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重点关注论文（完整分析）

1. 关于流体力学中双曲守恒律方程组广义解概念的等价性

On the equivalence of generalized solution concepts for systems of hyperbolic conservations laws in fluid dynamics

作者: Thomas Eiter, Robert Lasarzik, Emil Wiedemann

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00957v1

摘要翻译: 本文研究了流体力学中非线性偏微分方程方程组的几种广义解概念之间的关系。更准确地说，我们研究了测度值解、耗散弱解和能量变分解。对于不可压缩Euler方程组，我们证明了所有三个概念的等价性，前提是能量不等式以适当的方式表述。对于流体力学中出现的若干重要守恒律例子，我们建立了能量变分解与适当精化的耗散弱解之间的等价性，其中缺陷测度由能量缺陷精确控制。这些例子包括可压缩等熵Euler方程组、Euler-Korteweg方程组和Euler-Poisson方程组。

Comment: 无

链接: http://arxiv.org/abs/2604.00957v1

1. 研究对象和背景

广义解概念在偏微分方程理论中无处不在，尤其在流体动力学领域。众所周知，光滑解在一般情况下无法整体存在，激波乃至湍流现象均可能出现。不可压缩Euler方程和Navier-Stokes方程存在无穷多弱解这一事实表明，规则性的丧失是固有的内在问题。为处理这一问题，特别是在证明整体存在性时，学界发展了众多广义解概念。

论文研究对象为双曲守恒律系统中的三种核心广义解概念：

研究对象与方程设定：

  1. 不可压缩Euler方程：设$\Omega \subset \mathbb{R}^d$为有界Lipschitz区域，$T > 0$为时间区间，流体速度$v: \Omega \times (0,T) \to \mathbb{R}^d$和压力$p: \Omega \times (0,T) \to \mathbb{R}$满足

    $$ \partial_t v + (v\cdot\nabla)v + \nabla p = 0, \quad \operatorname{div} v = 0 \quad \text{在 } \Omega \times (0,T) \text{ 中}, $$

    $$ v\cdot n = 0 \quad \text{在 } \partial\Omega \times (0,T) \text{ 上}, $$

    $$ v(0) = v_0 \quad \text{在 } \Omega \text{ 中}. $$

  2. 可压缩等熵Euler系统：未知量为密度$\rho \geq 0$和速度$v$

    $$ \partial_t \rho + \operatorname{div}(\rho v) = 0, $$

    $$ \partial_t (\rho v) + \operatorname{div}(\rho v \otimes v) + \nabla p(\rho) = 0, $$

    其中压力$p(\rho) = \kappa \rho^\gamma$，$\kappa > 0$，$\gamma > 1$为绝热指数。

  3. Euler-Korteweg系统：描述具有表面张力的可压缩流体

    $$ \partial_t \rho + \operatorname{div}(\rho v) = 0, $$

    $$ \partial_t (\rho v) + \operatorname{div}(\rho v \otimes v) + \nabla p(\rho) = \rho \nabla \Delta \rho. $$

  4. Euler-Poisson系统：描述等离子体中的流体

    $$ \partial_t \rho + \operatorname{div}(\rho v) = 0, $$

    $$ \partial_t (\rho v) + \operatorname{div}(\rho v \otimes v) + \nabla p(\rho) = -\rho \nabla \Phi, \quad -\Delta \Phi = \rho. $$

问题的重要性：弱解作为非线性偏微分方程的标准概念存在若干缺陷。许多方程组缺乏必要的紧性性质以在弱形式中取极限，而凸积分方法允许构造无穷多弱解。因此，更一般的可解性概念受到关注。测度值解最早由Tartar引入，后由DiPerna和DiPerna-Majda作为适当的可解性框架定义。Lions引入的耗散解概念基于相对能量不等式，在从Boltzmann方程到Euler方程的极限过渡中发挥重要作用。能量变分解是最近引入的概念，允许通过极小化运动格式证明整体存在性。

本文试图填补的空白：尽管这些广义解概念在非线性发展型偏微分方程领域广泛应用，但它们之间的关联程度乃至等价性尚未得到充分回答。本文旨在建立流体动力学中不同偏微分方程系统广义解概念之间的特定等价结果，从而填补这一理论空白。

2. 主要定理或主要结果

论文的主要结果可分为两部分。

第一部分：不可压缩Euler方程三种解概念的等价性（定理3.5）

设$\Omega$为有界Lipschitz域。以下三条陈述等价：

(i) $(v, E)$是满足下式的能量变分解：存在$E \in BV([0,T];\mathbb{R})$使得$E(t) \geq E(v(t))$几乎处处成立，且对所有满足$\operatorname{div}\varphi = 0$和$\varphi \cdot n = 0$的测试函数$\varphi \in C^1(\Omega \times [0,T];\mathbb{R}^d)$，以及几乎所有$0 \leq s < t \leq T$（含$s = 0$），有

$$ \left[E - \int_\Omega v \cdot \varphi dx\right]_s^t + \int_s^t \int_\Omega \left(v \cdot \partial_t \varphi + v \otimes v : \nabla\varphi\right) dx d\tau + \int_s^t K(\varphi) [E(v) - E] d\tau \leq 0, $$

其中

$$ K(\varphi) = 2 \| (\nabla \varphi)_{\operatorname{sym},-} \|_{L^\infty(\Omega;\mathbb{R}^{d \times d})}. $$

(ii) $(v, E)$是耗散弱解：存在Reynolds缺陷$R \in L^\infty_{w^*}(0,T;\mathcal{M}(\Omega;\mathbb{R}^{d \times d}_{\operatorname{sym},+}))$使得对所有测试函数$\varphi$（同上），几乎所有$s, t \in (0,T)$（含$s = 0$），有

$$ \left[\int_\Omega v \cdot \varphi dx\right]_s^t = \int_s^t \int_\Omega \left(v \cdot \partial_t \varphi + v \otimes v : \nabla\varphi\right) dx d\tau + \int_s^t \int_\Omega \nabla\varphi : dR d\tau, $$

且$E$非增并满足

$$ E(v(t)) + \frac{1}{2} \int_\Omega I : dR \leq E(t) \quad \text{几乎处处}. $$

(iii) 存在测度值解$(\nu_0, \lambda, \nu_\infty)$使得$\langle \nu_0, \xi\rangle = v$，且

$$ E = \frac{1}{2} \int_\Omega \langle \nu_0, |\xi|^2\rangle dx + \frac{1}{2} \lambda_t(\Omega), $$

其中测度值解的定义涉及概率测度$\nu_0 \in L^\infty_{w^*}(\Omega \times (0,T);\mathcal{P}(\mathbb{R}^d))$、非负标量测度$\lambda \in L^\infty_{w^*}(0,T;\mathcal{M}_+(\Omega))$以及切向测度$\nu_\infty \in L^\infty_{w^*}(\Omega \times (0,T); \lambda \otimes L^\infty_{w^*}((0,T)); \mathcal{P}(S^{d-1})))$，并满足积分恒等式和能量不等式

$$ \frac{1}{2} \int_\Omega \langle \nu_0^t, |\xi|^2\rangle dx + \frac{1}{2} \lambda_t(\Omega) \leq \frac{1}{2} \int_\Omega \langle \nu_0^s, |\xi|^2\rangle dx + \frac{1}{2} \lambda_s(\Omega). $$

第二部分：可压缩流体系统能量变分解与耗散弱解的等价性

对于可压缩等熵Euler系统、Euler-Korteweg系统和Euler-Poisson系统，在适当精化的耗散弱解框架下建立等价性。这里的"精化"是指：常系数精确确定，且缺陷测度由能量缺陷精确控制。论文证明了对于这些系统，能量变分解与精化耗散弱解是等价的。

具体而言，在可压缩系统中，论文建立了精确的估计关系，缺陷测度$\mathcal{D}$满足类似

$$ \mathcal{D} \leq C \cdot (\text{能量缺陷}) $$

的估计，且常系数$C$被精确计算而非仅作为存在性参数。

3. 研究方法、关键技术和核心工具

论文的核心技术挑战在于建立不同广义解概念之间的等价性，主要障碍在于如何精确量化缺陷测度与能量缺陷之间的关系，并处理锥约束条件。

技术创新点一：锥值Radon测度的对偶刻画（引理2.2）

论文首先建立了锥值Radon测度的对偶特征化。设$\Omega \subset \mathbb{R}^d$为有界Lipschitz域，$C \subset \mathbb{R}^m$为闭凸锥。论文证明：$R \in \mathcal{M}(\Omega;\mathbb{R}^m)$满足$R(A) \in C$对所有可测集$A \subset \Omega$成立的充要条件是

$$ \int_\Omega B \cdot dR \leq 0 \quad \forall B \in C(\Omega;C^\circ), $$

其中$C^\circ = \{z \in \mathbb{R}^m \mid \forall y \in C: (z,y) \leq 0\}$为法锥。这一结果将锥值测度的约束转化为积分不等式，是后续精确估计的基础。

技术创新点二：矩阵范数的对偶关系（引理2.3）

论文详细分析了$\mathbb{R}^{d \times d}_{\operatorname{sym}}$上不同范数的对偶关系。对于对称矩阵$A$，定义

$$ |A|_2 = \max_i |\lambda_i(A)| \quad (\text{谱范数}), \quad |A|_{\operatorname{tr}} = \sum_i |\lambda_i(A)| \quad (\text{迹范数}), \quad |A|_F = \sqrt{\sum_i \lambda_i(A)^2} \quad (\text{Frobenius范数}). $$

论文证明这些范数关于Frobenius内积$A:B = \operatorname{tr}(A^\top B)$满足对偶关系

$$ |A|_{2,*} = |A|_{\operatorname{tr}}, \quad |A|_{\operatorname{tr},*} = |A|_2, \quad |A|_{F,*} = |A|_F. $$

这一精确的对偶关系对于消除估计中的绝对常数至关重要。

技术创新点三：等价性判据（引理2.4）

这是论文的核心技术工具，建立了线性泛函可表示为锥值测度作用与积分不等式之间的精确等价关系。设$Y \subset L^1(0,T;C^1(\Omega;\mathbb{R}^m))$为稠密线性子空间，$\ell: Y \to \mathbb{R}$为线性映射，$\zeta \in L^\infty(0,T)$满足$\zeta \geq 0$。设$C_1, C_2 \subset \mathbb{R}^{m \times d}$为两个锥，$L_i = \operatorname{span}(C_i)$。则以下两条等价：

(i) 存在$R_i \in L^\infty_{w^*}(0,T;\mathcal{M}(\Omega;L_i^\perp))$使得

$$ \langle \ell, \varphi\rangle = \int_0^T \int_\Omega \nabla\varphi : dR_1 + \int_0^T \int_\Omega \nabla\varphi : dR_2, \quad \forall \varphi \in Y, $$

且

$$ \frac{1}{\eta_1} \|R_1\|_{\mathcal{M}(\Omega;L_1^*)} + \frac{1}{\eta_2} \|R_2\|_{\mathcal{M}(\Omega;L_2^*)} \leq \zeta \quad \text{几乎处处}. $$

(ii) 对所有$\varphi \in Y$有

$$ \langle \ell, \varphi\rangle \leq \int_0^T \max\left\{ \eta_1 \|P_{C_1}(\nabla\varphi)\|_{L^\infty(\Omega;L_1)}, \eta_2 \|P_{C_2}(\nabla\varphi)\|_{L^\infty(\Omega;L_2)} \right\} \zeta dt. $$

这一引理的证明依赖于Hahn-Banach定理和Riesz表示定理，关键技巧是利用正交投影分解（由Moreau分解定理保证）将泛函表示转化为锥约束下的优化问题。

技术创新点四：不可压缩Euler方程的等价性证明框架

对于不可压缩Euler方程，论文将证明分为三个方向。

(1) 能量变分解 $\Rightarrow$ 耗散弱解：首先从能量变分不等式出发，通过取$\varphi = \alpha\psi$并令$\alpha \to \infty$得到线性泛函的积分估计，然后应用引理2.4，精确识别出Reynolds缺陷$R$。关键在于识别锥

$$ C_1 = \{ A \in \mathbb{R}^{d \times d}_{\operatorname{sym}} \mid A = -A^\top \text{ 或等价地 } A_{\operatorname{sym},+} = 0 \} $$

和相应空间，从而将负半定对称矩阵的约束纳入框架。

(2) 耗散弱解 $\Rightarrow$ 能量变分解：反向推导利用引理2.1(ii)的结论

$$ (z, y) \leq (P_C(z), y) \quad \forall y \in C, z \in H $$

将缺陷测度的估计转化为能量不等式中的$K(\varphi)$项。

(3) 与测度值解的等价：通过Young测度理论建立联系，测度值解中的$\nu_\infty$（支撑在$S^{d-1}$上的概率测度）恰对应于Reynolds缺陷的正交分解，而$\lambda$对应于标量缺陷测度。

技术创新点五：可压缩系统的适配

对于可压缩系统，论文将框架扩展到非齐次情形。特别地，对于可压缩等熵Euler系统，能量泛函$E(\rho, v) = \int_\Omega \frac{1}{2} \rho |v|^2 + \frac{\kappa}{\gamma-1} \rho^\gamma dx$需要适应变量$\rho$和$v$的联合演化。Euler-Korteweg系统中的$\nabla \Delta \rho$项需要额外的正则性假设，而Euler-Poisson系统中的势能项通过Newton势积分处理。在所有情形下，核心思想是利用引理2.4精确控制缺陷测度的范数，建立能量缺陷与缺陷测度之间的精确对应关系。

4. 与之前工作的比较

论文在以下几个维度上与既有工作形成对比和推进。

与Lions工作的比较：P.-L. Lions在[30]中引入了耗散解概念，特别用于从Boltzmann方程到Euler方程的极限过渡。本文考虑的耗散弱解和能量变分解构成Lions耗散解的子类：每个能量变分解、耗散弱解和测度值解的均值都是Lions意义下的耗散解，但反之不成立（参见[2, Remark 3.7]）。本文的贡献在于精确刻画了这些子类之间的等价关系，而Lions的原始框架不要求能量不等式的特定形式，因此涵盖范围更广但等价性较弱。

与DiPerna-Majda测度值解工作的比较：测度值解作为可解性框架最早由DiPerna和Majda在[13]中建立，随后被应用于可压缩Navier-Stokes系统[20]、Euler-Korteweg系统[14]和完全Euler系统[5]。本文首次建立了测度值解与能量变分解、耗散弱解之间的等价性。长期以来，学界普遍默认这些等价关系成立，但据作者了解，此前并无严格证明。本文利用引理2.2关于锥值测度的对偶刻画，填补了这一空白。特别地，测度值解中支撑在球面上的概率测度$\nu_\infty$与Reynolds缺陷的精确对应是本文的新发现。

与[16]的能量变分解工作的比较：第一作者等人在[16]中引入了能量变分解概念并证明了其与耗散弱解在不可压缩Euler情形下的等价性。本文不仅重复了这一结果（为读者方便），还将其扩展到两个新方向：(i) 将等价性推广到测度值解；(ii) 将框架扩展到可压缩系统（等熵Euler、Euler-Korteweg、Euler-Poisson）。此外，本文精确确定了缺陷测度的范数常数，而[16]中该常数仅作为存在性参数出现。

与凸积分理论工作的比较：凸积分理论由De Lellis和Székelyhidi等人发展（见[10]），允许构造不可压缩Euler方程的无数非唯一弱解。形式上，凸积分中的subsolutions与本文的耗散弱解等价，但凸积分框架通常要求额外的正则性假设（如连续性或分段连续性）。本文的等价性结果在无正则性假设的最小框架下成立，因此更一般化。

可压缩系统的推进：对于可压缩系统，既往工作如[19]研究耗散弱解，[22]研究Euler-Korteweg系统，但均未建立与能量变分解的等价性。本文的关键推进在于"精化"耗散弱解的定义：传统定义中估计缺陷测度的常数$C$满足$\mathcal{D} \leq C \cdot \text{能量缺陷}$，但$C$通常未指定；本文通过引理2.4精确确定了所有系统中的常数值（对于等熵Euler系统为$C = 1$，对于其他系统有相应计算），并给出缺陷测率的附加特征化。这一精确估计对于理论分析和数值格式的能量守恒识别具有重要意义（参见[28, 15]关于结构保持数值格式的工作）。

适用范围与时间尺度的扩展：本文的结果适用于整体时间区间$[0,T]$，而非局部时间。这依赖于能量变分解通过极小化运动格式建立的全局存在性[2, 17]。对于不可压缩Euler方程，等价性在有界Lipschitz域上成立；对于可压缩系统，域条件可能有额外假设（论文未详细展开）。

假设条件的变化：本文的核心技术假设是"能量不等式的适当表述"（见Remark 3.4）。存在其他解概念采用不同的能量不等式形式：仅与初始能量比较的类型，或考虑局部能量演化的类型。本文明确指出，这些类型与本文考虑的解概念不等价。因此，本文的等价性结果依赖于特定的能量不等式形式，而非一般性地对所有广义解概念成立。

了解领域论文（摘要翻译）

1. 纯p-拉普拉斯算子的非Ljusternik–Schnirelmann特征值的存在性

Non-Ljusternik–Schnirelman eigenvalues of the pure $p$-Laplacian exist

作者: Vladimir Bobkov

类别: math.AP, math.FA, math.SP

发布日期: 2026-04-01

arXiv ID: 2604.01138v1

链接: http://arxiv.org/abs/2604.01138v1

摘要翻译

中文标题: 纯p-拉普拉斯算子的非Ljusternik–Schnirelmann特征值的存在性

摘要翻译: 临界点理论中一个古老而著名的开放问题是：是否存在某个$p \neq 2$和某个有界区域$Ω$，使得$W_0^{1,p}(Ω)$中$L^p(Ω)$-球面上$p$-Dirichlet能量$\|\nabla u\|_p^p$的某个临界值位于Ljusternik–Schnirelman型临界值序列之外，后者被称为$p$-拉普拉斯算子的LS特征值。本文通过对$p>2$充分接近2且$Ω$为接近正方形的平面矩形的情形给出肯定回答，从而证明了非LS特征值的存在性。论证基于以下观察：拉普拉斯算子的单重特征值可作为$p$变化时$p$-拉普拉斯算子若干特征值分支的交汇点。由于LS特征值关于$p$连续且当$p=2$时穷尽整个谱，我们推知至少有一个分支必然包含非LS特征值。

2. 可压缩流体-结构相互作用系统在Navier-滑移边界条件下的最大正则性

Maximal regularity for a compressible fluid-structure interaction system with Navier-slip boundary conditions

作者: Kuntal Bhandari, Imene Aicha Djebour, Šárka Nečasová

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00973v1

链接: http://arxiv.org/abs/2604.00973v1

摘要翻译

中文标题: 可压缩流体-结构相互作用系统在Navier-滑移边界条件下的最大正则性

摘要翻译: 我们研究了一个流体-结构相互作用系统，其中流体的动力学由可压缩Navier-Stokes方程描述，而弹性结构由阻尼板方程建模。流体在三维有界域中演化，结构占据其边界的一部分。不同于标准的无滑移边界条件，我们在流体-结构界面以及固定边界处考虑Navier-滑移边界条件。我们在$L^{p}-L^{q}$框架下建立了强解的局部时间存在性和唯一性。该存在性结果通过解耦线性化系统并结合Tikhonov不动点定理的级联策略在小时间区间内获得，而唯一性则通过在Hilbert空间框架下推导相关线性耦合算子的弱正则性质来证明。这是关于可压缩流体与阻尼板在Navier-滑移边界条件下相互作用的强解的首个结果。

3. 环面上的对称双曲薛定谔方程

Symmetric hyperbolic Schrödinger equations on tori

作者: Baoping Liu, Xu Zheng

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00403v1

链接: http://arxiv.org/abs/2604.00403v1

摘要翻译

中文标题: 环面上的对称双曲薛定谔方程

摘要翻译: 本文研究周期情形下的对称双曲薛定谔方程。首先，通过推广Bourgain的大弧方法，我们在一般环面上证明了全范围的Strichartz估计，该结果在有理环面上是最优的。其次，在二维有理环面上，我们建立了两个双曲非线性薛定谔（HNLS）方程的最优局部适定性：七次HNLS和双曲-椭圆Davey-Stewartson系统。

4. 拟线性波动方程障碍问题的互补弱解与变分解存在性

Existence of Complementary and Variational Weak Solutions to Obstacle Problems for a Quasilinear Wave Equation

作者: João Paulo Dias, Wladimir Neves, José Francisco Rodrigues

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00336v1

链接: http://arxiv.org/abs/2604.00336v1

摘要翻译

中文标题: 拟线性波动方程障碍问题的互补弱解与变分解存在性

摘要翻译: 本文证明了一类一维空间中拟线性波动方程相关的单障碍问题弱解的存在性，推广了线性情形下已有的结果，并进一步讨论了双障碍问题。与线性情形不同，对于严格拟线性的情形，我们获得了互补弱意义下的连续解，这些解在弦不接触障碍物的自由区域还满足弱熵条件。我们进一步证明，在单障碍和双障碍情形下，这些解在双曲意义下是变分解，无需粘性项。

5. 动态随机环境中Hamilton–Jacobi方程遍历性的量化

Quantification of ergodicity for Hamilton–Jacobi equations in a dynamic random environment

作者: Xiaoqin Guo, Wenjia Jing, Hung Vinh Tran, Yuming Paul Zhang

类别: math.AP, math.OC, math.PR

发布日期: 2026-03-31

arXiv ID: 2604.00315v1

链接: http://arxiv.org/abs/2604.00315v1

摘要翻译

中文标题: 动态随机环境中Hamilton–Jacobi方程遍历性的量化

摘要翻译: 本文研究具有平稳遍历性且在时间上具有单位范围依赖的动态随机环境中Hamilton–Jacobi方程的定量大时间平均行为。我们的研究动机源于与无张力（不可压）KPZ方程相关的随机增长模型，该模型可表述为带有随机强迫力的Hamilton–Jacobi方程。理解解的大时间平均行为与理解此类增长过程中涨落与标度的基本问题密切相关。

6. 重整化在圆上正交Strichartz估计与NLS系统中的应用

Applications of renormalisation to orthonormal Strichartz estimates and the NLS system on the circle

作者: Sonae Hadama, Andrew Rout

类别: math.AP, math-ph

发布日期: 2026-03-31

arXiv ID: 2604.00252v1

链接: http://arxiv.org/abs/2604.00252v1

摘要翻译

中文标题: 重整化在圆上正交Strichartz估计与NLS系统中的应用

摘要翻译: 本文针对圆上非线性Schrödinger方程系统(NLSS)引入了与系统相关密度的重整化程序。我们证明了该重整化密度满足比非重整化密度（参见Nakamura (2020)）更好的正交Strichartz估计。关于这些Strichartz估计的最优指数范围，我们将其作为一个猜想留下。作为应用，我们确定了临界Schatten指数：低于该指数时，圆上的三次重整化NLSS是全局适定的；高于该指数时，则是不适定的。最后，我们证明了重整化密度在$d \geq 2$维环面$\mathbb{T}^d$上所满足的正交Strichartz估计的改进是微小的。

不相关论文（基本信息）

1. 随机椭圆算子在区域上的正则性定理

Regularity theorems for random elliptic operators on domains

作者: Peter Bella, Julian Fischer, Marc Josien, Claudia Raithel

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.01209v1

链接: http://arxiv.org/abs/2604.01209v1

摘要: Regularity theorems à la Avellaneda-Lin are an indispensable part of the modern quantitative theory of stochastic homogenization. While interior regularity results for random elliptic operators have been available for a while, on general smooth domains the existing theory has until recently remained limited to Lipschitz estimates. We establish $C^{1,α}$ regularity results for random elliptic operators on bounded sufficiently smooth domains, as well as for scalar problems on convex polytopes. We, furthermore, prove a number of auxiliary results typically employed in the derivation of fluctuation bounds, such as a weighted Meyers estimate.

2. 非线性对流扩散系统的Runge-Kutta间断Galerkin格式与SIAC后处理的后验误差分析

A Posteriori Error Analysis of Runge-Kutta Discontinuous Galerkin Schemes with SIAC Post-Processing for Nonlinear Convection-Diffusion Systems

作者: Jan Giesselmann, Kiwoong Kwon, Sebastian Krumscheid

类别: math.NA, math.AP

发布日期: 2026-04-01

arXiv ID: 2604.01200v1

链接: http://arxiv.org/abs/2604.01200v1

摘要: We develop reliable a posteriori error estimators for fully discrete Runge-Kutta discontinuous Galerkin approximations of nonlinear convection-diffusion systems endowed with a convex entropy in multiple spatial dimensions on the flat torus T^d, with a focus on the convection-dominated regime. In order to use the relative entropy method, we reconstruct the numerical solution via tensor-product Smoothness-Increasing Accuracy-Conserving (SIAC) filtering which has superconvergence properties. We then derive reliable a posteriori error estimators for the difference between the entropy weak solution and the reconstruction, with constants that are uniform in the vanishing viscosity limit. Our numerical experiments show that the a posteriori error bounds converge with the same order as the error of the reconstructed numerical solution.

3. 爆破分析与N维非局部交互泛函的极值函数

Blow-up analysis and extremal functions for nonlocal interaction functionals in dimension $N$

作者: A. Cannone, M. Yu

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.01162v1

链接: http://arxiv.org/abs/2604.01162v1

摘要: In this paper we study Moser-Trudinger type inequalities for some nonlocal energy functionals in presence of a logarithmic convolution potential, when the domain is a ball of $\mathbb{R}^N$ with $N \geq 2$. In particular, we perform a blow-up analysis to prove existence of extremal functions in the borderline case of critical growth. Using this, we extend the results in \cite{CiWeYu} to higher dimension and sharpen \cite{CC}.

4. Fabry-Pérot共振的频变电容矩阵公式化。第一部分：一维有限系统

Frequency-dependent capacitance matrix formulation for Fabry-Pérot resonances. Part I: One-dimensional finite systems

作者: Habib Ammari, Bowen Li, Ping Liu, Yingjie Shao

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.01159v1

链接: http://arxiv.org/abs/2604.01159v1

摘要: We study scattering resonances of finite one-dimensional systems of high-contrast resonators beyond the subwavelength regime. Introducing a novel tridiagonal frequency-dependent capacitance matrix, we derive quantitative asymptotic expansions of the hybridized Fabry-Pérot resonant frequencies in terms of the material contrast parameter. The leading-order shifts are governed by the eigenvalues of this matrix, while the corresponding eigenmodes are approximated, to leading order, by trigonometric functions on selected spacings between resonators. Our results extend the use of discrete approximations as a powerful tool for characterizing the resonant properties of a system of high-contrast resonators at arbitrarily high frequencies.

5. 随机约束的Koiter壳模型

Stochastically-constrained Koiter shell models

作者: Prince Romeo Mensah, Pierre Marie Ngougoue Ngougoue

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.01150v1

链接: http://arxiv.org/abs/2604.01150v1

摘要: We derive a stochastically-constrained Koiter shell model in line with the SALT (Stochastic Advection by Lie Transport) approach introduced by Holm [Proc. A. 471 (2015)]. First, we deduce the stochastic partial differential equations for the generalised nonlinearly- and linearly-elastic Koiter shell models with their abstract functional derivatives for their corresponding membrane and flexural energies. We then present a prototype for the stochastically-constrained (simplified) linearised Koiter shell models that encodes all necessary information on stiffness due to shell curvature, bending stress, membrane stress, interior and surface forces, and more generally, stochastic buckling.

6. 耦合于扩散场的标量微分方程中的振荡

Oscillations in a scalar differential equation coupled to a diffusive field

作者: Merlin Pelz, Arnd Scheel

类别: math.AP, nlin.PS

发布日期: 2026-04-01

arXiv ID: 2604.01135v1

链接: http://arxiv.org/abs/2604.01135v1

摘要: We study the emergence of periodic oscillations through a Hopf bifurcation in a scalar diffusion equation on the half line coupled to a dynamic boundary condition. Our results quantify the effect of delay through the buffering in the diffusive field on boundary kinetics, drawing a parallel to the emergence of oscillations in delay equations. Technically, the Hopf bifurcation occurs in the presence of essential spectrum induced by the diffusive field, preventing a simple approach via center-manifold reduction. The results are motivated by observations in biological systems where dynamic boundary conditions arise when modeling surface dynamics coupled to bulk diffusion.

7. Almgren-Pitts 与相变半体积谱的等价性

Equivalence of Almgren-Pitts and phase-transition half-volume spectra

作者: Talant Talipov

类别: math.DG, math.AP, math.MG

发布日期: 2026-04-01

arXiv ID: 2604.01091v1

链接: http://arxiv.org/abs/2604.01091v1

摘要: We prove that the Almgren-Pitts and phase-transition half-volume spectra of a closed Riemannian manifold are equal. This confirms a conjecture of Liam Mazurowski and Xin Zhou.

8. 非Lipschitz速度场的拓扑熵一般无穷

Topological entropy is generically infinite for non-Lipschitz velocity fields

作者: Carl Johan Peter Johansson, Giulia Mescolini

类别: math.DS, math.AP

发布日期: 2026-04-01

arXiv ID: 2604.01077v1

链接: http://arxiv.org/abs/2604.01077v1

摘要: We prove that for any Osgood non-Lipschitz modulus of continuity $ω$, flow maps associated with time-periodic $ω$-continuous velocity fields generically (in the sense of Baire) have infinite topological entropy.

9. 半经典爱因斯坦-克莱因-戈登系统的闵可夫斯基时空渐近分析

The Semiclassical Einstein-Klein-Gordon System: Asymptotic Analysis of Minkowski Spacetime

作者: Stefano Galanda, Paolo Meda, Simone Murro, Nicola Pinamonti, Gabriel Schmid

类别: math-ph, gr-qc, math.AP, math.DG

发布日期: 2026-04-01

arXiv ID: 2604.01047v1

链接: http://arxiv.org/abs/2604.01047v1

摘要: We establish the linear instability of the semiclassical Einstein-Klein-Gordon system linearised about the Minkowski vacuum spacetime. The proof relies on formulating a forcing problem for both metric and state perturbations within the space of past-compact sections. This geometric framework admits a unique tensor decomposition which, in conjunction with the quantum Møller operator, enables the decoupling of the linearised system into two distinct Cauchy problems. Consequently, the metric perturbations are shown to be governed by a higher-order, nonlocal hyperbolic partial differential equation. By relegating the nonlocal contributions to subleading order, we establish the well-posedness of this forcing problem. Furthermore, we provide a rigorous asymptotic analysis for physically admissible choices of the renormalisation constants. We prove that the system exhibits a late-time linear instability: the metric perturbations grow exponentially, bounded strictly by a universal scale H, thereby indicating a quantum backreaction-driven transition toward a de Sitter cosmological spacetime. Provided the parameters governing the system are restricted to a physically relevant regime, this universal scale is compatible with the measured expansion of our universe.

10. 耗散偏微分方程的严格C^1流

Rigorous $C_1$ integration of dissipative PDEs

作者: Jakub Banaśkiewicz

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.01046v1

链接: http://arxiv.org/abs/2604.01046v1

摘要: We introduce a new $C^1$ algorithm for the rigorous integration of dissipative partial differential equations. The algorithm is designed for computer-assisted proofs that require rigorous control of both solutions and their derivatives with respect to initial data. As applications, we establish the existence of locally attracting periodic orbits for initial and boundary value problems for two non-autonomous dissipative PDEs: the Chafee-Infante equation and the Burgers equation with a fractional Laplacian.

11. 关于任意维数拉普拉斯算子特征值的无效应切割方法

On the Effectless Cut Method for Laplacian Eigenvalues in any dimensions

作者: Vincenzo Amato, Nunzia Gavitone, Francesca de Giovanni

类别: math.SP, math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00976v1

链接: http://arxiv.org/abs/2604.00976v1

摘要: In this paper, we study the optimization of the first Laplacian eigenvalue on axisymmetric doubly connected domains under positive Robin boundary conditions. Under additional geometric constraints, we prove that spherical shells maximize this eigenvalue. Our approach combines known isoperimetric inequalities for mixed Laplacian eigenvalues with a higher-dimensional extension of the effectless cut technique introduced by Hersch to study multiply connected membranes of given area fixed along their boundaries.

12. 正则化最优传输的尖锐局部稀疏性

Sharp local sparsity of regularized optimal transport

作者: Albert González-Sanz, Rishabh S. Gvalani, Lukas Koch

类别: math.AP, math.NA, math.PR, math.ST

发布日期: 2026-04-01

arXiv ID: 2604.00843v1

链接: http://arxiv.org/abs/2604.00843v1

摘要: In recent years, the use of entropy-regularized optimal transport with $L^p$-type entropies has become increasingly popular. In this setting, the solutions are sparse, in the sense that the support of the regularized optimal coupling, $\mathrm{supp}(π_\varepsilon)$, shrinks to the support of the original optimal transport problem as $\varepsilon \to 0$. The main open question concerns the rate of this convergence. In this paper, we obtain sharp local results away from the boundary. We prove that the supports $\mathrm{supp}(π_\varepsilon(\cdot \mid x))$ of the conditional measures, $π_\varepsilon(\cdot \mid x)$, behave like balls of radius $\varepsilon^\frac 1 {d(p-1)+2}$. This allows us to show that the regularized potentials are uniformly strongly convex and to derive the rate of convergence of these potentials toward their unregularized limit. Our results generalize the results of (González-Sanz and Nutz, SIAM J.~Math.~Anal.) and (Wiesel and Xu, Ibid.) to the multivariate case and beyond the case of self-transport.

13. Adam型连续时间随机模型的Fokker-Planck分析与不变定律

Fokker-Planck Analysis and Invariant Laws for a Continuous-Time Stochastic Model of Adam-Type Dynamics

作者: Kaj Nyström

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00840v1

链接: http://arxiv.org/abs/2604.00840v1

摘要: We develop a continuous-time model for the long-term dynamics of adaptive stochastic optimization, focusing on bias-corrected Adam-type methods. Starting from a finite-sum setting, we identify a canonical scaling of learning rates, decay parameters, and gradient noise that yields a coupled, time-inhomogeneous stochastic differential equation for the parameters $x_t$, first-moment tracker $z_t$, and second-moment tracker $y_t$. Bias correction persists via explicit time-dependent coefficients, and the dynamics becomes asymptotically time-homogeneous. We analyze the associated Fokker-Planck equation and, under mild regularity and dissipativity assumptions on $f$, prove existence and uniqueness of invariant measures. Noise propagation is governed by $A(x)=\mathrm{Diag}(\nabla f(x))H_f(x)$. Hypoellipticity may fail on $\mathcal D_A\times\mathbb R^m\times(\mathbb R_+)^m$, where

$$ \mathcal D_A=\{x\in\mathbb R^m:\exists j,\ e_j^\top A(x)=0\}\subset\{x:\det A(x)=0\}=\mathcal D_A^\dagger, $$

and critical points of $f$ lie in $\mathcal D_A$. We show $\mathcal D_A^\dagger\neq\mathbb R^m$ and use this to prove exponential convergence of the Markov semigroup $μ_0P_t$ to a unique invariant measure, uniformly in $μ_0$. The proof uses a Harris-type argument, minorization on Lyapunov sublevel sets, control constructions, and hypoellipticity on $(\mathbb R^m\setminus\mathcal D_A)\times\mathbb R^m\times(\mathbb R_+)^m$. This provides a transparent continuous-time view of Adam-type dynamics.

14. 加权黎曼流形中Serrin型超定问题的对称性与刚性结果

Symmetry and rigidity results for Serrin’s overdetermined type problems in weighted Riemannian manifolds

作者: Laura Accornero, Giulio Ciraolo

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00740v1

链接: http://arxiv.org/abs/2604.00740v1

摘要: We study Serrin’s overdetermined boundary value problems in bounded domains on weighted Riemannian manifolds. When the closure of the domain is compact, we establish a rigidity result that characterizes both the solution and the geometry of the ambient manifold. We further address the case of domains with non-compact closure for manifolds conformally equivalent to the Euclidean space, possibly degenerating or becoming singular at a point, where both the weight and the conformal factor are radial functions.

15. 具有Riesz势驱动非局部相互作用的二物种系统

Two-species system with nonlocal interactions driven by Riesz potentials

作者: Simone Fagioli, Valeria Iorio

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00729v1

链接: http://arxiv.org/abs/2604.00729v1

摘要: This paper investigates a system of nonlocal continuity equations modelling the interaction of two species coupled through Riesz-type potentials. The model incorporates self- and cross-interaction kernels of possibly different fractional orders. By exploiting optimal transportation theory and the theory of gradient flows in Wasserstein spaces, we establish the existence of weak solutions under singularity assumptions on all interaction potentials, provided the cross-interaction ones satisfy a symmetry condition. Our analysis extends previous results available for either single-species equations or multi-species systems with smoother cross-interaction kernels.

16. 粗糙系数Helmholtz方程的适定性

Well-Posedness of the Helmholtz Equation with Rough Coefficients

作者: Peijun Li, Yichun Zhu

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00712v1

链接: http://arxiv.org/abs/2604.00712v1

摘要: We establish the well-posedness of the Helmholtz equation with rough and compactly supported coefficients in Rd under sharp regularity assumptions. Using a paraproduct calculus in rescaled weighted Besov spaces, we rigorously define the product between the solution and the coefficient at the lowest regularity level without renormalization. A rescaled Lippmann-Schwinger formulation is shown to be equivalent to the Helmholtz equation with the Sommerfeld radiation condition. We prove existence, uniqueness, and explicit wavenumber dependent resolvent estimates in a general Lp setting, including an L2 theory relevant to scattering amplitudes. The results provide a sharp analytic foundation for wave propagation and scattering in highly irregular media.

17. 多尺度随机系统的一致时间扩散逼近

Uniform-in-time diffusion approximations for multiscale stochastic systems

作者: Longjie Xie, Xicheng Zhang

类别: math.PR, math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00692v1

链接: http://arxiv.org/abs/2604.00692v1

摘要: This paper establishes a quantitative, uniform-in-time diffusion approximation for the joint law of a broad class of fully coupled multiscale stochastic systems. We derive a precise characterization of the limiting joint distribution as a specific skew-product of the conditional equilibrium of the fast process and the homogenized law of the slow component, thereby providing a rigorous uniform-in-time formulation of the adiabatic elimination principle. The convergence rate explicitly separates the initial relaxation of the fast dynamics from the long-time homogenized evolution and depends only on the regularity of the coefficients in the slow variable. As a consequence, we obtain the first quantitative identification of the limiting stationary distribution of the original multiscale system and prove the commutativity of the limits $\eps\to0$ and $t\to\infty$ for a large class of observables. Our framework accommodates unbounded and irregular coefficients, degenerate structures, and weakly mixing dynamics. We illustrate its scope with three applications: {\it (i)} a uniform-in-time averaging principle for fast-slow systems; {\it (ii)} a uniform Smoluchowski–Kramers approximation for degenerate Langevin systems, yielding convergence of the joint position-scaled velocity law and global-in-time asymptotics of key thermodynamic functionals (e.g., total energy, entropy production, free energy); and {\it (iii)} the first uniform-in-time periodic homogenization result for SDEs with distributional drifts.

18. 奇异与无界系数的随机微分方程的导数估计

Derivative estimates for SDEs with singular and unbounded coefficients

作者: Pengcheng Xia, Longjie Xie, Xicheng Zhang

类别: math.PR, math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00685v1

链接: http://arxiv.org/abs/2604.00685v1

摘要: We develop a unified PDE-probabilistic framework for pointwise gradient and Hessian estimates of Markov semigroups associated with stochastic differential equations with singular and unbounded coefficients. Under mild local structural assumptions on the diffusion matrix and integrability/regularity conditions on the drift, we obtain quantitative sharp short-time regularization estimates as well as long-time decay bounds (including exponential and polynomial rates) for the first and second spatial derivatives of the semigroup. A distinctive feature of our results is the explicit dependence of these estimates on local norms of the coefficients (through scale-invariant quantities), without requiring any global smoothness, boundedness or uniform ellipticity. In particular, our approach allows for degenerate or highly irregular behavior at infinity, subject to suitable local ellipticity and Lyapunov/ergodicity controls. As applications, we establish solvability and regularity results for Poisson equations on the whole space with singular coefficients, and we derive pointwise gradient estimates for SDEs with distributional drifts via a Zvonkin-type transform.

19. 二阶平均场博弈的弱-强唯一性

Weak-Strong Uniqueness for Second-Order Mean-Field Games

作者: Rita Ferreira, Diogo Gomes, Bashayer Majrashi

类别: math.AP, math.DS

发布日期: 2026-04-01

arXiv ID: 2604.00681v1

链接: http://arxiv.org/abs/2604.00681v1

摘要: We extend the weak-strong uniqueness principle for mean-field game (MFG) systems to a broad class of second-order stationary and time-dependent problems. Under standard monotonicity, growth, and coercivity assumptions on the Hamiltonian, and relying strictly on the integrability exponents guaranteed by the existing theory for monotone MFG systems, we show that any weak solution must coincide with a given strong solution. Our analysis covers models with spatially dependent scalar diffusion coefficients, using monotonicity arguments and a coefficient-adapted mollification strategy to manage the variable diffusion terms. We extend this strategy to establish weak-strong uniqueness in the corresponding second-order, initial-terminal, time-dependent setting. Finally, to address the critical quadratic growth regime, we derive a new a priori second-order estimate for a stationary MFG system with logarithmic coupling. This estimate provides quantitative bounds on the solution in terms of the data, and yields weak-strong uniqueness in the range where the improved integrability yields $L^2$ control of the density. Since numerical and approximation methods for MFGs naturally yield weak solutions in the monotonicity sense, whereas strong solutions are known to exist in many settings, our results identify any weak limit produced by such methods with the strong solution whenever one exists.

20. 道路网络上多类交通流模型的离散伴随梯度计算

Discrete adjoint gradient computation for multiclass traffic flow models on road networks

作者: Paola Goatin, Axel Klar, Carmen Mezquita-Nieto

类别: math.AP, math.OC

发布日期: 2026-04-01

arXiv ID: 2604.00670v1

链接: http://arxiv.org/abs/2604.00670v1

摘要: This paper applies a discrete adjoint gradient computation method for a multi-class traffic flow model on road networks. Vehicle classes are characterized by their specific velocity functions, which depend on the total traffic density, resulting in a coupled hyperbolic system of conservation laws. The system is discretized using a Godunov-type finite volume scheme based on demand and supply functions, extended to handle complex junction coupling conditions – such as merges and diverges – and boundary conditions with buffer lengths to account for congestion spillback. The optimization of different travel-related performance metrics, including total travel time and total travel distance, is formulated as a constrained minimization problem and is accomplished through the use of an adjoint gradient approach, allowing for an efficient computation of sensitivities with respect to the chosen time-dependent control variables. Numerical simulations on a sample network demonstrate the efficiency of the proposed framework, particularly as the number of control parameters increases. This approach provides a robust and computationally efficient solution, making it suitable for large-scale traffic network optimization.

21. 非局部扩散问题的Li-Yau和Harnack估计

Li-Yau and Harnack estimates for nonlocal diffusion problems

作者: Rico Zacher

类别: math.AP, math.PR

发布日期: 2026-04-01

arXiv ID: 2604.00645v1

链接: http://arxiv.org/abs/2604.00645v1

摘要: These notes give a brief introduction to differential Harnack inequalities and summarise the main results of the mini-course Li-Yau and Harnack estimates for nonlocal diffusion problems'', presented by the author at the Seasonal School on PDEs Oscillation Phenomena, PDEs, and Applications: A Comprehensive School in Mathematical Analysis’’, held at Ghent University in October 2025.

22. 使用凸优化验证线性偏微分方程的适定性

Verifying Well-Posedness of Linear PDEs using Convex Optimization

作者: Declan S. Jagt, Matthew M. Peet

类别: math.AP, eess.SY

发布日期: 2026-04-01

arXiv ID: 2604.00573v1

链接: http://arxiv.org/abs/2604.00573v1

摘要: Ensuring that a PDE model is well-posed is a necessary precursor to any form of analysis, control, or numerical simulation. Although the Lumer-Phillips theorem provides necessary and sufficient conditions for well-posedness of dissipative PDEs, these conditions must hold only on the domain of the PDE – a proper subspace of $L_{2}$ – which can make them difficult to verify in practice. In this paper, we show how the Lumer-Phillips conditions for PDEs can be tested more conveniently using the equivalent Partial Integral Equation (PIE) representation. This representation introduces a fundamental state in the Hilbert space $L_{2}$ and provides a bijection between this state space and the PDE domain. Using this bijection, we reformulate the Lumer-Phillips conditions as operator inequalities on $L_{2}$. We show how these inequalities can be tested using convex optimization methods, establishing a least upper bound on the exponential growth rate of solutions. We demonstrate the effectiveness of the proposed approach by verifying well-posedness for several classical examples of parabolic and hyperbolic PDEs.

23. 热方程在带扩散型动态边界条件的半空间中的基本解与扩散极限

Fundamental solution and diffusion limits for the heat equation in a half-space with a diffusive dynamical boundary condition

作者: Kazuhiro Ishige, Sho Katayama, Tatsuki Kawakami

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00497v1

链接: http://arxiv.org/abs/2604.00497v1

摘要: We derive an explicit representation of the fundamental solution to the heat equation in a half-space of ${\mathbb R}^N$ with a diffusive dynamical boundary condition, and establish sharp pointwise upper and lower bounds. We also investigate qualitative properties of the associated solutions, including precise decay estimates. Furthermore, we analyze the diffusion limits of solutions to the initial–boundary value problem, and reveal the role of the diffusive dynamical boundary condition in the behavior of solutions.

24. 布朗运动的陷阱行为

Trap behaviors for Brownian motions

作者: Raffaela Capitanelli, Mirko D’Ovidio

类别: math.PR, math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00474v1

链接: http://arxiv.org/abs/2604.00474v1

摘要: This paper investigates the relationship between the geometric properties of a domain and the diffusion dynamics of Brownian motion, with a specific focus on the phenomenon of “trapping” in terms of the behavior of stochastic processes.

25. 关于具有密度抑制运动性、间接信号产生和外部源的Keller-Segel系统

On a Keller–Segel System with Density-Suppressed Motility, Indirect Signal Production, and External Sources

作者: Yujiao Sun, Jie Jiang

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00378v1

链接: http://arxiv.org/abs/2604.00378v1

摘要: This paper investigates an initial-Neumann boundary value problem for a Keller–Segel system with parabolic-parabolic-ODE coupling. The model incorporates a signal-dependent, non-increasing motility function that, through indirect signal production, captures a self-trapping effect suppressing cellular movement at high densities. We establish the global existence of classical solutions in arbitrary spatial dimensions for a broad class of non-increasing motility functions, both with and without external source terms. Furthermore, we demonstrate that any external damping source exhibiting superlinear growth ensures uniform-in-time boundedness. Conversely, in the absence of such damping, solutions may become unbounded as time tends to infinity. More precisely, in the two-dimensional homogeneous case with the exponentially decaying motility function $γ(v) = e^{-v}$, a critical mass phenomenon emerges: classical solutions remain uniformly bounded for subcritical initial mass, while supercritical initial masses can lead to infinite-time blow-up. Our analysis relies on the construction of carefully designed auxiliary functions along with refined comparison methods and iteration arguments.

26. 超定值问题自由边界的存在性：精确条件、正则性及物理应用

Existence of free boundaries for overdetermined value problems: Sharp conditions, regularity, and physical applications

作者: Mohammed Barkatou, Samira Khatmi

类别: math.AP

发布日期: 2026-04-01

arXiv ID: 2604.00376v1

链接: http://arxiv.org/abs/2604.00376v1

摘要: This paper provides necessary and sufficient conditions for the existence of free boundaries in overdetermined value-problems (ODVP) for the Laplacian, and sufficient conditions for the bi-Laplacian, when the overdetermined boundary condition is non-constant. Using classical integral inequalities (Cauchy-Schwarz, Hölder, Hardy, eigenvalue bounds, Pohozaev and Reilly identities), we derive existence results for a broad class of free boundary problems arising in potential theory, plate theory, electromagnetism, and shape optimization. A regularity result for minimizers in the $C$-GNP class is established using the thickness function and the Wiener criterion, based on the geometric description of cusp points given in \cite{barkatou2002}. New results include refined estimates via interpolation inequalities, stability under perturbations, and connections with isoperimetric inequalities. Detailed analysis of the necessary and sufficient conditions for the existence of solutions to the ODVP is given. The physical interpretation of the bi-Laplacian problem $\mathcal{B}(f,g)$ in the Kirchhoff-Love theory of thin plates is emphasized.

27. 无界算子的局部化Davies生成元

Localised Davies generators for unbounded operators

作者: Jeffrey Galkowski, Maciej Zworski

类别: math-ph, math.AP

发布日期: 2026-03-31

arXiv ID: 2604.00306v1

链接: http://arxiv.org/abs/2604.00306v1

摘要: A classical Davies generator provides a Lindbladian for which the Gibbs state is stationary. Its construction involves precise knowledge of the Bohr spectrum or equivalently state evolution for all times. Recently Chen, Kastoryano and Gilyen proposed a construction involving localisation in time and carried out it out in the case of finite dimensional Hilbert spaces. The resulting generators are called quantum Gibbs samplers as the corresponding Lindblad is expected to settle to the Gibbs state. In this note, we show that the construction also works for classes of unbounded operators, including pseudodifferential operators used in the study of classical/quantum correspondence in Lindblad evolution.

28. Einstein标量场方程在ekpyrotic机制中的大爆炸稳定性与等向化

Big bang stability and isotropisation for the Einstein-scalar field equations in the ekpyrotic regime

作者: Florian Beyer, David Garfinkle, James Isenberg, Todd A. Oliynyk

类别: gr-qc, math.AP

发布日期: 2026-03-31

arXiv ID: 2604.00297v1

链接: http://arxiv.org/abs/2604.00297v1

摘要: It has been shown that, in spacetime dimensions $n\geq 3$, that the Kasner-scalar field solutions to the Einstein-scalar fields equations with potential $V_0 e^{-s φ}$, where $s<s_c=\sqrt{\frac{8(n-1)}{n-2}}$ and $V_0\in \mathbb{R}$, are nonlinearly stable to the past and terminate at a quiescent big bang singularity over the full range of sub-critical Kasner exponents. In particular, the spatially homogeneous and isotropic solutions, the Friedman-Lemaitre-Robertson-Walker (FLRW) spacetimes, to the Einstein-scalar field equations are stable in this sense for $s<s_c$ and $V_0\in\mathbb{R}$. While perturbations of the sub-critical Kasner-scalar field family of solutions, including the FLRW solutions, are asymptotically velocity term dominated (AVTD) near the big bang, they do not in general isotropise near the big bang singularity. Rather, they remain highly anisotropic, even for small perturbations of the isotropic FLRW solutions. In this article, we establish the stability of FLRW solutions to the Einstein-scalar field equations for the potential parameter values $s>s_c$ and $V_0<0$. Such scalar field potentials are known in the literature as \textit{ekpyrotic}. In particular, we prove that the FLRW solutions to the Einstein-scalar field equations are nonlinearly stable to the past and terminate at a quiescent, crushing AVTD big bang singularity. A distinguishing property of these perturbed spacetimes is that they isotropise towards the big bang.