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1. 基于能量变分解的欧拉-科特韦格与纳维-斯托克斯-科特韦格系统有限体积格式的收敛性分析
Convergence analysis for a finite-volume scheme for the Euler- and Navier-Stokes-Korteweg system via energy-variational solutions
作者: Thomas Eiter, Jan Giesselmann, Robert Lasarzik, Philipp Öffner, Robert Sauerborn
类别: math.NA, math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29880v1
摘要翻译: 本文针对欧拉-科特韦格(EK)方程和纳维-斯托克斯-科特韦格(NSK)方程，研究了一种保结构有限体积格式。我们证明了在网格加密过程中，该格式的数值解收敛于EK或NSK方程的能量变分解。能量变分是一种新型解概念，最近被引入到包括EK系统在内的双曲守恒律研究中，本文将其推广到NSK模型。我们的证明基于利用保结构格式的性质建立一致估计，并借助能量变分形式在自然能量空间弱收敛下的稳定性来完成。
Comment: 无
链接: http://arxiv.org/abs/2603.29880v1
1. 研究对象和背景
本研究论文关注的是可压缩流体动力学中一类重要的偏微分方程组——Euler-Korteweg（EK）系统和Navier-Stokes-Korteweg（NSK）系统。这些方程组描述了含有毛细效应的可压缩流体行为，是对经典Euler方程和Navier-Stokes方程的重要推广。经典流体方程在捕捉某些物理现象（如界面毛细效应）方面存在局限性，而Korteweg在1901年首次提出的弥散界面模型则能够有效地描述这类物理过程。
具体而言，论文考虑的EK和NSK系统具有如下形式的质量守恒方程和动量守恒方程。质量守恒方程描述了密度ρ和速度场u的演化关系，而动量方程则包含了压力项、粘性应力项以及由Korteweg应力张量引入的三阶导数项，该三阶项通过 capillary 系数κ来刻画界面效应。粘性应力由牛顿流变学定律给出，包含剪切粘性系数µ和体积粘性系数η。当µ和η为零时，系统退化为无粘性的EK系统；当它们为正时，则得到完整的NSK系统。压力函数p(ρ) = kργ（其中γ > 1）描述了流体的本构关系。
这些方程组在数学上具有显著的挑战性。即使在经典的双曲型守恒律层面，解通常仅局部时间存在，激波的形成要求引入弱解概念。近年来关于弱解非唯一性的严格证明从根本上挑战了这些方程在弱解层面的预测能力。对于不可压缩Euler方程和不可压缩Navier-Stokes方程的Leray解，已有关于非唯一性的研究结果。在可压缩情形下，缺乏毛细效应的理论更是面临着弱（熵）解的非唯一性和低正则性设定下的适定性困难。
为了保证大范围初始数据下解的整体存在性，文献中提出了多种广义解概念。测度值解是最早的推广之一，随后发展出更精细的耗散弱解概念，通过与测度值解相关联的振荡测度的质心来定义，并在动量方程的弱形式中引入缺陷测度。最近，能量变分解作为广义解家族中最新的成员被引入，它识别了众多偏微分方程系统中的共同凸性结构，使得可以在合适的近似中取极限。在某些情况下，高维缺陷测度可以等价地由一维辅助变量表示。
2. 主要定理或主要结果
本论文的核心结果是证明了有限体积格式的数值解在网格加密过程中收敛到EK或NSK系统的能量变分解。具体而言，主要定理7建立了以下收敛性结果。
在给定的假设条件下，包括κ > 0、压力函数p(ρ) = kργ（k > 0且γ > 1），以及初始数据满足有限总能量E(ρ₀, m₀) < ∞，假设对所有h > 0，近似系统在均匀时间区间(0, T)上存在解，且数值耗散参数λ_h满足lim_{h→0} h/λ_h = 0，则存在一个满足特定正则性条件的极限(ρ, m, E)。该极限具有以下正则性：ρ属于L∞(0, T; H¹(Ω) ∩ L^γ(Ω))，m属于L∞(0, T; L^{\frac{2γ}{γ+1}}(Ω; R²))，E属于BV([0, T])。
从收敛性结果来看，密度序列ρ_h在L∞(0, T; L^γ(T²))中弱星收敛于ρ，而梯度序列∇₊ρ_h、∇₋ρ_h和∇_cρ_h在L∞(0, T; L²(T²; R²))中弱收敛于∇ρ。密度本身在C([0, T]; L^p(T²))中对所有p ∈ [1, ∞)强收敛于ρ。动量ρ_h u_h在L∞(0, T; L^{\frac{2γ}{γ+1}}(T²; R²))中弱星收敛于m，离散能量E_h在BV([0, T])中弱星收敛于E。
当粘性系数µ和η均为正时，还可以建立速度场的收敛性。具体而言，u_h在L²(0, T; L²(T²; R²))中弱收敛于u，梯度∇₊u_h在L²(0, T; L²(T²; R^{2×2}))中弱收敛于∇u，且ρ_h u_h在L²(0, T; L^q(T²; R²))中对所有q ∈ [1, 2)强收敛于m。若密度从下方一致有界，则可以得到更强的收敛性：u_h在L²(0, T; L^q(T²; R²))中对所有q ∈ [1, 2)强收敛于u。
该极限(ρ, m, E)正是EK系统（当µ = η = 0时）或NSK系统（当µ, η > 0时）的能量变分解，其定义依赖于一个辅助变量E作为能量的上界E ≥ E(ρ, m)，并满足特定的变分不等式。这个定理的关键突破在于，据作者所知，这是首次在不假设强解存在性的情况下证明EK和NSK方程数值格式的收敛性。
3. 研究方法、关键技术和核心工具
本研究采用的研究方法建立在结构保持有限体积格式与能量变分解框架相结合的基础之上。首先，论文回顾了文献中已有的结构保持有限体积格式，该格式针对二维EK和NSK系统设计，具有以下基本性质：总质量和总动量守恒，以及自然能量耗散不等式满足。
格式构造采用周期边界条件的矩形计算域，通过笛卡尔网格将区域划分为有限体积单元。网格单元定义为Ω_{i,j} = (x_{i-1/2}, x_{i+1/2}] × (y_{j-1/2}, y_{j+1/2}]，其中网格步长h_x = L_x/M，h_y = L_y/N。数值解被构造为分片常函数，通过网格上的离散值(ρ_{i,j}, u_{i,j})表示。离散差分算子包括前向差分∇₊、后向差分∇₋和中心差分∇_c，以及拉普拉斯算子Δ_h的离散形式。
格式的常微分方程组（6）包含了质量方程、动量x分量方程和动量y分量方程。质量方程采用对流通量加上数值耗散项的形式。动量方程则更为复杂，包含对流项、压力梯度项、粘性项（由离散差分算子实现）以及Korteweg应力项的离散化。Korteweg项的处理利用了三阶导数的等价形式，通过恒等式将ρ∇Δρ表示为散度形式，从而实现弱形式的稳定离散化。
能量耗散性质的证明依赖于数值耗散参数λ的适当选择。论文证明了当λ取值为最大速度加声速的一半即λ = max{|u_{i,j}| + √{p’(ρ_{i,j})}/2时，离散能量（8）满足非负的耗散率（11）。具体而言，能量耗散率包含剪切变形张量dev(D⁺(u))的L²范数、体积膨胀率div⁺u的L²范数，以及capillary项λ_h|Δ_h ρ|²的积分。これらの项的组合确保了能量随时间单调递减，为建立一致先验估计奠定了基础。
能量变分解框架是本文的核心理论工具。为了定义能量变分解，引入辅助变量E作为能量E(ρ, m)的上界，其中总能量定义为E(ρ, m) = ∫[η̃(ρ, m) + (κ/2)|∇ρ|²] dx，函数η̃在ρ > 0时等于|m|²/(2ρ) + P(ρ)，在(ρ, m) = (0, 0)时为零，其余情况为无穷大，P由关系式ρP’(ρ) = p(ρ) + P(ρ)确定。
通过从能量耗散不等式中减去弱形式方程并添加由正则权重K乘以能量缺陷E(ρ, m) − E的项，得到了完整的变分不等式（5）。正则权重K: C²(T²; R²) → [0, ∞)被选择为K(φ) = 2‖(∇φ){sym,−}‖{L^∞(T²)} + max{(γ−1, 1)}‖divφ‖{L^∞(T²)}，其中(A){sym,−}表示对称矩阵A的负半定部分。这个权重函数的精心选择确保了变分不等式在变量(ρ, m, E)关于弱收敛是下半连续的。
收敛性证明首先从能量耗散不等式（11）推导出一系列一致先验估计。引理11建立了多个关键的有界性结果：ρ_h |u_h|²和P(ρ_h)在L∞(0, T; L¹(T²))中有界，|∇₊_h ρ_h|在L∞(0, T; L²(T²))中有界，离散速度梯度在适当的空间中也有界。这些估计独立于h、κ、µ和η。
进一步，通过时间导数的一致有界性（26），结合Aubin-Lions引理，可以从一致有界性获得紧性。具体而言，利用密度的时间正则性嵌入W^{−1, \frac{2γ}{γ+1}}(T²)和H^{−1}(T²)的对偶空间，结合空间嵌入，可以得到密度的强收敛性。
4. 与之前工作的比较
本研究在数值分析和偏微分方程理论两个层面都与前人的工作存在重要联系和显著进展。在数值分析方面，关于Euler方程和Navier-Stokes方程的有限体积格式收敛性已有丰富的研究历史。早期的研究如利用Lax-Friedrichs和Rusanov通量的经典有限体积格式收敛性证明，以及Godunov方法对完整Euler方程的收敛性结果，都为更复杂系统的分析奠定了基础。
近年来，高阶格式的收敛性分析取得了重要进展，包括残差分布格式、流通校正有限元方法以及间断Galerkin方法的收敛性结果。特别地，具有热力学兼容性的双曲有限体积格式的收敛性也得到了证明。在液晶Ericksen-Leslie方程的背景下，能量变分解框架已被成功用于证明结构继承有限元格式的收敛性。这些工作表明广义解框架在数值格式收敛性分析中具有重要价值。
本研究与这些工作的主要区别在于所针对的方程系统不同。EK和NSK系统包含三阶导数项（即Korteweg项），这增加了格式设计和分析的特殊性。此外，本论文首次将能量变分解框架应用于这类弥散界面模型，并将其从EK系统扩展到具有粘性效应的NSK模型。
在格式构造层面，本文采用的有限差分/有限体积格式源自先前工作[25]，该格式被设计为结构保持格式，即保持总质量和总动量守恒，并满足能量耗散不等式。本文的主要贡献在于建立了该格式的收敛性分析，填补了理论空白。
从解理论角度看，能量变分法作为一种广义解概念具有独特优势。相较于测度值解和Young测度解，能量变分解避免了引入复杂的测度空间，而是通过引入实值能量缺陷和一个精心选择的权重函数，就能在标准的能量空间中建立变分不等式。这种方法利用了能量变分 formulation 关于弱收敛的稳定性，无需借助多维缺陷测度来描述无法直接识别的极限项。
值得注意的是，能量变分解在其他物理系统中的应用已有成功案例，包括不可压缩和可压缩Euler方程、双相Navier-Stokes方程以及多凸弹性双曲系统。在这些情形下，能量变分解已被证明比变分解和Young测度值解更强。本文将这一框架进一步推广到NSK系统，丰富了能量变分法的应用范围。
关于数值格式，论文末尾简要提及了NSK系统的其他数值方法，如能量一致间断Galerkin方法和局部间断Galerkin方法，以及EK系统的熵稳定有限差分格式。然而，据作者声称，本论文首次在不强加强解存在性假设的前提下，证明了EK和NSK方程数值格式的收敛性，这代表了该领域的重要理论进展。
2. 气体动力学的欧拉系统
The Euler system of gas dynamics
作者: Eduard Feireisl
类别: math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29619v1
摘要翻译: 本文是一篇综述文章，重点介绍气体动力学欧拉系统适定性/不适定性方面的若干近期研究成果。系统的解被识别为一致逼近的极限，这些逼近由两类方式产生：一类是物理上更复杂的问题，特别是纳维-斯托克斯-傅里叶系统；另一类是数值实验中的近似格式。研究探讨了热力学第一定律和第二定律所蕴含的基本原理在确定唯一物理可容许解中所起的作用。
Comment: 无
链接: http://arxiv.org/abs/2603.29619v1
1. 研究对象和背景
本文的研究对象是气体动力学中的欧拉系统，这是一个描述理想气体在连续介质力学框架下时间演化的数学模型。欧拉系统由三个场方程构成，分别反映基本的物理原理：质量守恒、动量守恒和能量守恒。具体而言，质量守恒方程描述为 $\partial_t \rho + \text{div}_x \mathbf{m} = 0$，其中 $\rho = \rho(t,x)$ 是质量密度，$\mathbf{m} = \rho\mathbf{u}(t,x)$ 是动量；动量守恒方程为 $\partial_t \mathbf{m} + \text{div}_x \frac{\mathbf{m} \otimes \mathbf{m}}{\rho} + \nabla_x p = 0$，$p$ 为压强；能量守恒方程为 $\partial_t E + \text{div}_x \left(E + p\right)\frac{\mathbf{m}}{\rho} = 0$，$E$ 是总能。
从数学角度来看，欧拉系统是一类非线性守恒律系统的典型代表，具有重要的理论价值和研究意义。然而，该系统存在一个根本性的困难：对于相当广泛的一类初始数据，其经典解或强解会在有限时间内产生奇异性，即所谓的激波（shock waves）。因此，研究者们不得不将解的概念扩展到弱解的范畴，但即使在弱解的框架下，解的唯一性和存在性问题仍未得到完全解决。
本文的核心关注点在于探讨欧拉系统的适定性问题，特别是一致逼近（consistent approximation）在识别物理允许的唯一解中所起的关键作用。一致逼近可以通过两种途径产生：一是在物理上更为复杂的系统中取渐近极限，最典型的是Navier-Stokes-Fourier系统；二是数值计算中的近似格式。论文还特别强调了热力学第一定律和第二定律在确定唯一物理允许解方面的作用。
2. 主要定理或主要结果
本文综述了关于欧拉系统适定性的若干重要结果，主要包括以下几个方面的定理和发现。
关于经典解的存在性，Schochet的结果表明，对于足够正则的初始数据，欧拉系统在局部时间范围内是适定的。具体而言，若热力学函数 $p = p(\rho,\vartheta)$、$e = e(\rho,\vartheta)$、$s = s(\rho,\vartheta)$ 满足吉布斯定律且具有适当的光滑性，初始数据属于 $W^{3,2}$ 空间且满足相容性条件，则存在 $T > 0$ 使得欧拉系统在该时间段内存在经典解。
然而，论文着重指出的是弱解层面的不适定性。根据Klingenberg等人的结果，存在这样的黎曼数据，使得黎曼问题在任意时间区间内承认无穷多解，这些解都满足熵 admissibility条件。特别地，这些解在激波两侧分别取常数值，且在整个区域内满足一致的界。这一结果表明，即使是熵容许的弱解也远非唯一。
进一步地，论文证明了"wild data"（野数据）在 $L^q$ 拓扑下是稠密的。对于任意给定的正则初始数据，可以找到与之任意接近的野数据，使得从该数据出发存在无穷多个熵容许的弱解。这意味着不适定性是一个普遍现象而非例外情况。
论文的核心贡献在于引入了耗散测度值（DMV）解的概念。DMV解是欧拉系统的一种广义解，它通过一致逼近的渐近极限来定义，并具有一些基本性质。论文建立了DMV解的存在性理论，表明任何一致逼近的序列都存在收敛子列，其极限为某个DMV解。此外，论文还建立了DMV解与经典弱解之间的关系，并讨论了熵不等式在唯一性选择中的作用。
3. 研究方法、关键技术和核心工具
本文采用的核心方法是凸积分（convex integration）技术，该技术由De Lellis和Székelyhidi在研究欧拉方程的开创性工作中发展起来。凸积分方法能够构造出大量满足守恒律但违背物理直觉的解，从而揭示了欧拉系统弱解层面上的不适定性。
在DMV解的理论构建中，关键工具包括一致逼近（consistent approximation）的概念及其误差分析。一致逼近被定义为满足欧拉系统并在渐近极限 $\varepsilon \to 0$ 过程中误差消失的逼近序列。具体而言，论文定义了一致逼近需要满足的五个条件：近似总能量不等式、连续性方程的近似形式、动量平衡的近似形式、熵最小值原理以及近似熵不等式。
热力学框架的建立是另一核心技术。论文采用多方气体状态方程 $p = (\gamma - 1)\rho e$，其中 $\gamma > 1$ 为绝热指数。通过吉布斯定律 $\vartheta Ds = De + p D\frac{1}{\rho}$，可以推导出熵的表达式，并建立熵的传输方程。在存在奇异性时，熵方程被松弛为熵不等式 $\partial_t (\rho s) + \text{div}_x (\rho s \mathbf{u}) \geq 0$，这正是热力学第二定律的体现。
凸性和热力学稳定性在理论分析中扮演重要角色。总能量 $E(\rho,\mathbf{m},S) = \frac{|\mathbf{m}|^2}{2\rho} + \rho e(\rho,S)$ 被要求是下半连续且在其定义域上严格凸的凸函数，这与系统的热力学稳定性假设等价。
论文还讨论了Krylov、Cardona和Kapitanski的选择过程、基于凸性的两步选择程序，以及如何利用热力学第二定律作为单一选择准则来从众多DMV解中甄选出物理允许的解。
4. 与之前工作的比较
本文的主要贡献在于将近年来关于欧拉系统适定性的多个研究方向进行了系统性的整合与综述。
在经典解存在性方面，本文引用的Schochet定理代表了传统的正则性理论，该理论建立在先验估计和局部存在性结果之上。然而，本文指出这类结果的生命期可能有限，因为非线性双曲系统的特征线可能在有限时间内相交，导致解的破裂。
与DiPerna早期的测度值解理论相比，本文提出的DMV解概念有所发展和深化。DiPerna最初引入测度值解是为了处理含有振荡的逼近序列，而本文将这一思想扩展到更一般的渐近极限问题，特别关注了与Navier-Stokes-Fourier系统的联系。
与De Lellis和Székelyhidi的凸积分工作相比，本文不仅利用该技术揭示了弱解的不适定性，更重要的是探索了如何通过物理约束（特别是热力学第二定律）来克服这种不适定性。论文系统比较了耗散解与弱解之间的关系，指出DMV解概念如何弥合了理论解与物理现实之间的鸿沟。
在解的选择准则方面，本文对比了DiPerna最大性准则、Dafermos最大性准则以及基于凸性的选择程序。特别值得注意的是，本文提出了将热力学第二定律作为唯一选择准则的新视角，这与传统的admissibility条件有着本质区别。
最后，论文坦诚地指出，尽管在理解欧拉系统的适定性方面取得了显著进展，但对于任意有限能量初始数据是否存在全局定义的熵容许弱解这一基本问题仍然悬而未决，这为未来的研究指明了方向。
了解领域论文（摘要翻译）
1. 广义非守恒可压缩双流体模型的Cauchy问题的波的传播
Wave propagation of a generic non–conservative compressible two–fluid model
作者: Zhigang Wu, Weike Wang, Yinghui Zhang
类别: math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29799v1
链接: http://arxiv.org/abs/2603.29799v1
摘要翻译
中文标题: 广义非守恒可压缩双流体模型的Cauchy问题的波的传播
摘要翻译:
建立了三维空间中广义非守恒可压缩双流体模型Cauchy问题的广义Huygens原理。这项工作填补了该理论中的一个关键空白，因为先前的研究结果仅限于具有完整守恒律的系统，或Green函数中特定补偿相消所导致的"等效"守恒结构。事实上，这里研究的真正非守恒模型不属于这些范畴，并呈现出两个主要的分析挑战。首先，其固有的非守恒结构阻碍了直接使用有效应用于守恒系统的技术（如变量重构）。其次，其Green函数包含一个与分数密度相关的-1阶Riesz算子，由此产生的所谓Riesz波-IV与标准热核相比，呈现出更慢的时间衰减和更差的空间可积性，这要求与Huygens波进行新颖的精细卷积估计。为了克服这些困难，我们建立了一个精确非线性耦合的框架，包括Riesz波-IV与Huygens波的相互作用。一个关键的步骤是提取非守恒压力项的增强衰减率。通过将这些项重构为涉及分数密度和分数密度特定组合的乘积，然后证明该组合比单个密度衰减更快，我们满足了关键卷积估计的最低要求。这使得我们能够通过构造本质上全新的非线性估计来封闭非线性假设。我们分析的成功源于该模型的特殊结构，特别是等压条件。更广泛地说，这里发展的精细非线性估计可广泛应用于广泛类别的非守恒可压缩流体模型。
2. Vlasov-Poisson-Fokker-Planck方程在低正则性空间中的最优稳定性阈值
Optimal stability threshold in lower regularity spaces for the Vlasov-Poisson-Fokker-Planck equations
作者: Weiren Zhao, Ruizhao Zi
类别: math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29204v1
链接: http://arxiv.org/abs/2603.29204v1
摘要翻译
中文标题: Vlasov-Poisson-Fokker-Planck方程在低正则性空间中的最优稳定性阈值
摘要翻译: 本文研究了带有弱Fokker-Planck碰撞项的Vlasov-Poisson方程的最优稳定性阈值。我们证明了，若初始扰动在临界加权空间 $H_x^{\log}L^2_{v}(\langle v\rangle^m)$ 中的模为 $ν^{\frac{1}{2}}$，则该解在同一空间中保持相同量级。此外，时空型Landau阻尼成立，即 $\|E\|_{L^2_tL^2_x}\lesssim ν^{\frac{1}{2}}$；点态型Landau阻尼也成立，即 $\|E(t)\|_{L^2}\lesssim ν^{1/2}\langle t\rangle^{-N}$ 对任意 $N>0$ 在 $t\geq ν^{-1}$ 时成立。我们还证明了在 $H^{1}_xL^2_v(\langle v\rangle^m)$ 中存在初始扰动，其模为 $ν^{\frac12-\frac32ε_0}$（其中任意 $ε_0>0$），使得增强耗散在以下意义下失效：存在 $0<T\ll ν^{-\frac13}$ 使得 \begin{align*} |\langle v\rangle^m f_{\neq}(T)|{L^2_xL^2_v}\gtrsim \frac{1}{ν^{δ_1}}|\langle v\rangle^m f{\neq}(0)|_{ H^1_xL^2_v} \end{align*} 其中 $δ_1>0$。本文解决了[Bedrossian; arXiv: 2211.13707]中提出的关于低正则性空间中精确稳定性阈值的开问题。其核心思想是构造一个具有非常精确表达式的波算子 $\mathbf{D}$ 以吸收非局部项，即 \begin{align*} \mathbf{D}[\partial_tg+v\cdot \nabla_x g+E\cdot\nabla_v μ]=(\partial_t +v\cdot \nabla_x)\mathbf{D}[g]. \end{align*}
3. 渐近稳定化具有非简并迁移率和奇异势的相场方程的弱解
Asymptotic stabilization of weak solutions to phase-field equations with non-degenerate mobility and singular potential
作者: Maurizio Grasselli, Andrea Poiatti
类别: math.AP
发布日期: 2026-03-30
arXiv ID: 2603.29048v1
链接: http://arxiv.org/abs/2603.29048v1
摘要翻译
中文标题: 渐近稳定化具有非简并迁移率和奇异势的相场方程的弱解
摘要翻译:
具有奇异势的相场模型中的一个常见范式是：整体时间弱解仅在经历渐近正则化后才收敛到单一平衡态。然而，在arXiv:2510.17296中，我们引入了一种新方法来建立具有非简并迁移率和奇异势的Cahn-Hilliard方程及相关耦合系统的解收敛到单一平衡态的条件，在非常一般的假设下：我们仅要求存在满足能量不等式的整体弱解，然后利用Lojasiewicz-Simon不等式。在此基础上，我们迈出了重要的一步。我们放宽了证明轨迹预紧性所需的假设，这是完整证明的基本要素。由于这一推广，我们能够处理所有主要的相场模型，包括完全一般的奇异势，在三维空间中，其渐近行为至今仍是一个未解决的问题。具体而言，我们将新方法应用于具有非线性扩散的Cahn-Hilliard方程、守恒Allen-Cahn方程以及非局部Cahn-Hilliard方程。在二阶方程的情形，De Giorgi迭代论证对于证明弱解渐近一致地远离纯相（严格分离性质）至关重要，这是应用Lojasiewicz-Simon不等式的关键要素。我们预计这一新技术将在耦合问题中具有更广泛的应用范围，包括流体动力学模型，如守恒Allen-Cahn-Navier-Stokes系统或具有非简并迁移率的非局部Abels-Garcke-Grün系统。多组分模型的进一步应用也是可能的。
不相关论文（基本信息）
1. 几何正规性质下区域水平集的几何性质
Geometric Properties of Level Sets for Domains under Geometric Normal Property
作者: Mohammed Barkatou
类别: math.AP
发布日期: 2026-03-31
arXiv ID: 2603.30026v1
链接: http://arxiv.org/abs/2603.30026v1
摘要: This work is devoted to the study of the geometric properties of level sets for solutions of elliptic boundary value problems in domains satisfying the geometric normal property with respect to a convex set $C$ ($C$-GNP class). We prove that, for the classical Dirichlet problem as well as for the coupled system $\mathcal{B}(f,g)$ (related to the biharmonic plate equation), the level sets inherit the $C$-GNP structure. We establish their star-shapedness property, exact formulas for their mean curvature, and characterize their asymptotic behavior near singular contact points (cusps). We also study the stability of these level sets under the Hausdorff convergence of domains, establishing their convergence in the Hausdorff sense, in the compact sense, and in $L^1$. The analysis relies on adapted coarea formulas, leading to Faber-Krahn, Szegö-Weinberger, and Payne-Rayner type isoperimetric inequalities. In order to go beyond the purely qualitative framework of the $C$-GNP class, we introduce and analyze two new quantitative geometric measures: the thickness function $τ_Ω$ and the convexity gap $γ(Ω)$. We rigorously study their behavior, regularity, and continuity under Hausdorff convergence. These theoretical tools open up new perspectives for shape optimization under geometric constraints, the study of free boundary problems, and the geometric control of latent spaces in machine learning.
2. 具有Kato类位势的薛定谔算子的Brunn-Minkowski不等式
A Brunn-Minkowski inequality for Schrödinger operators with Kato class potentials
作者: Alessandro Carbotti
类别: math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29989v1
链接: http://arxiv.org/abs/2603.29989v1
摘要: In this paper we prove a Brunn-Minkowski inequality for the first Dirichlet eigenvalue of a Schrödinger type operator $\mathcal{H}_V:=-\operatorname{div}(A\nabla)+V$, where $V$ is convex and Kato decomposable, using the trace class property of the generated semigroup. As a consequence, using the ultracontractivity of the semigroup we obtain the log-concavity of the ground state.
3. 非可积薛定谔方程非平凡远场类多孤子解的构造
Construction of a multi-soliton-like solutions for non-integrable Schrödinger equations with non-trivial far field
作者: Jordan Berthoumieu
类别: math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29906v1
链接: http://arxiv.org/abs/2603.29906v1
摘要: This article provides a naturel sequel of previous works [6, 4] regarding the stability of travelling waves for a general one-dimensional Schrödinger equation (N LS) with non-zero condition at infinity. The aim of this article is twofold. First, we prove the asymptotic stability of well-prepared chains of dark solitons and secondly, we construct an asymptotic N -soliton-like solution, which is an exact solution of (N LS), the large-time dynamics of which is similar to a decoupled chain of solitons.
4. 分数阶Airy方程的位势方法
The Method of Potentials for the Airy Equation of Fractional Order
作者: Rakhimov Kamoladdin
类别: math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29887v1
链接: http://arxiv.org/abs/2603.29887v1
摘要: In this work, initial-boundary value problems for the time-fractional Airy equation are considered on different intervals. We study the properties of potentials for this equation and, using these properties, construct solutions to the considered problems. The uniqueness of the solution is proved using an analogue of the Gronwall-Bellman inequality and an a priori estimate.
5. 完全非线性logistic方程及其保护区
Fully nonlinear logistic equations with sanctuary
作者: Isabeau Birindelli, Giulio Galise, Fabiana Leoni
类别: math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29885v1
链接: http://arxiv.org/abs/2603.29885v1
摘要: For the fully nonlinear stationary logistic equation ${\mathcal F}(x,D^2u)+μu=k(x)u^p$ with $p>1$ and $k(x)\geq 0$, in a bounded domain with Dirichlet boundary condition, we determine, in terms of $μ$, the existence and uniqueness or the nonexistence of a positive solution. Furthermore, we study the asymptotic behavior of the solutions when $μ$ approaches the boundary points of the existence range.
6. 一个新的Duhamel型原理及其在几何（不）等式中的应用
A new Duhamel-type principle with applications to geometric (in)equalities
作者: Michele Caselli, Luca Gennaioli
类别: math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29823v1
链接: http://arxiv.org/abs/2603.29823v1
摘要: We introduce a simple new method, based on the Caffarelli-Silvestre extension and a Duhamel-type formula, to derive exact pointwise identities for fractional commutators and nonlinear compositions associated with the fractional Laplacian on general Riemannian manifolds. As applications, we obtain a pointwise fractional Leibniz rule, a fractional Bochner’s formula with an explicit Ricci curvature term, apparently the first of this kind, and exact remainders in the Córdoba-Córdoba and Kato inequalities for the fractional Laplacian. All these formulas are new even in the Euclidean space.
7. Alt-Phillips能量极小化约束映射的部分正则性
Partial regularity for minimizing constraint maps for the Alt-Phillips energy
作者: Rada Ziganshina
类别: math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29810v1
链接: http://arxiv.org/abs/2603.29810v1
摘要: In this paper, we establish an $\varepsilon$-regularity theorem for minimizers of an Alt-Phillips type functional subject to constraint maps. We prove that under sufficiently small energy, the minimizers exhibit regularity, and hence proving the smoothness of these maps. From here, we bootstrap to optimal regularity.
8. 一维带负类库仑势的薛定谔算子的色散估计
Dispersive estimates for Schrödinger operators with negative Coulomb-like potentials in one dimension
作者: Akitoshi Hoshiya, Kouichi Taira
类别: math.AP, math-ph
发布日期: 2026-03-31
arXiv ID: 2603.29731v1
链接: http://arxiv.org/abs/2603.29731v1
摘要: In this paper, we consider the dispersive estimates for Schrödinger operators with Coulomb-like decaying potentials, such as $V(x)=-c|x|^{-μ}$ for $|x|\gg 1$ with $0<μ<2$, in one dimension. As an application, we establish both the standard and orthonormal Strichartz estimates for this model. One of the difficulties here is that perturbation arguments, which are typically applicable to rapidly decaying potentials, are not available. To overcome this, we derive a WKB expression for the spectral density and use a variant of the degenerate stationary phase formula to exploit its oscillatory behavior in the low-energy regime.
9. 带退化乘性噪声的随机偏微分方程的强Feller性、不可约性与不变测度唯一性
Strong Feller property, irreducibility, and uniqueness of the invariant measure for stochastic PDEs with degenerate multiplicative noise
作者: Luca Scarpa, Margherita Zanella
类别: math.PR, math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29711v1
链接: http://arxiv.org/abs/2603.29711v1
摘要: We establish strong Feller property and irreducibility for the transition semigroup associated to a class of nonlinear stochastic partial differential equations with multiplicative degenerate noise. As a by-product, we prove uniqueness of the invariant measure under no strong-dissipativity assumptions. The drift of the equation diverges exactly where the noise coefficient vanishes, resulting in a competition between the dissipative effects and the degeneracy of the noise. We propose a method to measure the accumulation of the solution towards the potential barriers, allowing to give rigorous meaning to the inverse of the degenerate noise coefficient. From the mathematical perspective, this is one of the first contributions in the literature establishing strong Feller properties and irreducibility in the multiplicative degenerate case, and opens up novel investigation paths in the direction of regularisation effects and ergodicity in the degenerate-noise framework. From the application perspective, the models cover interesting scenarios in physics, in the context of evolution of relative concentrations of mixtures, under the influence of thermodynamically-relevant potentials of Flory-Huggins type.
10. 通过控制相互作用的$N$人微分对策的分布式均衡：存在性、唯一性及大$N$极限
Distributed Equilibria for $N$-Player Differential Games with Interaction through Controls: Existence, Uniqueness and Large $N$ Limit
作者: Hei Jie Lam, Alpár R. Mészáros
类别: math.AP, math.OC
发布日期: 2026-03-31
arXiv ID: 2603.29707v1
链接: http://arxiv.org/abs/2603.29707v1
摘要: We establish the existence and uniqueness of distributed equilibria to possibly nonsymmetric $N$ player differential games with interactions through controls under displacement semimonotonicity assumptions. Surprisingly, the nonseparable framework of the running cost combined with the character of distributed equilibria leads to a set of consistency relations different in nature from the ones for open and closed loop equilibria investigated in a recent work of Jackson and the second author. In the symmetric setting, we establish quantitative convergence results for the $N$ player games toward the corresponding Mean Field Games of Controls (MFGC), when $N\to+\infty$. Our approach applies to both idiosyncratic noise driven models and fully deterministic frameworks. In particular, for deterministic models distributed equilibria correspond to open loop equilibria, and our work seems to be the first in the literature to provide existence and uniqueness of these equilibria and prove the large $N$ convergence in the MFGC setting. The sharpness of the imposed assumptions is discussed via a set of explicitly computable examples in the linear quadratic setting.
11. Kantorovich位势的定量唯一性
Quantitative Uniqueness of Kantorovich Potentials
作者: William Ford
类别: math.OC, math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29595v1
链接: http://arxiv.org/abs/2603.29595v1
摘要: This paper studies the uniqueness of solutions to the dual optimal transport problem, both qualitatively and quantitatively (bounds on the diameter of the set of optimisers). On the qualitative side, we prove that when one marginal measure’s support is rectifiably connected (path-connected by rectifiable paths), the optimal dual potentials are unique up to a constant. This represents the first uniqueness result applicable even when both marginal measures are concentrated on lower-dimensional subsets of the ambient space, and also applies in cases where optimal potentials are nowhere differentiable on the supports of the marginals. On the quantitative side, we control the diameter of the set of optimal dual potentials by the Hausdorff distance between the support of one of the marginal measures and a regular connected set. In this way, we quantify the extent to which optimisers are almost unique when the support of one marginal measure is almost connected. This is a consequence of a novel characterisation of the set of optimal dual potentials as the intersection of an explicit family of half-spaces.
12. 分数阶薛定谔方程的正则性与海森堡群上的次拉普拉斯乘子
Regularity of fractional Schrödinger equations and sub-Laplacian multipliers on the Heisenberg group
作者: Aksel Bergfeldt
类别: math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29588v1
链接: http://arxiv.org/abs/2603.29588v1
摘要: We define functions of the sub-Laplacian $Δ$ on the Heisenberg group $\mathbb H^d$ as Fourier multipliers. In this setting, we show that the solution $u$ of the free fractional Schrödinger equation $i\partial_tu + (-Δ)^νu = 0, u|_{t=0} = u_0$, for any $ν > 0$, satisfies the Hardy space estimate that $\|u(t,\cdot)\|_{H^p(\mathbb H^d)}$ is estimated from above by $(1 + t)^{Q|1/p-1/2|}\|(1-Δ)^{νQ|1/p-1/2|}u_0\|_{H^p(\mathbb H^d)}$, with $Q = 2d + 2$, for all $p \in (0,\infty)$, and the corresponding estimate with $p=\infty$ in $\mathrm{BMO}(\mathbb H^d)$. This is done via a general regularity result for parameter dependent sub-Laplacian Fourier multipliers. We prove also that Bessel potential spaces on the Heisenberg group correspond to Sobolev spaces in the same way as in Euclidean space, also for Hardy spaces.
13. 磨光方法在分支运输与树形结构优化中的应用
A Mollification Approach to Ramified Transport and Tree Shape Optimization
作者: Alberto Bressan, Giacomo Vecchiato, Ludmil Zikatanov
类别: math.OC, math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29567v1
链接: http://arxiv.org/abs/2603.29567v1
摘要: The paper analyzes a mollification algorithm, for the numerical computation of optimal irrigation patterns. This provides a regularization of the standard irrigation cost functional, in a Lagrangian framework. Lower semicontinuity and Gamma-convergence results are proved. The technique is then applied to some numerical optimization problems, related to the optimal shape of tree roots and branches.
14. 感染年龄结构化竞争模型的全局渐近稳定性
On the global asymptotic stability of an infection-age structured competitive model
作者: Simon Girel, Quentin Richard
类别: math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29464v1
链接: http://arxiv.org/abs/2603.29464v1
摘要: We investigate an infection-age structured competitive epidemiological model involving multiple strains. While classical results establish competitive exclusion when a unique maximal basic reproduction number exists, we provide here a complete characterization of the asymptotic behavior for an arbitrary number of populations without assuming uniqueness of the maximal reproduction number. By means of integrated semigroups theory, persistence results, and Lyapunov functionals, we establish global asymptotic stability of equilibria and extend previous results obtained for simpler (ODE) models. A key contribution lies in overcoming technical difficulties related to the definition and differentiation of Lyapunov functionals, as well as in refining arguments based on the LaSalle invariance principle.
15. 反应扩散系统逼近快速扩散方程
Reaction-Diffusion System Approximation to the Fast Diffusion Equation
作者: Hideki Murakawa, Florian Salin
类别: math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29434v1
链接: http://arxiv.org/abs/2603.29434v1
摘要: This paper proposes a novel reaction-diffusion system approximation tailored for singular diffusion problems, typified by the fast diffusion equation. While such approximation methods have been successfully applied to degenerate parabolic equations, their extension to singular diffusion-where the diffusion coefficient diverges at low densities-has remained unexplored. To address this, we construct an approximating semilinear system characterized by a reaction relaxation parameter and a time-derivative regularizing parameter. We rigorously establish the well-posedness of this system and derive uniform a priori estimates. Using compactness arguments, we prove the convergence of the approximate solutions to the unique weak solution of the target singular diffusion equation under three distinct asymptotic regimes: the simultaneous limit, the limit via a parabolic-elliptic system, and the limit via a uniformly parabolic equation. This approach effectively transfers the diffusion singularity into the reaction terms, yielding a highly tractable system for both theoretical analysis and computation. Finally, we present numerical experiments that validate our theoretical convergence results and demonstrate the practical efficacy of the proposed approximation scheme.
16. 四维临界诺依曼系统在竞争机制下的爆破解
Blowing-up solutions to a critical 4D Neumann system in a competitive regime
作者: Qing Guo, Angela Pistoia, Shixin Wen
类别: math.AP
发布日期: 2026-03-31
arXiv ID: 2603.29329v1
链接: http://arxiv.org/abs/2603.29329v1
摘要: We build blowing-up solutions to the critical elliptic system with Neumann boundary condition, \begin{equation*} \begin{cases} -Δu_1 + λu_1 = u_1^{3} -βu_1u_2^2 & \text{in } Ω, -Δu_2 + λu_2 = u_2^{3} -βu_1^2u_2 & \text{in } Ω, \frac{\partial u_1}{\partialν} = \frac{\partial u_2}{\partialν} = 0, & \text{on } \partial Ω, \end{cases} \end{equation*} when $λ>0$ is sufficiently large in a competitive regime (i.e. $ β>0$) and in a domain $Ω\subset\mathbb R^4$ with smooth protrusions.
17. 利用厚度函数证明沙格霍里安定理的新自包含证明
A new, self-contained proof of Shahgholian’s theorem using the thickness function
作者: Mohammed Barkatou
类别: math.AP
发布日期: 2026-03-30
arXiv ID: 2603.29084v1
链接: http://arxiv.org/abs/2603.29084v1
摘要: This note presents a new, self-contained proof of Shahgholian’s geometric theorem on quadrature surfaces using the thickness function and level set methods. By relying on a radial parametrisation and fundamental maximum principles, the proof avoids the technical complexity of the moving plane method. It provides a more conceptual view, revealing that the overdetermined condition forces all level sets to be parallel to the convex hull of the support of the measure.
18. 关于 $\mathbb{R}^n$ 中 Allen-Cahn 方程的若干注记
Some remarks on the Allen-Cahn equation in $\mathbb{R}^n$
作者: Gabriele Ferla
类别: math.AP
发布日期: 2026-03-30
arXiv ID: 2603.29060v1
链接: http://arxiv.org/abs/2603.29060v1
摘要: In this short note we present new results on a higher-dimensional generalization of De~Giorgi’s conjecture for Allen–Cahn type equations, focusing on dimensions $n \ge 9$. Although counterexamples are known in this regime, our goal is to identify assumptions on solutions that still enforce one-dimensional symmetry. We prove an analogue of Savin’s theorem in arbitrary dimension: for energy-minimizing solutions whose level sets enjoy n-7 directions of monotonicity, we deduce one-dimensional symmetry. In the same spirit, we extend these ideas to nonlocal phase transitions, and we discuss an application to free boundary problems Finally, we establish a counterpart of the Ambrosio–Cabré theorem for solutions that are not necessarily energy minimizers and may lack bounded energy density, assuming instead n-2 directions of monotonicity everywhere. Overall, this note aims to further strengthen the connection between phase transition models and minimal surface theory.
19. 正方形区域中带有两个硬障碍物的第一狄利克雷本征值的最小化
Small-hole minimization of the first Dirichlet eigenvalue in a square with two hard obstacles
作者: Baruch Schneider, Diana Schneiderová, Yifan Zhang
类别: math.AP, math.SP
发布日期: 2026-03-30
arXiv ID: 2603.29015v1
链接: http://arxiv.org/abs/2603.29015v1
摘要: We study the small-hole minimization problem for the first Dirichlet eigenvalue in the square
where two equal disjoint hard circular obstacles of radius $r$ move inside $Q$. We prove that, as $r\to0$, every minimizing configuration consists, up to the dihedral symmetries of the square and interchange of the two holes, of two true corner-tangent obstacles located at adjacent corners. The argument is organized by geometric branches. On the side-tangent one-hole branch, odd reflection and simple-eigenvalue $u$-capacity asymptotics show that the true corner is the unique asymptotic minimizer. For configurations with holes near two distinct corners, an exact polarization argument proves that the adjacent true-corner pair strictly beats the opposite pair. For same-corner clusters, a reflected comparison principle reduces the two-hole cell problem to a scalar one-hole inequality, which is then closed by an explicit competitor. We also include a reproducible finite element validation that supports the analytic branch ordering.
20. 奇异测度的预设雅可比和散度的Lipschitz可解性
Lipschitz solvability of prescribed Jacobian and divergence for singular measures
作者: Luigi De Masi, Andrea Marchese
类别: math.AP, math.FA
发布日期: 2026-03-30
arXiv ID: 2603.28912v1
链接: http://arxiv.org/abs/2603.28912v1
摘要: Let $μ$ be a finite Radon measure on an open set $Ω\subset\mathbb{R}^d$, singular with respect to the Lebesgue measure. We prove Lusin-type solvability results for the prescribed divergence equation and the prescribed Jacobian equation with Lipschitz solutions. More precisely, for every $\varepsilon>0$ and every Borel datum $f \colon Ω\to \mathbb{R}$ there exists a vector field $V\in C^1_c(Ω;\mathbb{R}^d)$ such that $\operatorname{div} V=f$ on a compact set $K\subsetΩ$ with $μ(Ω\setminus K)<\varepsilon$, and $\operatorname{Lip}(V)\le (1+\varepsilon)\|f\|_{L^\infty(Ω,μ)}$. Similarly, for every Borel datum $g\colon Ω\to \mathbb{R}$ there exists a map $Φ$ with $Φ-\operatorname{Id}\in C^1_c(Ω;\mathbb{R}^d)$ such that $\det DΦ=g$ on a compact set $K\subsetΩ$ with $μ(Ω\setminus K)<\varepsilon$, and $\operatorname{Lip}(Φ-\operatorname{Id})\le (1+\varepsilon)\|g-1\|_{L^\infty(Ω,μ)}$. The maps $V$ and $Φ-\operatorname{Id}$ can be chosen arbitrarily small in supremum norm.
21. 余维数$d_*-d+1$的蒙日-安培方程组的完全柔性
Full flexibility of the Monge-Ampère system in codimension $d_*-d+1$
作者: Wentao Cao, Jonas Hirsch, Dominik Inauen, Marta Lewicka
类别: math.AP
发布日期: 2026-03-30
arXiv ID: 2603.28909v1
链接: http://arxiv.org/abs/2603.28909v1
摘要: We prove that $\mathcal{C}^{1,α}$ solutions to the Monge-Ampère system in dimension $d$ and codimension $k= d_*-d+1$, where $d_*$ denotes the Janet dimension, are dense in the space of continuous functions, for every Hölder exponent $α<1$. Our result strengthens the statement in [Lewicka 2022], obtained for $k = 2d_*$ and based on ideas from [Källen 1978] in the context of the isometric immersion system. It also generalizes the result of [Inauen-Lewicka 2025], where full flexibility was established in dimension $d=2$ and codimension $k=2$. The same proof scheme further yields local full flexibility of isometric immersions of $d$-dimensional Riemannian metrics into Euclidean space of dimension $d_* + 1$, generalizing the result in [Lewicka 2025] proved for $d=k=2$. By using techniques of [Conti-De Lellis-Szekelyhidi], the result can be extended to compact manifolds, in codimension $(d+1)d_*-d+1$.
22. Reissner–Nordström时空上的带电标量场 I：积分能量估计
Charged scalar fields on Reissner–Nordström spacetimes I: integrated energy estimates
作者: Dejan Gajic
类别: gr-qc, math-ph, math.AP, math.DG
发布日期: 2026-03-30
arXiv ID: 2603.28874v1
链接: http://arxiv.org/abs/2603.28874v1
摘要: This is the first part of a series of papers deriving the precise, late-time behaviour and (in)stability properties of charged scalar fields on near-extremal Reissner–Nordström spacetimes via energy estimates. In this paper, we establish global, weighted integrated energy decay and energy boundedness estimates for solutions to the charged scalar field equation on (near-)extremal Reissner–Nordström(–de Sitter) spacetimes. These estimates extend to Reissner–Nordström spacetimes away from extremality under the assumption of mode stability on the real axis. Together with the companion paper [Gaj26], this paper forms the first global quantitative analysis of the charged scalar field equation on asymptotically flat black hole spacetimes, without a smallness assumption on the scalar field charge. Due to a coupling of the degeneration of the red-shift effect with the presence of superradiance at the linearized level, charged scalar fields on Reissner–Nordström spacetimes also probe some of the main difficulties encountered when studying the (neutral) wave equation on extremal Kerr spacetimes.
23. 带电标量场在雷斯纳-诺德斯特洛姆时空中的演化 II：晚期尾部与不稳定性
Charged scalar fields on Reissner–Nordström spacetimes II: late-time tails and instabilities
作者: Dejan Gajic
类别: gr-qc, math-ph, math.AP, math.DG
发布日期: 2026-03-30
arXiv ID: 2603.28861v1
链接: http://arxiv.org/abs/2603.28861v1
摘要: This is the second part of a series of papers deriving the precise, late-time behaviour and (in)stability properties of charged scalar fields on near-extremal Reissner–Nordström spacetimes via energy estimates. In this paper, we use purely physical-space based methods to establish the precise late-time behaviour of solutions to the charged scalar field equation in the form of oscillating and decaying late-time tails that satisfy inverse-power laws, assuming global integrated energy decay estimates, which are proved in the companion paper [Gaj26]. This paper provides the first pointwise decay estimates for charged scalar fields on black hole backgrounds without an assumption of smallness of the scalar field charge. We also prove the existence of asymptotic instabilities for the radiation field along future null infinity and, in the extremal case, also along the future event horizon. Both the energy methods and the precise late-time asymptotics derived in this paper are expected to play an important role in future nonlinear studies of black hole dynamics in the context of the spherically symmetric (Einstein–)Maxwell–charged scalar field equations, as well as in the context of extremal Kerr spacetimes.