T}^2$ 上的二维散焦非线性薛定谔方程 $i \partial_t u + \Delta u = |u|^{2k} u$ 的流作用下，高斯测度的输运性质。特别地，我们证明了具有协方差逆算子为 $\|u\|_{H^s}^2$ 的高斯测度在流的作用下当 $s>2$ 时是拟不变的。此外，我们还证明了相应的Radon-Nikodym密度在空间上局部属于每个 $L^p$ 空间。证明依赖于文献[52]中引入的物理空间能量方法，以及一种新的抽象拟不变性论证，该方法使我们能够将流上的时空估计与测度支撑上的概率界相结合。

3. Almost sure global nonlinear smoothing for the 2D NLS

二维非线性薛定谔方程的几乎必然全局非线性光滑性

作者: Chenmin Sun, Nikolay Tzvetkov

类别: math.AP, math.PR

发布日期: 2025-12-15

arXiv ID: 2512.13088v1

链接: http://arxiv.org/abs/2512.13088v1

摘要翻译

中文标题: 二维非线性薛定谔方程的几乎必然全局非线性光滑性

摘要翻译: 本文证明了在二维环面上非线性薛定谔方程（NLS）具有几乎必然意义下关于时间的全局非线性光滑效应。对于确定性初值，这一现象早先由 Erdoğan 和 Tzirakis 在圆周上的 NLS 情形下得到证明，但在多维环面上该性质至今仍属未知。我们的论证基于协方差算子为 $(1-\Delta)^{-s}$（其中 $s>2$）的高斯测度的定量拟不变性。

4. Hyperbolic equations with fifth-order symmetries

具有五阶对称性的双曲型方程

作者: Rustem N. Garifullin

类别: math.AP, nlin.SI

发布日期: 2025-12-14

arXiv ID: 2512.12789v1

链接: http://arxiv.org/abs/2512.12789v1

摘要翻译

中文标题: 具有五阶对称性的双曲型方程

摘要翻译: 本文研究了双曲型方程的分类问题。我们考察一类形如

的方程，其中 $u(x,y)$ 是未知函数，$x,y$ 是自变量。该分类基于方程具有高阶五阶对称性的要求。结果得到了满足该条件的四个方程的列表。

5. Estimates for the wave equation on $β$-dimensional spaces of measures

关于作用在 $β$ 维测度空间上的波动方程的估计

作者: Riju Basak, Daniel Spector

类别: math.AP, math.FA

发布日期: 2025-12-14

arXiv ID: 2512.12618v1

链接: http://arxiv.org/abs/2512.12618v1

摘要翻译

中文标题: 关于作用在 $β$ 维测度空间上的波动方程的估计

摘要翻译: 本文建立了作用在 $β$ 维稳定测度空间上的波多乘子的 Miyachi-Peral 型固定时间估计。我们的估计结果对已知的 Hardy 空间估计进行了改进。由此类估计出发，我们进一步推导出具有测度初值数据的波动方程相应的估计结果。

6. Local Well-Posedness of the Motion of Inviscid Liquid Crystals with a Free Surface Boundary

具有自由边界的无粘性液晶运动的局部适定性

作者: Chenyun Luo, Hang Yu

类别: math.AP

发布日期: 2025-12-13

arXiv ID: 2512.12150v1

链接: http://arxiv.org/abs/2512.12150v1

摘要翻译

中文标题: 具有自由边界的无粘性液晶运动的局部适定性

摘要翻译: 本文证明了在拉格朗日坐标下，描述存在表面张力时无粘性向列型液晶运动的自由边界Lin-Liu方程组的局部适定性。众所周知，在涉及无粘性流体方程的自由边界问题中，仅凭先验能量估计不足以建立局部解的存在性，其主要原因在于线性化方程中对称性的丧失。主要困难在于构造一个有效的近似方程组，使其在渐近意义上与拉格朗日坐标下的自由边界Lin-Liu模型保持一致。该近似系统必须能够准确刻画流体运动与内部调和热流之间的耦合作用，同时充分反映移动边界的正则性。

不相关论文（基本信息）

1. Evolution equation with fractional Schrödinger operators: monotonicity and exponential decay of solutions in Morrey spaces

具有分数阶薛定谔算子的演化方程：Morrey空间中解的单调性与指数衰减

作者: Jan W. Cholewa, Anibal Rodriguez-Bernal

类别: math.AP

发布日期: 2025-12-15

arXiv ID: 2512.13499v1

链接: http://arxiv.org/abs/2512.13499v1

摘要: We consider evolution equation with fractional Schrödinger operators in Morrey spaces. We prove order preserving properties of the associated semigroup in Morrey scale. We prove monotonicity of the semigroup with respect to Morrey’s potentials and give some precise estimates of its exponential growth. We show that Arendt and Batty’s type condition on the potential is necessary for exponential decay of Morrey’s norms of the semigroup and find a large class of dissipative potentials for which it is also sufficient.

2. Estimating parameters of the diffusion model via asymptotic expansions

通过渐近展开估计扩散模型的参数

作者: Konstantinos Kalimeris, Leonidas Mindrinos

类别: math.AP, math-ph

发布日期: 2025-12-15

arXiv ID: 2512.13419v1

链接: http://arxiv.org/abs/2512.13419v1

摘要: A broad class of inverse problems deals with determining certain parameters, from measurement data, in models which are associated to certain partial differential equations. In this work we focus on the heat equation on a finite interval and we determine the dimensionless diffusion parameter from a single measurement. Our results extend to estimating additional parameters of the initial-boundary value problem, such as the length of the interval and/or the time required for the solution to achieve a specific state. Our approach relies on the asymptotic solution of an integral equation: The formulation of this integral equation is based on the solution of the direct problem via the Fokas method; the solution of this equation is achieved through the asymptotic evaluation of the associated integrals which yield an effective approximate solution, supported by numerical verifications. We apply these approximations to well-established problems in soil science and we compare our results with existing ones, displaying clear improvement.

3. Fractional and Integer Order Sobolev Spaces for Compact Metric Graphs

紧致度量图的分数阶与整数阶索博列夫空间

作者: Elsiddig Awadelkarim, David Bolin, Alexandre B. Simas

类别: math.AP, math.PR

发布日期: 2025-12-15

arXiv ID: 2512.13308v1

链接: http://arxiv.org/abs/2512.13308v1

摘要: Given a compact metric graph $Γ$ and the Laplacian $Δ_Γ$ coupled with standard (Kirchhoff) vertex conditions, solutions to fractional elliptic partial differential equations of the form $(κ^2 - Δ_Γ)^{α/2}u=f$ on $Γ$ exhibit a distinctive regularity structure: even-order derivatives are continuous across vertices, while odd-order derivatives may be discontinuous. This non-standard smoothness property precludes the direct application of classical tools from real functional analysis. Because of this, we introduce and systematically study new families of Sobolev spaces tailored to this setting. We define these spaces, denoted $W^{α,p}(Γ)$ and $H^α(Γ)$, to respect the continuity constraints on even-order derivatives at vertices, while permitting discontinuities in odd-order derivatives. We establish their fundamental properties, including characterizations, embedding theorems into Hölder and Lebesgue spaces, and compactness results. A central contribution in this investigation is the derivation of uniform bounds on the supremum norm of eigenfunctions for a class of Laplacians on metric graphs, a result of independent interest. Finally, we demonstrate that these spaces provide a natural framework for analyzing the regularity of solutions to fractional elliptic PDEs and SPDEs driven by Gaussian white noise on metric graphs, in particular, establishing a general characterization of the domain of the fractional powers of $(κ^2-Δ_Γ)$ and $(κ^2-\nabla(a\nabla))$ in terms of the Sobolev spaces we introduce, thereby extending all previously known characterizations in the literature, and improving the regularity results previously obtained to their sharp counterparts (with general fractional powers). We also show that these spaces are fundamental to the characterization of Gaussian free fields on metric graphs.

4. Generic regularity and Lipschitz metric for a two-component Novikov system

一类双组分Novikov系统的通有正则性与Lipschitz度量

作者: Kenneth H. Karlsen, Yan Rybalko

类别: math.AP

发布日期: 2025-12-15

arXiv ID: 2512.13305v1

链接: http://arxiv.org/abs/2512.13305v1

摘要: We investigate the Cauchy problem for a two-component generalization of the Novikov equation with cubic nonlinearity – an integrable system whose solutions may develop strong nonlinear phenomena such as gradient blow-up and interactions between peakon-like structures. Our study has two main objectives: first, to analyze the generic regularity of global conservative solutions; and second, to construct a new metric that guarantees the Lipschitz continuity of the flow. Building on the geometric framework developed by Bressan and Chen for quasilinear second-order wave equations, we prove that the solution retains $C^k$ regularity away from a finite number of piecewise $C^{k-1}$ characteristic curves. Furthermore, we provide a description of the solution behavior in the vicinity of these curves. By introducing a Finsler norm on tangent vectors in the space of solutions, expressed in the transformed Bressan-Constantin variables, we introduce a Lipschitz metric representing the minimal energy transportation cost between two solutions.

5. A non-local quenching system with coupled and uncoupled singular absorption terms

具有耦合与非耦合奇异吸收项的非局部熄灭系统

作者: Sergio Junquera

类别: math.AP

发布日期: 2025-12-15

arXiv ID: 2512.13291v1

链接: http://arxiv.org/abs/2512.13291v1

摘要: In this paper we study a non-local diffusion system of two equations with both coupled and uncoupled singular absorption terms of the type $u^{-p}$. We prove that there exist necessary and sufficient conditions for the existence of both stationary and quenching solutions. We also characterize in terms of the exponents of the absorption terms when the quenching is simultaneous or non-simultaneous and obtain the quenching rates.

6. Stability and Regularization of Quasi-Variational Inequalities under Monotone Operator Perturbations

单调算子扰动下拟变分不等式的稳定性与正则化

作者: M. H. M. Rashid

类别: math.FA, math.AP

发布日期: 2025-12-15

arXiv ID: 2512.13275v1

链接: http://arxiv.org/abs/2512.13275v1

摘要: This paper establishes comprehensive stability results for quasi-variational inequalities (QVIs) under monotone perturbations of the governing operator. We prove strong convergence of both minimal and maximal solutions when sequences of operators converge pointwise while preserving fundamental properties including homogeneity, strong monotonicity, Lipschitz continuity, and T-monotonicity. Our analysis extends to regularization techniques, finite-dimensional approximations, and non-monotone nonlinearities, providing explicit convergence rates under appropriate conditions. The theory encompasses applications to $p$-Laplacian operators, elliptic regularizations, and optimal control problems with QVI constraints. By developing a unified framework that bridges classical monotone operator theory with contemporary computational challenges, this work provides essential mathematical foundations for the robust numerical approximation and sensitivity analysis of quasi-variational problems arising in materials science, physics, and engineering applications.

7. The principal eigenvalue of an age-structured operator with diffusion and advection: qualitative analysis and an application

具有扩散与对流的年龄结构算子的主特征值：定性分析及其应用

作者: Hao Kang, Rui Peng, Maolin Zhou

类别: math.AP

发布日期: 2025-12-15

arXiv ID: 2512.13234v1

链接: http://arxiv.org/abs/2512.13234v1

摘要: In this paper, we investigate an eigenvalue problem associated with an age-structured operator incorporating random diffusion and advection. Our primary focus is on examining the asymptotic behaviors of the principal eigenvalue with respect to large advection and small or large diffusion rates. We subsequently apply these results to a nonlinear age-structured model, providing a better understanding of how diffusion and advection influence the spatial distribution of species. Among other ingredients, our approach involves constructing various types of super- and sub-solutions to tackle the novel challenges posed by the nonlocal terms in the problems under consideration.

8. An abstract framework for a class of nonlocal structured population models: existence, uniqueness and stability of steady states

一类非局部结构种群模型的抽象框架：稳态解的存在性、唯一性与稳定性

作者: Jérôme Coville, Léo Girardin

类别: math.AP, math.DS

发布日期: 2025-12-15

arXiv ID: 2512.13232v1

链接: http://arxiv.org/abs/2512.13232v1

摘要: This paper is concerned with the study of a class of nonlinear nonlocal functional evolution problems defined in an abstract Banach algebra. We introduce an abstract functional setting that encompasses a wide range of structured population models appearing in biomathematical literature. Within this framework, we analyze the well-posedness of the Cauchy problem and the existence of stationary solutions in the positive cone of the Banach algebra. By reviewing a large number of approaches, we also derive conditions for the local and global stability of these stationary solutions. Additionally, we explore the limits of these conditions by exhibiting explicit counterexamples. In particular, for mutation–selection models with symmetric mutation operators, we uncover both sufficient conditions for existence, uniqueness and stability, and counterexamples to existence or stability.

9. Rigidity of weighted manifolds via classification results for semilinear equations

通过半线性方程的分类结果研究加权流形的刚性

作者: Giulio Ciraolo, Alberto Farina, Troy Petitt

类别: math.AP, math.DG

发布日期: 2025-12-15

arXiv ID: 2512.13181v1

链接: http://arxiv.org/abs/2512.13181v1

摘要: We study model semilinear equations on complete and non-compact weighted Riemannian manifolds with non-negative Bakry-Émery Ricci curvature. Our main goal is to classify positive solutions of the equation at the Sobolev-critical exponent, and furthermore to prove that the existence of such solutions implies rigidity of the manifold and triviality of the weight. This is possible when the weighted manifold has non-negative finite dimensional Bakry-Émery Ricci curvature, and even under the weaker condition of non-negative infinite dimensional Bakry-Émery Ricci curvature, up to imposing some additional conditions in the latter case. To exhibit the sharpness of these additional conditions, we construct a non-trivial positive solution of the critical problem on a weighted manifold with positive infinite dimensional curvature. We also obtain a corresponding rigidity result for solutions of the Liouville equation on weighted Riemannian surfaces. Finally, we prove some non-existence theorems when the nonlinearity is sub-critical or simply under certain volume growth conditions. In particular, the latter rules out all positive solutions on shrinking gradient Ricci solitons.

10. PFEM-GP-dpHs : a finite element framework for combining Gaussian processes and infinite-dimensional port-Hamiltonian systems

PFEM-GP-dpHs：一种结合高斯过程与无限维端口-哈密顿系统的有限元框架

作者: Florian Courteville, Iain Henderson, Denis Matignon, Sylvain Dubreuil

类别: math.AP

发布日期: 2025-12-15

arXiv ID: 2512.13163v1

链接: http://arxiv.org/abs/2512.13163v1

摘要: In order to learn distributed port-Hamiltonian systems (dpHs) using Gaussian processes (GPs), the partitioned finite element method (PFEM) is combined with the Gp-dpHs method. By following a late lumping approach, the discretization of the functional hyperparameters of the GP prior over the Hamiltonian functional is chosen independently from the discretization of the dpHs, thus reducing the numerical complexity of our method. We next model the mean of the GP prior of the Hamiltonian as a quadratic form, enabling the GP kernel to focus on the nonlinear part of a given dpHs. We illustrate our method on a nonlinear one dimensional wave equation with unknown physical parameters (tension and linear mass).

11. Convection Effects and Optimal Insulation: Modelling and Analysis

对流效应与最优隔热：建模与分析

作者: Harbir Antil, Alex Kaltenbach, Keegan L. A. Kirk

类别: math.AP, math.OC

发布日期: 2025-12-15

arXiv ID: 2512.13098v1

链接: http://arxiv.org/abs/2512.13098v1

摘要: In this paper, we study an insulation problem that seeks to determine the optimal distribution of a given amount $m>0$ of insulating material coating an insulated boundary part $Γ_I\subseteq \partialΩ$ of a thermally conducting body $Ω\subseteq \mathbb{R}^d$, $d\in \mathbb{N}$, subject to convective heat transfer. The `$\textit{thickness}$’ of the insulating layer $Σ_{I}^{\varepsilon}\subseteq \mathbb{R}^d$ is given locally via $\varepsilon \mathtt{d}$, where $\varepsilon>0$ denotes the (arbitrarily small) conductivity and $\mathtt{d}\colon Γ_{I}\to [0,+\infty)$ the (to be determined) distribution of the insulating material. Then, the physical process is modelled by the stationary heat equation in the insulated thermally conducting body $Ω_{I}^{\varepsilon}:= Ω\cupΣ_{I}^{\varepsilon}$ with Robin-type boundary conditions on the interacting insulation boundary $Γ_I^{\varepsilon}\subseteq \partialΩ_{I}^{\varepsilon}$ (reflecting convective heat transfer between the thermally conducting body $Ω$ and its surrounding medium) as well as Dirichlet and Neumann boundary conditions at the remaining boundary parts, $\textit{i.e.}$, $\partialΩ_{I}^{\varepsilon}\setminus Γ_I^{\varepsilon}$. More precisely, we establish $Γ(L^2(\mathbb{R}^d))$-convergence of the heat loss formulation (as ${\varepsilon \to 0^+}$), in the case that the thermally conducting body $Ω$ is a bounded Lipschitz domain having a $C^{1,1}$-regular or piece-wise flat insulated boundary $Γ_I$.

12. Stability analysis for a plate-membrane system without geometric condition

无几何条件的板-膜系统稳定性分析

作者: Bienvenido Barraza Martínez, Robert Denk, Jonathan González Ospino, Jairo Hernández Monzón

类别: math.AP

发布日期: 2025-12-15

arXiv ID: 2512.13087v1

链接: http://arxiv.org/abs/2512.13087v1

摘要: In this work, we consider a transmission problem describing a thermoelastic plate surrounding a membrane without any mechanical damping. The main results consist of the lack of exponential stability for this problem and the polynomial stability without the usual geometric condition.

13. Local controllability in finite time and the controllable time of the Korteweg-De Vries equation using the right Neumann controls

利用右端诺伊曼控制实现有限时间内的局部可控性及Korteweg-de Vries方程的可控时间

作者: Hoai-Minh Nguyen

类别: math.OC, math.AP

发布日期: 2025-12-15

arXiv ID: 2512.13066v1

链接: http://arxiv.org/abs/2512.13066v1

摘要: We investigate the local boundary controllability of the Korteweg-de Vries (KdV) equation with right Neumann boundary controls at critical lengths. We show that the KdV system is not locally null-controllable in small time for all critical lengths for which the unreachable subspace of the linearized system has dimension at least two. This result extends the work of Coron, Koenig, and Nguyen, who established it for a subclass of these lengths. We also obtain a new controllability time for such systems for all but two critical lengths. It is worth noting that the latest results on the controllability time prior to this work date back to the work of Cerpa (2007) and Cerpa and Crepeau (2009).

14. Conservation laws of nonlinear PDEs arising in elasticity and acoustics in Cartesian, cylindrical, and spherical geometries

源自弹性力学和声学中笛卡尔、柱面与球面几何下非线性偏微分方程的守恒律

作者: Willy Hereman, Rehana Naz

类别: math.AP, math-ph, nlin.SI

发布日期: 2025-12-15

arXiv ID: 2512.13062v1

链接: http://arxiv.org/abs/2512.13062v1

摘要: Conservation laws are computed for various nonlinear partial differential equations that arise in elasticity and acoustics. Using a scaling homogeneity approach, conservation laws are established for two models describing shear wave propagation in a circular cylinder and a cylindrical annulus. Next, using the multiplier method, conservation laws are derived for a parameterized system of constitutive equations in cylindrical coordinates involving a general expression for the Cauchy stress. Conservation laws for the Khokhlov-Zabolotskaya-Kuznetsov equation and Westervelt-type equations in various coordinate systems are also presented.

15. Heat kernel estimates for Markov processes in bounded sets with jump kernels decaying at the boundary

具有在边界衰减的跳跃核的有界集内马尔可夫过程的热核估计

作者: Soobin Cho, Panki Kim, Renming Song, Zoran Vondraček

类别: math.PR, math.AP

发布日期: 2025-12-15

arXiv ID: 2512.12991v1

链接: http://arxiv.org/abs/2512.12991v1

摘要: In this paper, we study two types of purely discontinuous symmetric Markov processes $X$ in bounded smooth subsets of $\mathbb R^d$: conservative processes and processes killed either upon approaching the boundary of the set or by a killing potential $κ$. The jump kernel of $X$ is of the form $J(x,y)={\cal B}(x,y)|x-y|^{-d-α}$, $α\in (0,2)$, where the function ${\cal B}(x,y)$ decays to 0 at the boundary and is described in terms of two $O$-regularly varying functions and one slowly varying function. Under the conditions, introduced in \cite{CKSV24}, on ${\cal B}(x,y)$ and on the killing potential $κ$, we establish sharp two-sided estimates on the heat kernel of $X$: in Lipschitz sets when $X$ is conservative, and in $C^{1,1}$ open sets for the killed process.

16. On the variational dual formulation of the Nash system and an adaptive convex gradient-flow approach to nonlinear PDEs

关于Nash系统的变分对偶表述及非线性偏微分方程的自适应凸梯度流方法

作者: Dmitry Vorotnikov, Amit Acharya

类别: math.AP, math.NA, math.OC

发布日期: 2025-12-14

arXiv ID: 2512.12878v1

链接: http://arxiv.org/abs/2512.12878v1

摘要: We investigate the influence of base states on the consistency of the dual variational formulation for quadratic systems of PDEs, which are not necessarily conservative (typical examples include the noise-free Nash system with a quadratic Hamiltonian and multiple players). We identify a sufficient condition under which consistency holds over large time intervals. In particular, in the single-player case, there exists a sequence of base states (each exhibiting full consistency) that converges in mean to zero. We also prove existence of variational dual solutions to the noise-free Nash system for arbitrary base states. Furthermore, we propose a scheme based on Hilbertian gradient flows that, starting from an arbitrary base state, generates a sequence of new base states that is expected to converge to a solution of the original PDE.

17. Ergodicity for the Dean–Kawasaki Equation with Dirichlet Boundary Conditions: Taming the Square-Root

具有Dirichlet边界条件的Dean–Kawasaki方程的遍历性：驯服平方根项

作者: Shyam Popat, Zhengyan Wu

类别: math.PR, math.AP

发布日期: 2025-12-14

arXiv ID: 2512.12861v1

链接: http://arxiv.org/abs/2512.12861v1

摘要: In this paper, we establish the ergodicity of generalized Dean–Kawasaki equations with correlated noise and Dirichlet boundary conditions. In contrast to the ergodicity results of Fehrman, Gess, and Gvalani arXiv:2206.14789, our analysis accommodates irregular, square-root type noise coefficients. For such irregular coefficients, we prove that the law of the classical Dean–Kawasaki equation converges exponentially fast to equilibrium, while for the porous medium type Dean–Kawasaki equation, the convergence occurs at a polynomial rate. Furthermore, we obtain a regularization by noise effect, showing that the polynomial convergence rate improves to an exponential one whenever the noise coefficient is sufficiently regular, including the case of conservative multiplicative linear noise. Our approach relies on establishing a supercontraction property in a suitably weighted Lebesgue space, achieved through a refined doubling of variables argument. The construction of the weight function crucially exploits the specific structure of the Dean–Kawasaki-type correlated noise.

18. A bilinear pointwise tracking optimal control problem for a semilinear elliptic PDE

一类半线性椭圆偏微分方程的双线性逐点追踪最优控制问题

作者: Enrique Otarola, Daniel Quero, Matias Sasso

类别: math.OC, math.AP

发布日期: 2025-12-14

arXiv ID: 2512.12854v1

链接: http://arxiv.org/abs/2512.12854v1

摘要: We consider a bilinear optimal control problem with pointwise tracking for a semilinear elliptic PDE in two and three dimensions. The control variable enters the PDE as a (reaction) coefficient and the cost functional contains point evaluations of the state variable. These point evaluations lead to an adjoint problem with a linear combination of Dirac measures as a forcing term. In Lipschitz domains, we derive the existence of optimal solutions and analyze first and necessary and sufficient second order optimality conditions. We also prove that every locally optimal control $\bar u$ belongs to $H^1(Ω)$. Finally, assuming that the domain $Ω\subset \mathbb{R}^2$ is a convex polygon, we prove that $\bar u \in C^{0,1}(\bar Ω)$.

19. Directional Spectral Analysis: Dimension Reduction for Periodic Elliptic Operators

方向谱分析：周期性椭圆算子的降维方法

作者: Ruming Zhang

类别: math.AP, math.SP

发布日期: 2025-12-14

arXiv ID: 2512.12848v1

链接: http://arxiv.org/abs/2512.12848v1

摘要: The study of the limiting absorption principle for elliptic equations with periodic structures is very challenging when the dimension is greater than 1. The fundamental reason for the dimensional barrier is the mismatch between directional physical reality and the direction-independent classic spectral analysis. In this paper, we introduce a new approach which introduce the direction into classic spectral analysis. With the new approach, the solution obtained by the limiting absorption principle can be formulated in a semi-analytic form, which not only gives an explicit representation of the solutions, but also reflects the phenomenon in physics. The new approach resolves the mismatch between mathematics and physics, and also breaks the dimensional barriers. It also opens a door to a lot of further possibilities, ranging from the analysis of solutions and numerical simulations for the solutions.

20. Structure versus regularity of set-valued maps in convex generalized Nash equilibrium problems in Banach spaces

Banach空间中凸广义纳什均衡问题下集值映射的结构与正则性

作者: Marcelo Bongarti, Michael Hintermüller

类别: math.OC, math.AP

发布日期: 2025-12-14

arXiv ID: 2512.12831v1

链接: http://arxiv.org/abs/2512.12831v1

摘要: A generalized Nash equilibrium problem (GNEP) in Banach space consists of $N>1$ optimal control problems with couplings in both the objective functions and, most importantly, in the feasible sets. We address the existence of equilibria for convex GNEPs in Banach space. We show that the standard assumption of lower semicontinuity of the set-valued constraint maps - foundational in the current literature on GNEPs - can be replaced by graph convexity or the so-called Knaster-Kuratowski-Mazurkiewicz (KKM) property. Lower semicontinuity is often essential for obtaining upper semicontinuity of best response maps, crucial for the existence theory based on Kakutani-Fan fixed-point arguments. However, in function spaces or PDE-constrained settings, verifying lower semicontinuity becomes much more challenging (even in convex cases), whereas graph convexity, for example, is often straightforward to check. Our results unify several existence theorems in the literature and clarify the structural role of constraint maps. We also extend Rosen’s uniqueness condition to Banach spaces using a multiplier bias framework. Additionally, we present a geometric counterpart to our analytic framework using preference maps. This geometric is intended as a complement to, rather than a replacement for, the analytic theory developed in the main body of the paper.

21. Newton Methods for Mean Field Games: A Numerical Study

均值场博弈的牛顿方法：一项数值研究

作者: Elisabetta Carlini, Ahmad Zorkot

类别: math.NA, math.AP

发布日期: 2025-12-14

arXiv ID: 2512.12752v1

链接: http://arxiv.org/abs/2512.12752v1

摘要: We address the numerical solution of second-order Mean Field Game problems through Newton iterations in infinite dimensions, introduced in [14], where quadratic convergence of the method was rigorously established. Building upon this theoretical framework, we develop new numerical discretization techniques, including both a finite difference and a semi-Lagrangian scheme, that enable an effective computational implementation of the infinite-dimensional iterations. The proposed methods are tested on several benchmark problems, and the resulting numerical experiments demonstrate their robustness, accuracy, and efficiency. A comparative analysis between the two schemes and existing approaches from the literature is also presented, highlighting the potential of Newton-based solvers for MFG systems.

22. Sharp Time Estimates for Some Nonlocal Evolution Equations

某些非局部发展方程的精确时间估计

作者: Xiaolei Yang

类别: math.AP

发布日期: 2025-12-14

arXiv ID: 2512.12700v1

链接: http://arxiv.org/abs/2512.12700v1

摘要: In this paper, we consider the Cauchy problem for the nonlocal evolution equation $u_t = d(J*u) - u$ in $\R$. We present a new method to prove that $e^{(d-1)t}$ is the sharp time bound.

23. Poisson Kernels and Hilbert Transforms for Trigonometric Heckman-Opdam Polynomials of type $A_1$

$A_1$型三角Heckman-Opdam多项式的Poisson核与Hilbert变换

作者: B. Amri, A. Guesmi

类别: math.CA, math.AP

发布日期: 2025-12-14

arXiv ID: 2512.12659v1

链接: http://arxiv.org/abs/2512.12659v1

摘要: In this paper, we investigate the trigonometric Heckman-Opdam polynomials of type $A_1$. We establish connections with ultraspherical polynomials and derive an explicit expression for the associated Poisson kernel. Using the product formula, we introduce a natural convolution structure and develop a theory of fractional integrals associated with these polynomials. We also define a generalized Hilbert transform in the framework of the Cherednik operator and prove its boundedness on $L^p$-spaces. This work provides an alternative perspective on the approach of B. Muckenhoupt and E.M. Stein \cite{MS}.

24. Uniformity of Maximal Hypoellipticity on Graded Lie Groups: From Pointwise to Global

分次李群上极大亚椭圆性的统一性：从逐点到整体

作者: Shiqi Liu, Edward McDonald, Fedor Sukochev, Dmitriy Zanin

类别: math.AP, math.FA

发布日期: 2025-12-14

arXiv ID: 2512.12646v1

链接: http://arxiv.org/abs/2512.12646v1

摘要: On graded Lie groups, we develop a mechanism that transfers the uniformity of maximal hypoellipcity from the frozen coefficients principal part of a differential operator to the full operator. Our approach brings the century-old “freeze-unfreeze” strategy into the hypoelliptic setting, and offers a transparent and flexible framework for lifting symbol-level hypoelliptic properties to global elliptic estimates, without relying on pseudodifferential calculus. In addition, we prove that symmetric operators of hypoelliptic type on a graded Lie group are self-adjoint.

25. Existence and nonexistence results for a nonlocal isoperimetric problem on $\mathbb{H}^n$

关于 $\mathbb{H}^n$ 上一类非局部等周问题的存在性与非存在性结果

作者: Haizhong Li, Bo Yang

类别: math.AP, math-ph, math.DG

发布日期: 2025-12-14

arXiv ID: 2512.12621v1

链接: http://arxiv.org/abs/2512.12621v1

摘要: In Euclidean space $\mathbb{R}^n$, the minimization problem of a nonlocal isoperimetric functional with a competition between perimeter and a nonlocal term derived from the negative power of the distance function, has been extensively studied. In this paper, we investigate this nonlocal isoperimetric problem in hyperbolic space $\mathbb{H}^n$, we prove that the geodesic balls are unique minimizers (up to hyperbolic isometries) for small volumes $m$ and obtain nonexistence results for large volumes $m$ under certain ranges of the exponent in the nonlocal term.

26. Parabolic Equations with Singular Coefficients and Boundary Data: Analysis and Numerical Simulations

具有奇异系数和边界数据的抛物方程：分析与数值模拟

作者: Arshyn Altybay, Alibek Yeskermessuly

类别: math.AP

发布日期: 2025-12-14

arXiv ID: 2512.12612v1

链接: http://arxiv.org/abs/2512.12612v1

摘要: We investigate linear parabolic equations in divergence form with singular coefficients and non-smooth boundary data. When the diffusion, drift, or potential terms, as well as the initial or boundary conditions, are distributions rather than functions, classical and weak solution concepts become inadequate due to the ill-posedness of products involving distributions. To overcome this, we introduce a framework of very weak solutions based on regularization techniques and the theory of moderate nets. Existence of very weak solutions is established under minimal regularity assumptions. We further prove consistency with classical solutions when the data are smooth and demonstrate uniqueness via negligibility arguments. Finally, we present numerical computations that illustrate the robustness of the very weak solution framework in handling highly singular inputs, including delta-type potentials and distributional boundary traces.

27. The quermassintegral inequalities for horo-convex domains in the sphere

球面中关于horo-凸区域的混合体积不等式

作者: Shujing Pan, Julian Scheuer

类别: math.DG, math.AP

发布日期: 2025-12-14

arXiv ID: 2512.12565v1

链接: http://arxiv.org/abs/2512.12565v1

摘要: We study a new notion of convexity for subsets of the unit sphere, which closely resembles the horo-convexity for subsets of the hyperbolic space. We call this notion, accordingly, horo-convexity. For horo-convex hypersurfaces of the unit sphere, we prove the smooth convergence of the classical Guan/Li flow of inverse type and use this result to prove the full set of quermassintegral inequalities for horo-convex hypersurfaces of the unit sphere.

28. Global controllability of the Cahn-Hilliard equation

Cahn-Hilliard方程的整体可控性

作者: Víctor Hernández-Santamaría, Subrata Majumdar, Luz de Teresa

类别: math.AP

发布日期: 2025-12-14

arXiv ID: 2512.12562v1

链接: http://arxiv.org/abs/2512.12562v1

摘要: This paper deals with the global control properties of the Cahn-Hilliard equation posed on the $d$-dimensional flat torus $\mathbb{T}^d$. We first prove that the system is small-time globally approximately controllable using a control supported on finitely many Fourier modes, following an approach based on geometric control theory techniques. Next, we show that the corresponding linearized problem is null-controllable with a localized control, where the control region is an arbitrary measurable set of positive Lebesgue measure. Our analysis is based on quantitative propagation of smallness estimates for the free solution. Furthermore, when $d\in \{1,2,3\},$ we ensure the local null controllability for the nonlinear system via a fixed-point argument. Finally, by combining these two results, we establish the global null controllability of the Cahn-Hilliard equation. In the first phase, the control is localized in Fourier modes, whereas in the second phase, it is spatially localized on a set of positive Lebesgue measure.

29. Steady Solutions to the Relativistic Boltzmann Equation in a Slab

相对论性玻尔兹曼方程在平板中的稳态解

作者: Jin Woo Jang, Seok-Bae Yun

类别: math.AP

发布日期: 2025-12-14

arXiv ID: 2512.12540v1

链接: http://arxiv.org/abs/2512.12540v1

摘要: We study steady solutions to the relativistic Boltzmann equation with hard-sphere interactions in a slab geometry. Under a spatial symmetry assumption in the transverse variables $x_2$ and $x_3$, the problem reduces to a one-dimensional spatial slab $x_1 \in [0,1]$ while retaining full three-dimensional momentum dependence. For non-negative inflow boundary conditions prescribed at $x_1=0$ and $x_1=1$, we prove the existence and uniqueness of a stationary solution in a weighted $L^1_p L^\infty_{x_1}$ framework, together with exponential decay in momentum. Our analysis treats the full slab domain and does not rely on any smallness assumption on the slab width. We establish sharp coercivity and continuity estimates for the collision frequency, together with weighted convolution and pointwise bounds for the nonlinear gain term. These estimates generate and propagate a $(-Δ_p)^{-1}$-type regularity within the solution framework, which plays a crucial role in the existence and uniqueness argument. In addition, we obtain uniform weighted integrability of the solution over arbitrary two-dimensional hyperplanes through the origin. This hyperplane estimate is derived as a genuinely a posteriori regularity property, without imposing any a priori hyperplane bounds, and follows from a Lorentz-invariant geometric reduction.

30. Adiabatic approximation of Abelian Higgs models

阿贝尔希格斯模型的绝热近似

作者: Amirmasoud Geevechi, Robert L Jerrard

类别: math.AP

发布日期: 2025-12-14

arXiv ID: 2512.12525v1

链接: http://arxiv.org/abs/2512.12525v1

摘要: We construct novel solutions in $d\ge 3$ space dimensions of a family of nonlinear evolutions equations that includes the critical hyperbolic Abelian Higgs model (AHM). For the AHM, these solutions exhibit an ensemble of $N\ge 1$ slowly-moving, nearly parallel vortex filaments, whose leading-order dynamics are described by a wave map from ${\mathbb R}^{d-2}$ into the Abelian Higgs moduli space, a manifold carrying a natural Riemannian structure that parametrizes stationary 2d solutions of the AHM. We also prove extremely similar results that relate the critical Abelian Higgs heat flow, modeling certain superconductors, to the harmonic map heat flow into the Moduli space, as well as some parallel results for near-critical equations. When $d=3$, these results allow for the study of the poorly-understood phenomenon of vortex reconnection in this setting.

31. Existence and stability for traveling waves of fourth order semilinear wave and Schrodinger equations

四阶半线性波动方程与薛定谔方程行波的存在性与稳定性

作者: Vishnu Iyer, Ross Parker, Atanas G. Stefanov

类别: math.AP, nlin.PS

发布日期: 2025-12-13

arXiv ID: 2512.12390v1

链接: http://arxiv.org/abs/2512.12390v1

摘要: We investigate the existence and spectral stability of traveling wave solutions for a class of fourth-order semilinear wave equations, commonly referred to as beam equations. Using variational methods based on a constrained maximization problem, we establish the existence of smooth, exponentially decaying traveling wave profiles for wavespeeds in the interval $(0, \sqrt{2})$. We derive precise spectral properties of the associated linearized operators and prove a Vakhitov-Kolokolov (VK) type stability criterion that completely characterizes spectral stability. Furthermore, we determine the sharp exponential decay rate of the traveling waves and demonstrate that it matches the decay rate of the Green’s function for the linearized operator. Our analysis extends to fourth-order nonlinear Schrodinger equations, for which we establish analogous existence and stability results. The theoretical findings are complemented by numerical computations that verify the stability predictions and reveal the transition from unstable to stable regimes as the wavespeed varies. These results provide a comprehensive mathematical framework for understanding wave propagation phenomena in structural mechanics, particularly suspension bridge models.

32. A third-order conservation law for the Kirchhoff-Pokhozhaev equation

Kirchhoff-Pokhozhaev方程的一个三阶守恒律

作者: Chiara Boiti, Renato Manfrin

类别: math.AP

发布日期: 2025-12-13

arXiv ID: 2512.12380v1

链接: http://arxiv.org/abs/2512.12380v1

摘要: We prove that the special Kirchhoff equation studied by Pokhozhaev admits a third-order conservation law. We further show that if the energy of the solution is sufficiently small, then the $L^2$-norms of the derivatives up to third order of the solution remain uniformly bounded with respect to time.

33. Nontrivial solutions for nonlinear problems driven by a superposition of fractional p-Laplacians with Neumann boundary conditions

由分数阶 p-拉普拉斯算子叠加驱动且具诺伊曼边界条件的非线性问题的非平凡解

作者: Yergen Aikyn

类别: math.AP

发布日期: 2025-12-13

arXiv ID: 2512.12351v1

链接: http://arxiv.org/abs/2512.12351v1

摘要: In this paper, we investigate existence results for nonlinear nonlocal problems governed by an operator obtained as a superposition of fractional $p$-Laplacians, subject to Neumann boundary conditions. A spectral analysis of the main operator leads us to apply different variational tools to establish our results. Specifically, we will use either the mountain pass method or the technique of linking over cones. Due to the generality of the setting, the resulting theory applies to a wide range of specific situations.

34. Lower Bound of Nodal Sets in Elliptic Homogenization and Functions with Strong Maximum Principle

椭圆型均匀化与强极值原理函数中节点集的下界

作者: Jiahuan Li, Zhichen Ying

类别: math.AP

发布日期: 2025-12-13

arXiv ID: 2512.12305v1

链接: http://arxiv.org/abs/2512.12305v1

摘要: In this note, we first try to prove a uniform lower bound of nodal volume in elliptic homogenization setting. This lower bound is far from optimal. But, we can prove a constant lower bound in dimension two. Motivated by the proof, we extend this results to more general settings. To be more specific, we prove that the nodal volume has a constant lower bound for all continuous functions with strong maximum princple. Our result works for general functions beyond solutions to elliptic PDEs.

35. From cohesive to brittle debonding: the quasistatic framework

从黏聚性到脆性脱粘：准静态框架

作者: Filippo Riva

类别: math.AP

发布日期: 2025-12-13

arXiv ID: 2512.12261v1

链接: http://arxiv.org/abs/2512.12261v1

摘要: The approximation of brittle laws via steeper and steeper cohesive profiles is validated within the mechanical setting of debonding models, which describe the detachment process of a peeled elastic adhesive membrane. In a quasistatic framework, energetic solutions to a suitably rescaled cohesive debonding problem, formulated in terms of displacements, are proved to converge to a limit evolution of shapes solving its brittle counterpart. The proposed approach relies on an equivalent and recently introduced free-boundary reformulation of this latter model.

36. Taylor polynomials on left-quotients of Carnot groups

Carnot群左商上的泰勒多项式

作者: Alessandro Ottazzi

类别: math.AP

发布日期: 2025-12-13

arXiv ID: 2512.12239v1

链接: http://arxiv.org/abs/2512.12239v1

摘要: We prove classical Taylor polynomial theorems for sub-Riemannian manifolds that are obtained as the submetric image of a Carnot group. For these theorems we also prove a sufficient condition for real analyticity and a result on L-harmonicity of Taylor polynomials.

37. Discrete-to-continuum convergence of the density of states for Mathieu’s equation

Mathieu方程态密度的离散到连续收敛性研究

作者: Peter Hofhansel, Alexander B. Watson

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

发布日期: 2025-12-12

arXiv ID: 2512.12039v1

链接: http://arxiv.org/abs/2512.12039v1

摘要: The density of states of a self-adjoint operator generalizes the eigenvalue distribution of a Hermitian matrix. We prove convergence of the density of states for a tight-binding model with a slowly-varying periodic potential to the density of states of its continuum approximation, a Mathieu-type equation.

38. The energy scaling behavior of a class of incompatible two-well problems

一类非相容双势阱问题的能量标度行为

作者: Noah Piemontese-Fischer

类别: math.AP

发布日期: 2025-12-12

arXiv ID: 2512.12014v1

链接: http://arxiv.org/abs/2512.12014v1

摘要: In this article, we study scaling laws for singularly perturbed two-well energies with prescribed Dirichlet boundary data in settings where the wells and/or the boundary data are incompatible. Our main focus is the geometrically linear two-well problem, for which we characterize the energy scaling in two dimensions for nearly all combinations of linear boundary data and stress-free strains. In particular, we prove that if the boundary data enforces oscillations and the weight $ε$ of the surface energy is small, the minimal energy upon subtracting the zeroth-order contribution scales either as $ε^{{4}/{5}}$ or as $ε^{{2}/{3}}$, depending on whether the wells differ by a rank-one or a rank-two matrix, respectively. For the gradient and divergence-free two-well problem, we obtain analogous results, showing an $ε^{{2}/{3}}$-scaling behavior in two dimensions whenever oscillations are energetically favored. These results follow by deriving matching upper and lower scaling bounds. The lower scaling bounds are established in a general $\mathcal{A}$-free framework for incompatible two-well problems, which allows us to compute the excess energy and characterize boundary data which enforce oscillations. The upper scaling bounds are obtained by branching constructions which are adapted to the incompatible setting.

39. The Gevrey class of the Euler-Bernoulli beam model with singularities

具有奇性的欧拉-伯努利梁模型的Gevrey类

作者: Jaime E. Munoz Rivera, Maria Grazia Naso, Bruna T. Silva Sozzo

类别: math.AP

发布日期: 2025-12-12

arXiv ID: 2512.11789v1

链接: http://arxiv.org/abs/2512.11789v1

摘要: We study the Euler-Bernoulli beam model with singularities at the points $x=ξ_1$, $x=ξ_2$ and with localized viscoelastic dissipation of Kelvin-Voigt type. We assume that the beam is composed by two materials; one is an elastic material and the other one is a viscoelastic material of Kelvin-Voigt type. Our main result is that the corresponding semigroup is immediately differentiable and also of Gevrey class $4$. In particular, our result implies that the model is exponentially stable, has the linear stability property, and the smoothing effect property over the initial data.