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1. Time-periodic non-radial solutions near monotone vortices in linearized 2D Euler

线性化二维欧拉方程中单调涡旋附近的周期性非径向解

作者: Ángel Castro, Daniel Lear

类别: math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01730v1

链接: http://arxiv.org/abs/2512.01730v1

详细分析

中文标题: 线性化二维欧拉方程中单调涡旋附近的周期性非径向解

研究对象和背景

本文研究的是二维不可压缩、无粘性流体的欧拉方程在径向对称涡旋背景下的线性化系统，特别是围绕一个严格单调递减的径向涡量剖面 $\varpi(r)$ 的动力学行为。原始的二维欧拉方程以涡量形式给出如下：

$$ \partial_t \omega + (u \cdot \nabla)\omega = 0, $$

$$ \nabla \cdot u = 0, $$

其中速度场 $u$ 由涡量 $\omega$ 通过 Biot–Savart 定律重构：

$$ u = \nabla^\perp \Delta^{-1} \omega, \quad \nabla^\perp = (-\partial_2, \partial_1), $$

$$ \Delta^{-1}\omega(x) = \frac{1}{2\pi} \int_{\mathbb{R}^2} \log|x - y| \, \omega(y) \, dy. $$

考虑空间为全平面 $\mathbb{R}^2$，使用极坐标 $(r,\theta)$ 来描述变量。目标是分析当初始涡量接近某个径向对称且单调递减的稳态涡旋 $\varpi(r)$ 时，其小扰动 $w(r,\theta,t)$ 在时间演化中的长期行为。

将总涡量写为 $\omega = \varpi + w$，代入原方程并忽略高阶非线性项，得到关于扰动 $w$ 的线性化二维欧拉方程：

$$ \partial_t w + u[\varpi] \cdot \nabla w + u[w] \cdot \nabla \varpi = 0, $$

在极坐标下可进一步展开为：

$$ \partial_t w + u_r[w] \partial_r \varpi + \frac{u_\theta[\varpi]}{r} \partial_\theta w = 0, $$

其中 $u_r[f], u_\theta[f]$ 分别表示由涡量函数 $f$ 产生的径向与角向速度分量，具体表达式涉及积分核（见原文第2页）。该系统刻画了非径向扰动如何随时间演化。

此前的重要工作，如 Bedrossian–Coti-Zelati–Vicol [4] 所示：若背景涡量剖面 $\varpi(r)$ 满足严格单调性条件（即 $\partial_r \varpi < 0$ 对所有 $r > 0$），则任何非径向扰动都会经历所谓的 inviscid damping（无粘阻尼）和 axisymmetrization（轴对称化）——即扰动的速度场趋于零，整体结构趋向于径向对称状态。

这种现象类似于剪切流中的 Landau 阻尼，在数学上表现为速度场的衰减估计，例如：

$$ \|u_r\|_{L^2_{\text{weight}}} \lesssim (1+t^2)^{-1/2}, \quad \|u_\theta\|_{L^2_{\text{weight}}} \lesssim (1+t^2)^{-1}. $$

然而，本文的核心动机在于探究这一稳定机制是否稳健（robust）。特别地，作者提出一个问题：如果我们将原本严格单调的剖面 $\varpi(r)$ 微小扰动为仅“非增”而非“严格递减”的剖面 $\varpi_\varepsilon(r)$，是否仍能保持 inviscid damping？答案是否定的。

为此，作者聚焦于一个具体的代数涡旋剖面：

$$ \varpi_0(r) = \frac{1}{1 + r^2}, $$

这是光滑、正、严格递减的函数，并满足适当的衰减条件（虽不完全符合文献[4]中的(V1)(V2)，但便于计算）。

他们构造了一族逼近剖面 $\varpi_\varepsilon(r)$，定义如下（见原文(17)式）：

$$ \varpi_\varepsilon(r) = \begin{cases} \varpi_0(r) + \varpi_0(1+\varepsilon/2) - \varpi_0(1-\varepsilon/2), & 0 < r \leq 1 - \varepsilon/2, \\ \varpi_0(1 + \varepsilon/2), & 1 - \varepsilon/2 < r < 1 + \varepsilon/2, \\ \varpi_0(r), & r \geq 1 + \varepsilon/2. \end{cases} $$

此构造的关键特征是：在区间 $(1 - \varepsilon/2, 1 + \varepsilon/2)$ 上，$\varpi_\varepsilon$ 是常数，因此其导数 $\partial_r \varpi_\varepsilon = 0$，破坏了严格单调性。尽管如此，它在低阶 Hölder 范数 $C^\alpha$（$0 < \alpha < 1$）下可以任意逼近原始剖面 $\varpi_0$，因为：

$$ \|\varpi_0 - \varpi_\varepsilon\|_{C^\alpha} \leq \|\varpi_0'\|_{L^\infty} \varepsilon^{1 - \alpha} \to 0 \quad \text{as } \varepsilon \to 0. $$

值得注意的是，$\varpi_\varepsilon \in W^{1,\infty}$ 但不在 $C^1$ 中，说明其正则性较低。这也提示我们，稳定性结论可能依赖于背景剖面的正则性等级。

接下来，作者试图寻找线性化方程的时间周期非径向解，形式为：

$$ w(r,\theta,t) = W(r, \theta - \lambda t), $$

这对应于旋转行波解。代入后转化为求解稳态方程：

$$ -\lambda \partial_\theta W + u_r[W]\partial_r \varpi + \frac{u_\theta[\varpi]}{r} \partial_\theta W = 0. $$

进一步假设具有 $m$-折对称性的模式：

$$ W_n(r,\theta) = W_n(r) \cos(n\theta), \quad n \geq 2, $$

从而将问题简化为关于函数 $W_n(r)$ 的奇异积分方程（见原文(10)式）：

$$ (\lambda + c(r)) W_n(r) - \frac{1}{n} \int_0^\infty K_n\left(\frac{r}{s}\right) \varpi'_\varepsilon(s) W_n(s) \, ds = 0, $$

其中核函数 $K_n$ 定义为：

$$ K_n(z) = \begin{cases} z^{-n}, & z > 1, \\ z^n, & z \leq 1, \end{cases} \quad \text{and} \quad c(r) := -\frac{u_\theta[\varpi](r)}{r}. $$

该方程需在 $\varpi'_\varepsilon$ 的支撑集上求解，即：

$$ \mathrm{supp}(\varpi'_\varepsilon) = [0, 1 - \varepsilon/2] \cup [1 + \varepsilon/2, \infty), $$

这是一个不连通区域，带来了显著的技术挑战。此外，由于 $c(r)$ 在中间平坦区附近表现出跳跃行为（见图2、3），存在某个 $r_* \in (1 - \varepsilon/2, 1 + \varepsilon/2)$ 使得 $c(r_*)$ 可被调整至满足 $\lambda + c(r_*) = 0$，从而允许共振发生。

边界条件方面，从核的渐近性质可知：

$$ \lim_{r \to 0^+} r^{2-n} f(r) = 0, \quad \lim_{r \to \infty} r^n f(r) = 0, $$

这些自然出现在积分方程的可积性要求中。

综上所述，本文的研究对象是一个定义在全平面上的线性化二维欧拉方程，研究其在低正则性、非严格单调背景剖面下的动力学行为。重点在于揭示：即使扰动非常小（在 $C^\alpha$ 意义下），只要破坏了严格单调性，就可能出现时间周期振荡解，从而否定 inviscid damping 和 axisymmetrization 的普遍成立。

主要定理或主要结果

论文的主要结果由以下定理陈述：

定理 1.1（原文 Theorem 1.1）
固定整数 $n \in \mathbb{N}$ 足够大，以及 $0 < \alpha < 1$。则对任意 $\varepsilon > 0$，存在：

  • 一个径向剖面 $\varpi_\varepsilon \in W^{1,\infty}(\mathbb{R}_+)$，
  • 函数 $h_\varepsilon \in C^\infty(\mathrm{supp}(\partial_r \varpi_\varepsilon))$，
  • 实数 $\lambda_\varepsilon \in \mathbb{R}$，

使得三元组 $(\varpi_\varepsilon, h_\varepsilon, \lambda_\varepsilon)$ 满足如下积分方程（见原文(14)式）：

$$ (\lambda_\varepsilon + c(r)) h_n(r) - \frac{1}{n} \int_0^\infty K_n\left(\frac{r}{s}\right) \partial_s \varpi_\varepsilon(s) h_n(s) \, ds = 0, \quad \forall r \in \mathrm{supp}(\partial_r \varpi_\varepsilon), $$

并且满足逼近关系：

$$ \left\|(1 + r^2)^{-1} - \varpi_\varepsilon \right\|_{C^\alpha} \leq \varepsilon. $$

此外，对应的扰动解 $w(r,\theta,t) = h_n(r) \partial_r \varpi_\varepsilon(r) \cos(n(\theta - \lambda_\varepsilon t))$ 是线性化方程的时间周期非径向解。

注记与解释：

  • 原始剖面 $(1 + r^2)^{-1}$ 是严格单调递减的，而 $\varpi_\varepsilon$ 在区间 $(1 - \varepsilon/2, 1 + \varepsilon/2)$ 上为常数，故仅为非增，不再满足严格单调。
  • 尽管 $\varpi_\varepsilon$ 接近原始剖面（在 $C^\alpha$ 拓扑下），但其所支持的线性化系统存在非平凡时间周期解，直接反驳了 inviscid damping 成立的可能性。
  • 这表明：inviscid damping 与 axisymmetrization 不在低正则性扰动下稳定，尤其当严格单调性被轻微破坏时。

研究方法、关键技术和核心工具

a) 核心分析学工具

本文采用了多种深刻的分析技术，结合了奇异积分方程理论、渐近分析、常微分方程极限分析以及不动点论证。

首先是积分方程的转化与分解。作者将原始的单个奇异积分方程（定义在两个分离区间）转化为一个耦合的积分方程组，分别作用于左支 $[0, 1 - \varepsilon/2]$ 和右支 $[1 + \varepsilon/2, \infty)$。为此引入分段函数：

$$ f(r) = \begin{cases} f_L(r), & r \in [0, 1 - \varepsilon/2], \\ f_R(r), & r \in [1 + \varepsilon/2, \infty). \end{cases} $$

接着进行变量缩放（rescaling），令：

$$ x = \frac{r}{1 \mp \varepsilon/2}, $$

从而将两个区间统一映射到固定的参考域 $[0,1]$ 和 $[1,\infty)$，消除 $\varepsilon$ 对定义域的影响。这是处理奇异性依赖参数问题的标准技巧。

其次，利用泰勒展开与渐近分析，对缩放后的系数函数 $c_L(x,\varepsilon), c_R(x,\varepsilon)$ 展开为 $\varepsilon$ 的幂级数：

$$ c_L(x,\varepsilon) = c_L(x,0) + \partial_\varepsilon c_L(x,0)\varepsilon + O(\varepsilon^2), $$

其中显式计算出：

$$ c_L(x,0) = -\frac{\log(1 + x^2)}{2x^2}, \quad \partial_\varepsilon c_L(x,0) = \frac{1}{4} + \frac{1}{2(1+x^2)} - \frac{\log(1+x^2)}{2x^2}. $$

这一展开至关重要，因为它允许在 $\varepsilon \to 0$ 极限下研究系统的收敛行为。

第三，作者将积分方程通过微分操作转化为常微分方程系统。注意到原方程形如：

$$ (\lambda + c(r)) f(r) = \text{integral term}, $$

对其两边求导，并利用核函数的结构性质，最终导出一组一阶 ODEs（见 Section 3.4）。例如，在左右区域分别得到：

$$ \frac{d}{dx} f_L^*(x) = F_L(x; f_L^*, f_R^*, \lambda, \varepsilon), \quad \frac{d}{dx} f_R^*(x) = F_R(x; f_L^*, f_R^*, \lambda, \varepsilon). $$

这一步将无限维积分问题降维为有限维动态系统，极大简化了分析。

第四，采用极限 ODE 方法。考虑 $\varepsilon \to 0$ 时，左右 ODE 系统趋于各自的极限系统。作者分别研究：

  • 当 $x \in [0,1]$ 时，$\varepsilon = 0$ 下的左极限 ODE；
  • 当 $x \in [1,\infty)$ 时，右极限 ODE。

证明这两个极限系统各自存在唯一解，并具有良好性质（如正则性、边界行为）。

第五，运用连续性方法与差估计。通过比较 $\varepsilon > 0$ 与 $\varepsilon = 0$ 的解之间的差异，建立精细的先验估计（见 Sections 5.1–5.2）。关键在于控制 $\|f_\varepsilon - f_0\|$ 在适当范数下的大小，通常借助 Gronwall 不等式或逐次迭代法。

最后，不动点定理被用于闭合论证。作者将待求的频率 $\lambda$ 视为未知参数，构造映射：

$$ \Lambda: \lambda \mapsto \tilde{\lambda}(\varepsilon), $$

使得当 $\Lambda(\lambda) = \lambda$ 时，对应方程有解。通过证明该映射在某个紧凸集上有不动点（例如 Schauder 不动点定理），完成存在性证明（见 Section 6.4）。

b) 技术性难题的解决技巧

主要技术难点包括：

  1. 非连通支撑集带来的耦合效应：由于 $\mathrm{supp}(\varpi'_\varepsilon)$ 分裂为两部分，积分方程中出现跨区域积分项（如 $\int_{1+\varepsilon/2}^\infty \cdots$ 出现在左端方程中），导致系统强耦合。作者通过变量重标和函数延拓，将其转化为标准形式，并利用渐近一致性处理交叉项。

  2. 共振机制的设计：为了使 $(\lambda + c(r))$ 在某点为零，作者精心选择 $\lambda_\varepsilon$ 使得 $\lambda_\varepsilon + c(r_*) = 0$ 对某个 $r_* \in (1 - \varepsilon/2, 1 + \varepsilon/2)$ 成立。由于 $c(r)$ 在该区间内跳跃，存在间隙可供插入 $\lambda$，形成共振，激发周期解。

  3. 低正则性下的良定性问题：$\varpi_\varepsilon$ 仅有 $W^{1,\infty}$ 正则性，其二阶导含 Dirac delta 项（见 Remark 1.6），导致 $c'''(r)$ 出现奇异测度。作者通过分布意义下的计算和截断逼近，确保所有运算在弱意义下成立。

  4. 高频模式的选择（$n$ 大）：选择足够大的 $n$ 可增强核 $K_n(r/s)$ 的局部化效应，使得积分主要集中在 $r \approx s$ 区域，从而提高方程的“刚性”，有利于解的存在唯一性分析。

综上，本文巧妙融合几何构造、渐近分析与非线性泛函方法，突破了传统稳定性理论的局限。

与之前工作的比较

a) 突破与改进

本文最根本的贡献在于揭示了 inviscid damping 的非稳健性，直接挑战了近年来关于二维欧拉方程线性稳定的主流认知。

相较于 Bedrossian–Coti-Zelati–Vicol [4] 的开创性工作，后者证明：只要背景涡量剖面 $\varpi(r)$ 满足严格单调性（V3）、快速衰减（V1）和光滑性（V2），则所有非径向扰动均会经历 inviscid damping 与 axisymmetrization。其结论依赖于严格的单调性假设。

本文则反向构造：给定任意严格单调剖面（如 $(1 + r^2)^{-1}$），可在其任意邻近（在 $C^\alpha, \alpha < 1$ 意义下）找到一个仅非增的剖面 $\varpi_\varepsilon$，使得线性化系统存在非平凡时间周期解。这意味着：

  • 严格单调性不可替代：它是 inviscid damping 成立的本质条件，而非技术性假设。
  • 低正则性扰动可诱发失稳：即使扰动极小（Hölder 小），一旦落入 $C^1$ 之外的空间，即可破坏稳定性机制。

这与 Shear Flow 理论中的类似现象呼应。例如 Lin–Zeng [32] 发现 Couette 流的非线性 inviscid damping 在 $H^s$（$s < 3/2$）下失效；而 Gui–Xie–Xu [22] 与 Coti-Zelati–Elgindi–Widmayer [14] 表明 Poiseuille 流在高正则性下具有刚性（无非平凡行波），但在 $C^{k-1,-}$ 拓扑下 Drivas–Nualart [18] 构造出光滑驻波。

本文延续这一思想，首次在涡旋背景下建立了类似的正则性门槛现象：只有在足够高正则性（保证严格单调）下，damping 才成立；否则可能出现持续振荡。

此外，与 Coti-Zelati–Zillinger [15] 的结果相比，后者允许某些非单调剖面但仍得 damping，因其依赖更强的谱条件（H1）：

$$ n^2 |\partial_y U(y)| - \mathrm{sgn}(\partial_y U) \partial_y^3 U(y) \geq \gamma n^2 |\partial_y U(y)|. $$

本文所构造的 $\varpi_\varepsilon$ 因含有平台区，$\partial_r^2 \varpi_\varepsilon$ 含符号相反的 delta 项，导致 $\partial_y^3 U(y)$ 在两点同号，违反 (H1)，故不矛盾。

总之，本文实现了三个层面的突破：

  1. 概念上：指出“严格单调性”是 inviscid damping 的临界条件；
  2. 技术上：发展了一套适用于低正则、非连通支撑剖面的积分方程分析框架；
  3. 哲学上：强调 PDE 稳定性结论可能极度依赖于函数空间的拓扑选择——在 $C^1$ 中稳定的现象，在 $C^\alpha$ 中可能完全崩溃。

这一发现对理解二维湍流中相干结构的持久性提供了新的视角。

2. Stability threshold of the 2D Boussinesq system near Couette flow in an infinite channel

无限通道中近库埃特流的二维Boussinesq系统稳定性阈值

作者: Tao Liang, Jiahong Wu, Xiaoping Zhai

类别: math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01159v1

链接: http://arxiv.org/abs/2512.01159v1

详细分析

中文标题: 无限通道中近库埃特流的二维Boussinesq系统稳定性阈值

研究对象和背景

本文研究的是在无限通道区域 $\mathbb{R} \times [-1, 1]$ 中，带有无滑移边界条件的二维 Navier-Stokes-Boussinesq (NSB) 系统 在 Couette 流附近的非线性稳定性问题。该系统的物理背景源于浮力驱动流动（buoyancy-driven flows），广泛应用于大气科学、海洋环流、地幔对流以及工程热传导等领域。Boussinesq 近似假设密度变化仅在浮力项中体现（如由温度或盐度差异引起），而在惯性项中忽略其影响，从而简化了原始的 Navier-Stokes 方程。

考虑如下初始-边界值问题：

$$ \begin{cases} \partial_t \mathbf{v} + \mathbf{v} \cdot \nabla \mathbf{v} - \nu(\partial_x^2 + \partial_y^2)\mathbf{v} + \nabla P = \rho \mathbf{e}_2, \\ \partial_t \rho + \mathbf{v} \cdot \nabla \rho - \mu(\partial_x^2 + \partial_y^2)\rho = 0, \\ \mathrm{div}\,\mathbf{v} = \partial_x v_1 + \partial_y v_2 = 0, \\ \mathbf{v}(t,x,\pm1) = (\pm1, 0),\quad \rho(t,x,\pm1) = c_0, \\ \mathbf{v}(0,x,y) = \mathbf{v}^{\mathrm{in}}(x,y),\quad \rho(0,x,y) = \rho^{\mathrm{in}}(x,y). \end{cases} \tag{1.1} $$

其中：

  • $\mathbf{v} = (v_1, v_2)$ 是速度场，
  • $\rho$ 是温度（或密度扰动），
  • $P$ 是压力，
  • $\nu > 0$ 是粘性系数，
  • $\mu > 0$ 是热扩散系数，
  • $\mathbf{e}_2 = (0,1)$ 是竖直方向单位向量，
  • 边界上速度满足无滑移条件：$\mathbf{v}(t,x,\pm1) = (\pm1,0)$，即上下壁面以相反方向匀速运动，形成经典的 Couette 剪切流背景；
  • 温度在边界保持常数 $c_0$。

为研究稳定性，将稳态解 $\mathbf{v}_s = (y, 0),\ \rho_s = 1,\ P_s = y + c$ 作为参考，定义扰动变量：

$$ \mathbf{u} = \mathbf{v} - \mathbf{v}_s,\quad \theta = \rho - \rho_s,\quad p = P - P_s. $$

代入原系统后得到扰动方程组：

$$ \begin{cases} \partial_t \mathbf{u} + y\partial_x \mathbf{u} + \mathbf{u} \cdot \nabla \mathbf{u} - \nu(\partial_x^2 + \partial_y^2)\mathbf{u} + \nabla p = \begin{pmatrix} 0 \\ \theta \end{pmatrix}, \\ \partial_t \theta + y\partial_x \theta + \mathbf{u} \cdot \nabla \theta - \mu(\partial_x^2 + \partial_y^2)\theta = 0, \\ \mathrm{div}\,\mathbf{u} = 0, \\ \mathbf{u}(t,x,\pm1) = 0,\quad \theta(t,x,\pm1) = 0. \end{cases} \tag{1.2} $$

进一步引入涡量 $\omega = \nabla \times \mathbf{u} = \partial_y u_1 - \partial_x u_2$ 和流函数 $\psi$ 满足 $\mathbf{u} = \nabla^\perp \psi = (\partial_y \psi, -\partial_x \psi),\ \Delta \psi = \omega$，则系统可转化为：

$$ \begin{cases} \partial_t \omega + y\partial_x \omega + \mathbf{u} \cdot \nabla \omega - \nu(\partial_x^2 + \partial_y^2)\omega = -\partial_x \theta, \\ \partial_t \theta + y\partial_x \theta + \mathbf{u} \cdot \nabla \theta - \mu(\partial_x^2 + \partial_y^2)\theta = 0, \\ \mathbf{u} = \nabla^\perp \psi,\quad \Delta \psi = \omega, \\ \psi(t,x,\pm1) = \partial_y \psi(t,x,\pm1) = 0,\quad \theta(t,x,\pm1) = 0, \\ \omega(0,x,y) = \omega^{\mathrm{in}}(x,y),\quad \theta(0,x,y) = \theta^{\mathrm{in}}(x,y). \end{cases} \tag{1.3} $$

本工作的核心目标是确定 稳定性阈值（stability threshold）：即当初始扰动足够小时，解是否会全局存在并渐近趋于零（即原稳态稳定）。特别地，在高雷诺数（小 $\nu$）极限下，即使线性部分稳定，有限幅值扰动仍可能引发湍流转捩。因此，“稳定性阈值”指的是使得非线性稳定性成立的初始扰动大小关于 $\nu$ 的最优幂次。

一个关键困难来自浮力项中的导数损失：方程中 $\partial_x \theta$ 出现在 $\omega$ 的演化方程右侧，但 $\theta$ 本身没有额外正则性补偿，导致非线性耦合难以控制。此外，由于区域为 $\mathbb{R} \times [-1,1]$，水平波数 $k$ 是连续谱（而非周期情形下的离散整数），这带来了新的挑战——特别是当 $k \to 0$（长波扰动）时，增强耗散效应（enhanced dissipation）显著减弱，衰减速率趋近于零，使得非线性相互作用更易激发不稳定模式。

文章采用傅里叶分析方法，对水平变量 $x$ 做傅里叶变换，令：

$$ f_k(t,y) = \int_{\mathbb{R}} f(t,x,y) e^{-ikx} dx, $$

从而将系统分解为每个波数 $k \in \mathbb{R}$ 对应的模态 $(\omega_k, \theta_k)$，满足：

$$ \begin{cases} \partial_t \omega_k + iky \omega_k - \nu(\partial_y^2 - k^2)\omega_k = -ik\theta_k - (\mathbf{u} \cdot \nabla \omega)_k, \\ \partial_t \theta_k + iky \theta_k - \nu(\partial_y^2 - k^2)\theta_k = - (\mathbf{u} \cdot \nabla \theta)_k, \\ \mathbf{u}_k = (\partial_y \psi_k, -ik \psi_k),\quad (-k^2 + \partial_y^2)\psi_k = \omega_k, \\ \psi_k(t,\pm1) = \partial_y \psi_k(t,\pm1) = 0,\quad \theta_k(t,\pm1) = 0. \end{cases} \tag{1.4} $$

由于 $\nu = \mu$ 被标准化，且关注的是无限通道结构，文中需精细处理不同频率区间（低频 $|k| \ll \nu$、中频 $10\nu \leq |k| \leq 1$、高频 $|k| \geq 1$）的行为，并利用 Airy 函数分析、预解估计（resolvent estimates）、能量方法与半群估计相结合的技术来克服长波极限下的弱耗散问题。

主要定理或主要结果

论文的核心结果是以下定理：

定理 1.1（稳定性阈值） 设 $\nu = \mu$。令 $(\mathbf{u}, \theta)$ 为系统 (1.2) 的解，对应初值 $(\mathbf{u}^{\mathrm{in}}, \theta^{\mathrm{in}})$。则存在与 $\nu$ 无关的正常数 $\nu_0, \varepsilon_0, \varepsilon_1, C > 0$，使得当 $0 < \nu \leq \nu_0$ 且初始数据满足小性条件：

$$ \|\mathbf{u}^{\mathrm{in}}\|_{H^2} \leq \varepsilon_0 \nu^{1/2}, \quad \|\theta^{\mathrm{in}}\|_{H^1} + \big\||\partial_x|^{1/3} \theta^{\mathrm{in}}\big\|_{H^1} \leq \varepsilon_1 \nu^{5/6}, $$

则系统 (1.2) 存在全局解 $(\mathbf{u}, \theta)$，并且满足一致的稳定性估计：

$$ \sum_k \|E_k[\omega_k]\|_{L^1(\mathbb{R})} \leq C \varepsilon_0 \nu^{1/2}, \quad \sum_k \|E_k[\theta_k]\|_{L^1(\mathbb{R})} \leq C \varepsilon_1 \nu^{5/6}, $$

其中能量泛函 $E_k[\omega_k]$ 和 $E_k[\theta_k]$ 分段定义如下：

对于 $E_k[\omega_k]$：

$$ E_k[\omega_k] = \begin{cases} \|\omega_k\|_{L^\infty_t L^2_y} + \nu^{1/2}\|\omega_k\|_{L^2_t L^2_y} + \|\mathbf{u}_k\|_{L^\infty_t L^\infty_y} + \nu^{1/2}\|\mathbf{u}_k\|_{L^2_t L^2_y}, & 0 \leq |k| \leq 10\nu; \\ \|(1-|y|)^{1/2} \omega_k\|_{L^\infty_t L^2_y} + \nu^{1/4}|k|^{-1/4}\|\omega_k\|_{L^\infty_t L^2_y} + \nu^{1/4}|k|^{1/4}\|\omega_k\|_{L^2_t L^2_y} \\ \quad + \|\mathbf{u}_k\|_{L^\infty_t L^\infty_y} + |k|^2 \|\mathbf{u}_k\|_{L^2_t L^2_y}, & 10\nu \leq |k| \leq 1; \\ \|(1-|y|)^{1/2} \omega_k\|_{L^\infty_t L^2_y} + \nu^{1/4}|k|^{1/2}\|\omega_k\|_{L^2_t L^2_y} + |k|^{1/2} \|\mathbf{u}_k\|_{L^\infty_t L^\infty_y} + |k| \|\mathbf{u}_k\|_{L^2_t L^2_y}, & |k| \geq 1; \end{cases} $$

对于 $E_k[\theta_k]$：

$$ E_k[\theta_k] = \begin{cases} \|\theta_k\|_{L^\infty_t L^2_y} + \nu^{1/2}\|\theta_k\|_{L^2_t L^2_y}, & 0 \leq |k| \leq 10\nu; \\ \|\theta_k\|_{L^\infty_t L^2_y} + \nu^{1/6}|k|^{1/3}\|\theta_k\|_{L^2_t L^2_y}, & 10\nu \leq |k| \leq 1; \\ |k|^{1/3}\|\theta_k\|_{L^\infty_t L^2_y} + \nu^{1/6}|k|^{2/3}\|\theta_k\|_{L^2_t L^2_y}, & |k| \geq 1. \end{cases} $$

此定理表明：只要初始扰动的速度部分在 $H^2$ 范数下不超过 $O(\nu^{1/2})$，而温度部分满足更强的正则性要求（包含 $|\partial_x|^{1/3}$ 导数），其总范数不超过 $O(\nu^{5/6})$，则系统是渐近稳定的，且解具有一致的能量衰减。

研究方法、关键技术和核心工具

a) 核心分析学工具

  1. 傅里叶模态分解与频率分段估计
    文章将整个问题按水平波数 $k \in \mathbb{R}$ 分解为三个区域进行分析：

    • 低频区：$0 \leq |k| \leq 10\nu$，
    • 中频区：$10\nu \leq |k| \leq 1$，
    • 高频区：$|k| \geq 1$。

    不同区域表现出不同的动力学行为。例如，在高频区，剪切流引起的“增强耗散”效应较强；而在低频区（尤其是 $k \to 0$），耗散率趋于零，必须借助其他机制（如边界层分析、Airy 函数估计）来控制解。

  2. 预解估计（Resolvent Estimates）与 Airy 函数分析
    针对线性算子 $A = -\nu(\partial_y^2 - k^2) + iky$，作者引用并改进了 [7,9] 中的方法，通过构造 Airy 函数解来获得 sharp 的 $L^2 \to H^2$ 和 $H^{-1} \to H^1$ 类型的预解估计。这些估计对于控制非齐次项至关重要。

    特别地，命题 2.2 给出了如下关键不等式：

    $$ \nu^{2/3}|k|^{1/3}\|\partial_y f\|_{L^2_y} + (\nu k^2)^{1/3}\|f\|_{L^2_y} + |k|\|(y-\lambda)f\|_{L^2_y} \lesssim \|F\|_{L^2_y}, $$

    这种估计揭示了当 $k$ 很小时，虽然整体耗散弱，但仍可通过 $(\nu k^2)^{1/3}$ 保留一定的控制能力。

  3. 能量方法与空间-时间范数设计
    作者设计了一套精细的空间-时间混合范数 $E_k[\cdot]$，根据不同频率范围调整权重，以匹配实际的衰减速率。这种范数不仅包含标准能量项（如 $L^\infty_t L^2_y$），还加入了反映增强耗散效应的加权 $L^2_t L^2_y$ 项，以及速度的 $L^\infty$ 控制项，确保非线性能被有效吸收。

  4. 拟线性化技巧（Quasi-linearization）与误差系统分离
    对温度方程，作者将其拆分为齐次部分 $\theta_k^H$ 与非齐次部分 $\theta_k^I$：

    $$ \theta_k = \theta_k^H + \theta_k^I, $$

    其中 $\theta_k^H$ 满足不含非线性项的方程，$\theta_k^I$ 初始为零。这样可以利用半群估计直接控制 $\theta_k^I$，避免对主系统施加额外小性条件。

  5. Plancherel 定理与时域傅里叶变换
    在估计非齐次项时，作者对时间变量做傅里叶变换，将微分方程转为代数方程，再结合 Plancherel 定理还原回时域，从而获得 $L^2_t$ 估计。这是处理耗散方程中强迫项的标准高效手段。

b) 技术性难题的应对策略

  1. 导数损失的补偿机制
    浮力项 $-\partial_x \theta$ 在 $\omega$ 方程中造成一阶导数损失。常规方法会导致温度稳定性阈值比速度低 $\nu^{1/2}$。但本文通过要求初始温度具有额外 $|\partial_x|^{1/3}$ 正则性，即控制 $\big\||\partial_x|^{1/3} \theta^{\mathrm{in}}\big\|_{H^1}$，成功将损失降低至 $\nu^{1/3}$ 差距，从而提升整体阈值。

  2. 长波极限 ($k \to 0$) 下的弱耗散问题
    在无限通道中，$k$ 可任意接近零，导致增强耗散率 $\sim \nu^{1/3}|k|^{2/3} \to 0$。为此，作者在低频区使用更弱的耗散项 $\nu^{1/2}\|\theta_k\|_{L^2_t L^2_y}$，并在能量泛函中加入边界权重 $(1-|y|)^{1/2}$ 来捕捉边界层行为。

  3. 非线性项的卷积结构处理
    非线性项如 $(\mathbf{u} \cdot \nabla \omega)_k = \int_{\mathbb{R}} ik u_{1,\ell} \omega_{k-\ell} d\ell + \cdots$ 形成卷积。作者通过将非线性项表示为 $-ik f_k^1 - \partial_y f_k^2$，然后分别估计 $f_k^1, f_k^2$，再代入线性估计框架中完成闭合。

  4. 边界条件的影响建模
    无滑移边界导致速度在 $y=\pm1$ 处为零，进而影响流函数及其导数。这通过设定 $\psi_k(\pm1)=\partial_y \psi_k(\pm1)=0$ 显式体现，并在能量估计中引入相应的加权项（如 $(1-|y|)^{1/2}$）加以控制。

与之前工作的比较

a) 突破与改进

本文最显著的贡献在于提升了温度扰动的稳定性阈值，从先前工作中已知的 $O(\nu^{11/12})$ 改进到 $O(\nu^{5/6})$，同时首次在无限通道 $\mathbb{R} \times [-1,1]$ 上建立了严格的非线性稳定性结果。

具体而言，Masmoudi-Zhai-Zhao [J. Funct. Anal., 2023] 在周期通道 $\mathbb{T} \times [-1,1]$ 中证明了当：

$$ \|\mathbf{v}^{\mathrm{in}} - (y,0)\|_{H^2} \lesssim \nu^{1/2},\quad \|\rho^{\mathrm{in}} - 1\|_{H^1} \lesssim \nu^{11/12} $$

时系统稳定。他们通过利用增强耗散 $\nu^{1/4}|k|^{1/4}\|\omega_k\|$ 和 $\nu^{1/6}|k|^{1/3}\|\theta_k\|$ 来吸收非线性，但由于浮力项含 $\partial_x \theta$，仍需用 $|\partial_x|^{1/6}$ 正则性来补偿剩余导数损失。

相比之下，本文在无限通道中面临更大的技术挑战：当 $k \to 0$ 时，增强耗散完全消失，传统方法失效。然而，作者通过更精细的频率分区和更强的初始正则性假设（$|\partial_x|^{1/3}$ 而非 $|\partial_x|^{1/6}$），反而实现了更高的幂次 $5/6 > 11/12 \approx 0.916$？注意：此处需澄清数值关系。

实际上，$\nu^{5/6} \approx \nu^{0.833} < \nu^{11/12} \approx \nu^{0.916}$，所以表面上看似乎是“更差”。但注意：稳定性阈值越小，允许的扰动越小，结果越强。然而，作者强调的是“threshold”指“临界尺度”，即若扰动大于该尺度则可能失稳。因此，能容忍更大的扰动（即更高幂次）才是更强的结果。

但 $5/6 = 0.833 < 11/12 \approx 0.916$，说明 $\nu^{5/6} > \nu^{11/12}$ 当 $\nu \ll 1$，即允许的扰动更大！举例：$\nu=10^{-6}$，则 $\nu^{5/6}=10^{-5}$，$\nu^{11/12} \approx 10^{-5.5}$，前者更大。因此，$\nu^{5/6}$ 是比 $\nu^{11/12}$ 更宽松的条件，意味着本文结果允许更大的初始扰动仍保持稳定，故为实质性改进。

这一改进的关键在于：作者意识到，虽然无限通道中 $k\to0$ 导致耗散弱，但通过施加稍强的正则性（$|\partial_x|^{1/3}$），可以在高频部分更好地利用增强耗散，从而在整体上实现更优的非线性控制。

此外，本文的方法适用于连续谱设置，推广了此前仅限于周期情形的工作，具有更强的普适性。这也为未来研究三维或分数耗散情形提供了新思路。

综上所述，本文的主要突破体现在：

  • 将稳定性阈值从 $O(\nu^{11/12})$ 提升至 $O(\nu^{5/6})$（实为放宽条件）；
  • 首次在无限通道中建立严格非线性稳定性；
  • 发展了一套适用于连续波数谱的分频估计框架；
  • 揭示了通过增加初始正则性可补偿导数损失的新机制。

3. Local and Global Results on Three Dimensional Rarefaction Waves in Spherical Symmetry

球对称情形下三维稀疏波的局部与整体结果

作者: Ruotong Zhang

类别: math.AP

发布日期: 2025-11-29

arXiv ID: 2512.00353v1

链接: http://arxiv.org/abs/2512.00353v1

详细分析

中文标题: 球对称情形下三维稀疏波的局部与整体结果

研究对象和背景

本文研究的是三维球对称情形下的可压缩欧拉方程组中一类特殊解——中心稀疏波（centered rarefaction wave）的局部存在性与整体存在性。所考虑的流体为完美气体，满足等熵条件（homentropic flow），即熵恒定，且状态方程为伽马律（gamma-law）：

$$ p = K\rho^\gamma, $$

其中 $ p $ 是压力，$ \rho $ 是密度，$ K > 0 $ 为常数，$ \gamma > 1 $ 为绝热指数。

在三维空间中，假设流动具有球对称性，即所有物理量仅依赖于时间 $ t $ 和径向坐标 $ r = |x| $，且速度场为纯径向：$ v = v(t,r) e_r $。在此设定下，三维可压缩欧拉方程组可写为如下形式：

$$ \begin{cases} \partial_t \rho + v \partial_r \rho + \rho \partial_r v = -2\rho \frac{v}{r}, \\ \partial_t v + v \partial_r v + \frac{c^2}{\rho} \partial_r \rho = 0, \end{cases} $$

其中声速 $ c $ 定义为：

$$ c^2 = \frac{d p}{d\rho}\bigg|_{\text{const entropy}} = K\gamma \rho^{\gamma-1}. $$

该系统的两个主要变量是径向速度 $ v $ 和密度 $ \rho $（或等价地，声速 $ c $）。由于问题涉及非线性双曲守恒律中的稀疏波现象，作者引入了黎曼不变量（Riemann invariants）以简化分析。定义如下两个函数：

$$ w = \frac{1}{\gamma-1}c - \frac{1}{2}v, \quad w = \frac{1}{\gamma-1}c + \frac{1}{2}v. $$

利用这两个变量，原欧拉系统可以重写为关于 $ w $ 和 $ w $ 的一阶偏微分方程组。特别地，在球对称情况下，其演化方程包含额外的几何项（来自 $ \nabla \cdot v $ 中的 $ 2v/r $ 项），导致其结构不同于经典的一维情形：

$$ \begin{cases} \partial_t w + (v - c)\partial_r w = -c \frac{v}{r}, \\ \partial_t w + (v + c)\partial_r w = -c \frac{v}{r}. \end{cases} $$

这些方程揭示了特征方向上的输运行为，并带有源项，反映了球对称带来的曲率效应。

本文的核心问题是构造一个从初始球面出发的中心稀疏波解，该解连接一个给定的外部背景流（exterior background flow）与内部可能趋于真空或低密度区域的状态。这种设置类似于经典的黎曼问题（Riemann problem），但在三维球对称框架下推广：不考虑两点间的间断，而是考虑一个从初始时刻 $ t=0 $、半径 $ r=r_0 $ 的球面上发散出的稀疏波扇形区域。

具体来说，设 $ C_0 = \{(t,r) \mid r = r_{C_0}(t)\} $ 是一条从 $ (0,r_0) $ 出发的外向特征曲面，其外部已知一个光滑解 $ (v_{\text{ext}}, c_{\text{ext}}) $ 满足上述欧拉方程。目标是在 $ C_0 $ 内侧构造一个稀疏波区域 $ D_{\text{rf}} = \{(t,r) \mid t \in [0,t^*],\, r_{C_u}(t) \leq r \leq r_{C_0}(t)\} $，其中 $ C_u $ 是一族参数化为 $ u \in [0,u^*] $ 的外向声学零锥（outgoing null cones），并使得整个解在 $ C_0 $ 上连续但横向导数不连续，从而形成典型的稀疏波结构。

值得注意的是，该问题的初始数据在 $ (t,r)=(0,r_0) $ 处是奇异的：所有稀疏波特征线汇聚于一点，导致标准柯西问题无法直接应用。因此，传统的初值问题理论不再适用，必须采用渐近匹配法或边界数据延拓法来处理这一奇异性。

此外，文中使用的函数空间并非通常的 Sobolev 或 BV 空间，而是在声学坐标（acoustical coordinates）$ (t,u,\theta^A) $ 下进行分析，其中 $ u $ 是沿内向声学特征定义的“声学函数”，满足 $ Lu = 0 $，$ L = \partial_t + (v+c)\partial_r $ 为外向声算子。在此坐标系下，稀疏波区域被参数化为 $ (t,u) \in [0,t^*] \times [0,u^*] $，并在该坐标下证明解的光滑性。

边值条件方面，要求解在边界 $ C_0 $ 上与外部背景流连续，即：

$$ (w_{\text{rf}}, w_{\text{rf}}) = (w_{\text{ext}}, w_{\text{ext}}) \quad \text{on } C_0 \setminus \{(0,r_0)\}, $$

但其横向导数（即沿 $ T = \partial_u $ 方向）不连续，体现稀疏波的本质特征。而在初始点附近，则要求解趋向于一维稀疏波的结构，即当 $ t \to 0 $ 时，

$$ w_{\text{rf}}(0,u) = w_{\text{ext}}(0,r_0), \quad w_{\text{rf}}(0,u) = w_{\text{ext}}(0,r_0) - \frac{2}{\gamma+1}u, $$

这对应于一维稀疏波中 $ x/t $ 结构的类比。

综上所述，本文的研究对象是一个在奇点处发散的非线性双曲系统解，其数学挑战在于如何在缺乏经典适定性框架的情况下，通过几何分析方法构造出具有指定边界行为和正则性的局部乃至全局解。

主要定理或主要结果

定理 3.1（稀疏波解的局部存在性）

设 $ (v_{\text{ext}}, c_{\text{ext}}) $ 是定义在外部区域 $ D_{\text{ext}} = \{(t,r) \mid t \in [0,t_0],\, r \geq r_{C_0}(t)\} $ 上的光滑球对称解，满足三维球对称可压缩欧拉方程，且 $ r_{C_0}(t) $ 是一条外向声学特征曲线，起始于 $ (0,r_0) $。令 $ w,w $ 为其对应的黎曼不变量。

则存在 $ t^* > 0 $ 和 $ u^* \in \left(0, \frac{\gamma+1}{\gamma-1}c_0\right) $，使得可在扇形区域 $ D_{\text{rf}} = \{(t,r) \mid t \in [0,t^*],\, r_{C_u}(t) \leq r \leq r_{C_0}(t)\} $ 上构造唯一的稀疏波解 $ (w_{\text{rf}}, w_{\text{rf}}) $，满足以下性质：

  1. 连续性：在 $ C_0 \setminus \{(0,r_0)\} $ 上，$ (w_{\text{rf}}, w_{\text{rf}}) = (w_{\text{ext}}, w_{\text{ext}}) $；
  2. 横向导数不连续：在 $ C_0 \setminus \{(0,r_0)\} $ 上，$ \partial_r w_{\text{rf}} \neq \partial_r w_{\text{ext}} $；
  3. 坐标光滑性：$ (w_{\text{rf}}, w_{\text{rf}}, r) $ 在声学坐标 $ (t,u) \in [0,t^*] \times [0,u^*] $ 下是光滑函数；
  4. 初始行为：当 $ t \to 0 $ 时，有
    $$ \lim_{t \to 0} \frac{r_{C_u}(t) - r_0}{t} = (v_{\text{ext}} + c_{\text{ext}})(0,r_0) - u, $$
    且
    $$ w_{\text{rf}}(0,u) = w_{\text{ext}}(0,r_0), \quad w_{\text{rf}}(0,u) = w_{\text{ext}}(0,r_0) - \frac{2}{\gamma+1}u. $$

此外，$ t^* $ 的大小取决于 $ u^* $ 及外部解至多四阶导数的大小。

定理 4.6（稀疏波解的整体存在性）

若外部背景解 $ (v_{\text{ext}}, c_{\text{ext}}) $ 足够接近某个常状态，且其偏离量以适当速率衰减（例如代数衰减），则上述构造的稀疏波解可延拓到所有正时间，即 $ t^* = \infty $。进一步地，特征线之间的分离尺度满足 $ \kappa \sim \ln t $，表明稀疏过程长期持续。

此结果的关键在于对能量估计的精细控制，允许初始扰动非紧支但需满足一定的可积性条件。

研究方法、关键技术和核心工具

a) 核心分析工具

本文采用的主要数学工具是基于声学几何（acoustical geometry）的能量估计方法，灵感来源于 Christodoulou 关于相对论欧拉方程中激波形成的开创性工作 [Chr07]，以及后续 Luo–Yu 在多维稀疏波构造中的进展 [LY25a][LY25b]。

首先，作者引入声学度量（acoustical metric）：

$$ g = -c^2 dt^2 + (dx^i - v^i dt)^2, $$

这是一个洛伦兹度量，其零测地方向对应声波传播路径。在此度量下，定义外向零向量场：

$$ L = \partial_t + (v + c)\partial_r, $$

并构建外向零叶状结构（outgoing null foliation），由声学函数 $ u(t,r) $ 参数化，满足 $ Lu = 0 $，且在初始时刻 $ t = 0 $ 上单调递减。

接着定义内向类空向量场：

$$ T = \partial_u^{(t,u)}, $$

其在伽利略坐标下的表达为 $ T = (\partial_u r) \partial_r $，长度记为 $ \kappa = |\partial_u r| $，它衡量了叶状结构的空间密度。同时定义 $ \mu = c\kappa $，用于能量估计中的时间密度校正。

在此框架下，原始欧拉方程被改写为关于 $ w,w $ 和 $ r $ 的输运方程组：

$$ \begin{cases} L w = -\kappa v, \\ L w = -c v, \\ L r = v + c. \end{cases} $$

为了建立先验估计，作者使用应力-能量张量方法（stress-energy tensor method）推导能量恒等式。对于标量波方程 $ \Box_g \psi = \tilde{\rho} $，定义能量动量张量：

$$ T_{\mu\nu} = \partial_\mu \psi \partial_\nu \psi - \frac{1}{2} g_{\mu\nu} g^{\alpha\beta} \partial_\alpha \psi \partial_\beta \psi, $$

并选取乘子向量场 $ X = aL + bL $，其中 $ L $ 为归一化的内向零向量。由此得到能量流 $ P_\mu = -T_{\mu\nu} X^\nu $，并通过散度定理获得能量恒等式。

最终得出的关键命题是：

命题 2.1（能量估计框架）
设 $ \psi $ 满足 $ \Box_g \psi = \tilde{\rho} $，在区域 $ [t_1,t_2] \times [u_1,u_2] \times S^2 $ 上，有：

$$ F[u](t_2) - F[u](t_1) + E[t_1,t_2](u_2) - E[t_1,t_2](u_1) = \int_{t_1 \leq t \leq t_2,\, u_1 \leq u \leq u_2} Q \, d\mu_g, $$

其中 $ F,E $ 分别为时间和空间能量通量，$ Q = Q_0 + Q_1 + Q_2 + Q_3 $ 为误差项，具体形式为：

$$ Q_0 = -\frac{1}{2\mu} \tilde{\rho} (a L\psi + b L\psi), $$

$$ Q_1 = -\frac{1}{2} L(a tr\chi + b tr\bar{\chi}) |L\psi|^2, $$

等等（详见原文公式 (2.2)）。这一框架允许作者对高阶导数进行逐阶能量估计。

b) 技术性技巧

  1. 边界数据的确定：由于初始数据奇异，作者不在 $ t=0 $ 上赋值，而是在边界 $ C_0 $ 上通过求解一系列沿 $ L $ 的常微分方程系统来确定横向导数 $ T^n w, T^n w, T^{n-1}\kappa $。这些 ODE 来自将输运方程反复作用 $ T $ 导出的高阶方程组（如公式 (3.2) 所示）。初始值在 $ (0,0) $ 处按一维稀疏波模式设定，确保解在奇点附近逼近经典结构。

  2. 近似解的构造：取小参数 $ \delta > 0 $，在时间切片 $ \Sigma_\delta = \{t = \delta\} $ 上利用泰勒展开构造光滑的近似初值，然后从 $ t = \delta $ 开始求解标准柯西问题，得到一族近似解 $ (w^\delta, w^\delta) $。

  3. 一致先验估计：通过对近似解进行高阶能量估计（使用命题 2.1 中的框架），证明能量范数在 $ \delta \to 0 $ 过程中保持一致有界，从而保证生命跨度 $ t^* $ 不依赖于 $ \delta $。

  4. 收敛论证：利用紧性原理（如 Ascoli–Arzelà 或弱紧性）提取极限，并验证极限满足原始方程和边界条件，完成精确解的构造。

  5. 整体解的延拓：对于接近常状态的背景流，作者显式计算边界 $ C_0 $ 上的数据衰减率，并通过放松对 $ \kappa $ 等量的衰减要求，建立关于大时间 $ T $ 的一致能量估计，从而实现全局延拓。

与之前工作的比较

a) 突破与改进

本文是对 Alinhac [Ali89a][Ali89b] 和 Luo–Yu [LY25a][LY25b] 工作的重要推进，主要体现在以下几个方面：

  1. 首次在球对称下处理非平凡曲率界面：先前的多维稀疏波构造（如 Luo–Yu）集中在平坦或渐近平坦几何中，未考虑初始奇点为球面的情形。本文明确处理了 $ \nabla \cdot v = 2v/r $ 带来的几何曲率项，这在能量估计中引入新的耦合项，需更精细的控制。

  2. 避免高技术复杂性的同时获得更强结果：虽然 Luo–Yu 使用了类似 Christodoulou 的几何方法，但其框架高度抽象且技术繁复。本文借助球对称性极大简化了几何结构，使得许多量（如 $ \chi, \bar{\chi} $）可显式表达，从而避开 Nash–Moser 迭代等复杂工具，仍能获得全范数控制（full control of norms），优于 Alinhac 加权空间中退化的估计。

  3. 从局部到整体的跨越：以往工作大多止步于局部存在性。本文首次在三维设置下证明了整体存在性结果，条件是外部流足够接近常状态并具合理衰减。这一结果类似于 Wang [Wan24] 的结论，但应用于稀疏波而非激波。

  4. 降低对初始数据的要求：相比需要极高正则性的早期方法，本文允许外部解仅为光滑函数，无需解析性或其他强限制。同时，通过 ODE 方法自动匹配边界数据，避免人为设定不自然的初值。

  5. 提供未来无对称情形的参考：尽管本文依赖球对称，其所发展的能量估计策略和对横向导数的处理方式，可作为将来研究一般三维稀疏波的“横向部分”的模板，具有方法论上的指导意义。

综上，本文不仅完善了多维稀疏波理论的图景，也为理解非线性双曲系统中奇性化解的构造机制提供了新的视角。

了解领域论文（摘要翻译）

1. Mathematical and numerical study of a model for navigation in stratified waters

分层水域中航行模型的数学与数值研究

作者: Zeina Rammal, Matthieu Brachet, Germain Rousseaux, Morgan Pierre

类别: math.AP, physics.class-ph

发布日期: 2025-12-01

arXiv ID: 2512.01445v1

链接: http://arxiv.org/abs/2512.01445v1

摘要翻译

中文标题: 分层水域中航行模型的数学与数值研究

摘要翻译: 我们推导了一个船舶速度可变的两层流体中航行的线性模型，并分析了包含瑞利阻尼项的谱形式模型。我们证明了当速度及初始数据足够正则时，柯西问题存在唯一解。对匀速航行的情形进行了深入研究，并指出了一个临界速度的重要性，该速度将系统划分为两种不同类型的运行机制。我们提出了一种数值格式：空间离散采用离散傅里叶变换，时间离散采用指数积分法。我们对指数积分法证明了误差估计。最后，通过一维和二维的数值实验验证了理论结果的正确性。

2. Higher derivative estimates for Stokes equations with closely spaced rigid inclusions in three dimensions

三维空间中含紧密排列刚性夹杂的斯托克斯方程的高阶导数估计

作者: Hongjie Dong, Haigang Li, Huaijun Teng, Peihao Zhang

类别: math.AP

发布日期: 2025-11-30

arXiv ID: 2512.00866v1

链接: http://arxiv.org/abs/2512.00866v1

摘要翻译

中文标题: 三维空间中含紧密排列刚性夹杂的斯托克斯方程的高阶导数估计

摘要翻译: 本文研究了在包含两个紧密排列刚性夹杂的三维区域中，斯托克斯方程解的高阶导数估计问题。我们通过归纳构造一系列辅助函数，以分离出夹杂间狭窄区域内高阶导数的主导奇异项。对于一类具有一般形状的凸夹杂，三维辅助函数的构造——与二维情形不同——依赖于一类具有奇异系数的二维偏微分方程解的衰减性质。利用这一特性，我们对一般形状的夹杂建立了直至七阶导数的逐点上界估计；在附加对称性条件下，进一步获得了任意阶导数的最优估计。由此，我们得到了夹杂间狭窄区域内柯西应力及其高阶导数奇性增长速率的精确刻画。

3. Error analysis of an acceleration corrected diffusion approximation of Langevin dynamics with background flow

带背景流的朗之万动力学加速度修正扩散近似的误差分析

作者: Yoichiro Mori, Chanoknun Sintavanuruk, Truong-Son P. Van

类别: math.NA, math.AP, math.PR

发布日期: 2025-11-30

arXiv ID: 2512.00685v1

链接: http://arxiv.org/abs/2512.00685v1

摘要翻译

中文标题: 带背景流的朗之万动力学加速度修正扩散近似的误差分析

摘要翻译: 本文研究了惯性粒子在背景流输运下的朗之万动力学的近似问题。特别地，我们考察了一种对朗之万动力学进行加速度修正的对流-扩散近似，该近似在湍流输运研究中被广泛采用。我们在平均化框架下建立了误差估计，其中无量纲松弛时间尺度 $\varepsilon$ 为小参数。我们证明，在任意有限时间区间内，该近似的强意义误差为 $\mathcal{O}(\varepsilon)$ 阶，弱意义误差为 $\mathcal{O}(\varepsilon^2)$ 阶，并通过数值实验验证了这些收敛阶的最优性。此外，我们提供了数值证据，表明该近似还能有效捕捉朗之万动力学的长时间行为。

4. Vanishing layer thickness limit of convection in multilayer porous media

多层多孔介质中对流的零层厚极限

作者: Kaijian Sha, Xiaoming Wang

类别: math.AP

发布日期: 2025-11-29

arXiv ID: 2512.00430v1

链接: http://arxiv.org/abs/2512.00430v1

摘要翻译

中文标题: 多层多孔介质中对流的零层厚极限

摘要翻译: 在多层多孔介质对流的达西-布辛涅斯克（Darcy-Boussinesq）框架下，我们研究其中一层厚度趋于零的奇异极限问题。我们证明了完整系统的解收敛于相应少一层的极限模型的解。该收敛性从两个互补的角度得以确立：(i) 在任意有限时间区间上的强 $L^2$ 收敛性；以及 (ii) 描述长时间渐近行为的全局吸引子的上半连续性。

不相关论文（基本信息）

1. Weak coupling for Schrödinger operators with complex potentials

复势薛定谔算子的弱耦合

作者: Jussi Behrndt, Markus Holzmann, Petr Siegl, Nicolas Weber

类别: math.SP, math-ph, math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01941v1

链接: http://arxiv.org/abs/2512.01941v1

摘要: We study the discrete eigenvalues emerging from the threshold of the essential spectrum of one or two-dimensional Schrödinger operators with complex-valued $ L^p $-potentials in a weak coupling regime. We derive necessary and sufficient conditions on the potential for the existence or absence of discrete eigenvalues in this regime and also analyze their uniqueness and algebraic multiplicity. Our results can be viewed as natural non-self-adjoint extensions of the well-known classical weak coupling phenomenon for self-adjoint Schrödinger operators with real-valued potentials going back half a century to Simon’s famous paper [Simon 1976].

2. Prescribed energy solutions of concave-convex type problems involving sign-changing or vanishing weights

涉及变号或消失权函数的凹凸型问题的预定能量解

作者: Kanishka Perera, Humberto Ramos Quoirin, Kaye Silva

类别: math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01931v1

链接: http://arxiv.org/abs/2512.01931v1

摘要: We provide an abstract approach to find couples $(λ,u) \in \mathbb{R} \times X$ satisfying

$$ Φ_λ(u)=c \quad \mbox{and} \quad Φ'_λ(u)=0, $$

for some suitable values of $c \in \mathbb{R}$. Here $Φ_λ$ is a $C^1$ functional (set on a Banach space $X$) whose main prototype is the energy functional associated to a concave-convex problem with sign-changing or vanishing weights. This approach allows us to derive several existence, multiplicity and bifurcation type results for the equation $Φ'_λ(u)=0$ with $λ$ fixed.

3. Dirichlet heat kernel estimates for parabolic nonlocal equations

抛物型非局部方程的狄利克雷热核估计

作者: Philipp Svinger, Marvin Weidner

类别: math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01919v1

链接: http://arxiv.org/abs/2512.01919v1

摘要: In this article we establish the optimal $C^s$ boundary regularity for solutions to nonlocal parabolic equations in divergence form in $C^{1,α}$ domains and prove a higher order boundary Harnack principle in this setting. Our approach applies to a broad class of nonlocal operators with merely Hölder continuous coefficients, but our results are new even in the translation invariant case. As an application, we obtain sharp two-sided estimates for the associated Dirichlet heat kernel. Notably, our estimates cover nonlocal operators with time-dependent coefficients, which had remained open in the literature.

4. Biharmonic Equations

双调和方程

作者: Dirk Pauly, Alberto Valli

类别: math.AP, math.FA

发布日期: 2025-12-01

arXiv ID: 2512.01860v1

链接: http://arxiv.org/abs/2512.01860v1

摘要: In this note we devise and analyse well-posed variational formulations and operator theoretical methods for boundary value problems associated to the biharmonic operator. Of particular interest are Neumann type and over- and underdetermined (maximal and minimal) boundary value problems.

5. Cauchy data for multiple collapsing boson stars

多个坍缩玻色子星的柯西数据

作者: Elena Giorgi, Dawei Shen, Jingbo Wan

类别: gr-qc, math.AP, math.DG

发布日期: 2025-12-01

arXiv ID: 2512.01844v1

链接: http://arxiv.org/abs/2512.01844v1

摘要: We construct Cauchy initial data for the Einstein-Maxwell-Klein-Gordon (EMKG) system, which evolves in finite time into spacetimes containing multiple trapped surfaces. From a physical perspective, this corresponds to preparing multiple well-separated boson stars, each of which collapses to form a spacelike black hole region. In particular, this extends the result of the second and third named authors on the formation of multiple trapped surfaces in vacuum to the EMKG system.

6. Conservative Formulations of the Standard Enskog and Povzner Equations

标准恩斯科格方程与波兹纳方程的守恒型表述

作者: Zhe Chen

类别: math.AP, math-ph

发布日期: 2025-12-01

arXiv ID: 2512.01800v1

链接: http://arxiv.org/abs/2512.01800v1

摘要: This article introduces a conservative formulation of the Standard Enskog equation and the Povzner equation, both of which generalize the Boltzmann equation by incorporating the contribution of particle volume in collisions. The primary result expresses these collision integrals as the divergence with respect to the velocity variable v of a mass current. Moreover, the terms v C[f,f] and |v|^2 C[f,f], where C[f,f] denotes the Standard Enskog or Povzner collision integral, are represented as phase-space divergences (that is, divergences in both position and velocity) of corresponding momentum and energy currents. This work extends Villani’s earlier result (Math. Modelling Numer. Anal. M2AN 33 (1999), 209–227) for the classical Boltzmann equation to the case of dense gases.

7. Solutions to Sobolev Supercritical Nonlinear Schrodinger Equations on an Annulus via a Hopf Reduction Method

通过Hopf约化方法求解环形区域上Sobolev超临界非线性薛定谔方程的解

作者: Jian Liang, Hua-Yang Wang

类别: math.AP, math.FA

发布日期: 2025-12-01

arXiv ID: 2512.01794v1

链接: http://arxiv.org/abs/2512.01794v1

摘要: This paper investigates the existence of positive solutions with a prescribed mass for nonlinear Schrodinger equations on an annulus, possibly in the Sobolev supercritical regime. A reduction method based on the Hopf fibration is used to transform the problem into a lower-dimensional one. We obtain a new mass critical threshold and we show that in the new mass subcritical or critical regimes there exists a positive solution which corresponds to a global minimizer, while in the mass supercritical regime, there exists two positive solutions which correspond to a local minimizer and a mountain pass solution respectively. Some other problems are also discussed in this paper.

8. Global and local existence of solutions for a novel type of parabolic Kirchhoff system with singular term

一类含奇异项的新型抛物型Kirchhoff系统解的整体存在性与局部存在性

作者: Aberqi Ahmed, Abdesslam Ouaziz, Maria Alessandra Ragusa

类别: math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01792v1

链接: http://arxiv.org/abs/2512.01792v1

摘要: In this paper, we investigate solutions for a fractional system involving a novel class of Kirchhoff functions and logarithmic nonlinearity: \begin{equation*} \left{\begin{array}{lll} \displaystyle \mathfrak{u}_{t}+\mathcal{K}\left([\mathfrak{u}]_p^s\right) \mathscr{L}_p^s u=\vert \mathfrak{v} \vert^{σ}\vert \mathfrak{u} \vert^{σ-2} u \log | \mathfrak{u} \mathfrak{v}|, , , & \mbox{in}\quad &\mathcal{U} \times[0, T),\ \mathfrak{v}_t+\mathcal{K}\left([\mathfrak{v}]_q^s\right) \mathscr{L}_q^s \mathfrak{v}=\vert \mathfrak{u} \vert^{σ}|\mathfrak{v}|^{σ-2} \mathfrak{v} \log | \mathfrak{u} \mathfrak{v}|, & \text { in } & \mathcal{U} \times[0, T), \ \mathfrak{u}(\mathrm{x}, t)=\mathfrak{v}(\mathrm{x}, t)=0, & \text { in } & \partial \mathcal{U} \times[0, T), \ \mathfrak{u}(\mathrm{x}, 0)=\mathfrak{u}_0(\mathrm{x}), \mathfrak{v}(\mathrm{x}, 0)=\mathfrak{v}_0(\mathrm{x}), & \text { in } & \mathcal{U}, \end{array}% \right. \end{equation*} where $\mathcal{K}$ is Kirchhoff function, and $\mathscr{L}_{p}^{s}$ is the fractional $p-$ Laplacian operator. We prove the existence of a weak solution using the Faedo-Galerkin method under suitable assumptions on the Kirchhoff function. We investigate the finite-time blow-up and global existence of solutions based on critical, subcritical, and supercritical initial energy levels. Subsequently, we establish the stabilization of the solution with positive initial energy by applying Komornik’s integral inequality.

9. A decoupled, unconditionally stable and second-order integrator for the Landau–Lifshitz–Gilbert equation with magnetoelastic effects

一种用于含磁弹性效应Landau-Lifshitz-Gilbert方程的解耦、无条件稳定且二阶精度的积分方法

作者: Martin Kružík, Hywel Normington, Michele Ruggeri

类别: math.NA, math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01741v1

链接: http://arxiv.org/abs/2512.01741v1

摘要: We consider the numerical approximation of a nonlinear system of partial differential equations modeling magnetostriction in the small-strain regime consisting of the Landau–Lifshitz–Gilbert equation for the magnetization and the conservation of linear momentum law for the displacement. We propose a fully discrete numerical scheme based on first-order finite elements for the spatial discretization. The time discretization employs a combination of the classical Newmark-$β$ scheme for the displacement and the midpoint scheme for the magnetization, applied in a decoupled fashion. The resulting method is fully linear, formally of second order in time, and we prove that it is unconditionally stable. Finally, we assess the stability, accuracy, and energy conservation properties of the proposed method in a collection of numerical experiments.

10. Resonance analysis of one-dimensional acoustic media: a propagation matrix approach

一维声学介质的共振分析：一种传播矩阵方法

作者: Yi Huang, Bowen Li, Ping Liu, Yingjie Shao

类别: math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01734v1

链接: http://arxiv.org/abs/2512.01734v1

摘要: This work analyzes the scattering resonances of general acoustic media in a one-dimensional setting using the propagation matrix approach. Specifically, we characterize the resonant frequencies as the zeros of an explicit trigonometric polynomial. Leveraging Nevanlinna’s value distribution theory, we establish the distribution properties of the resonances and demonstrate that their imaginary parts are uniformly bounded, which contrasts with the three-dimensional case. In two classes of high-contrast regimes, we derive the asymptotics of both subwavelength and non-subwavelength resonances with respect to the contrast parameter. Furthermore, by applying the Newton polygon method, we recover the discrete capacitance matrix approximation for subwavelength Minnaert resonances in both Hermitian and non-Hermitian cases, thereby establishing its connection to the propagation matrix framework.

11. Theory And Applications Of One-Sided Coupled Operator Matrices

单边耦合算子矩阵的理论与应用

作者: Marjeta Kramar, Delio Mugnolo, Rainer Nagel

类别: math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01719v1

链接: http://arxiv.org/abs/2512.01719v1

摘要: The theory of one-sided coupled operator matrices, recently introduced by K.-J. Engel, is an abstract framework for concrete initial value problems and allows complete information on well-posedness, and stability of solutions. These notes are meant as a survey on this rich theory, with a particular stress on applications to initial-boundary value problems with unbounded boundary feedbacks. A diffusion-transport system with dynamical boundary conditions is discussed, and its well-posedness and various other properties are investigated. As a by-product, the well-posedness of a wave equation with dynamical boundary condition is also obtained.

12. Normalized solutions for the planar Schrödinger-Poisson system with two electrons interaction

平面薛定谔-泊松系统在双电子相互作用下的归一化解

作者: Baihong Li, Yuanhong Wei, Xiangjian Zeng

类别: math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01655v1

链接: http://arxiv.org/abs/2512.01655v1

摘要: This paper focuses on the normalized solutions for the planar Schrödinger-Poisson system with a two-electron interaction, which models the effect between electrons and the electrostatic potential they generate. As the parameters vary, some existence results are established. Specifically, a ground state solution is obtained for some general cases. The existence of two solutions is established for the mass-supercritical case, one of which is a ground state solution and the other one is an excited state solution. We develop a compactness method to deal with the functionals involving logarithmic convolution terms. The Pohožaev identity for the coupled Schrödinger-Poisson system with a logarithmic convolution term is also shown, which is crucial for addressing the mass-supercritical problem.

13. Completeness of reparametrization-invariant Sobolev metrics on the space of surfaces

关于曲面空间上重参数化不变Sobolev度量的完备性

作者: Martin Bauer, Cy Maor, Benedikt Wirth

类别: math.DG, math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01566v1

链接: http://arxiv.org/abs/2512.01566v1

摘要: We study reparametrization-invariant Sobolev-type Riemannian metrics on the space of immersed surfaces and establish conditions ensuring metric and geodesic completeness as well as the existence of minimizing geodesics. This provides the first extension of completeness results for immersed curves, originating from works of Bruveris, Michor, and Mumford, and validates an earlier conjecture of Mumford on completeness properties of general spaces of immersions in this important case. The result is obtained by recasting earlier approaches to completeness on manifolds of mappings as a general completeness criterion for infinite-dimensional Riemannian manifolds that are open subsets of a complete Riemannian manifold and by combining it with geometric estimates based on the Michael–Simon–Sobolev inequality to establish the completeness for specific Sobolev metrics on immersed surfaces.

14. Existence and Nonlocal-to-Local Convergence for Singular, Anisotropic Nonlocal Cahn-Hilliard Equations

奇异各向异性非局部Cahn-Hilliard方程的存在性与非局部到局部收敛性

作者: Helmut Abels, Yutaka Terasawa

类别: math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01545v1

链接: http://arxiv.org/abs/2512.01545v1

摘要: We study the nonlocal-to-local convergence for a nonlocal Cahn-Hilliard equation with anisotropic and singular kernels. In particular, we show convergence of weak solutions of the nonlocal Cahn-Hilliard equation to weak solutions of a corresponding anisotropic Cahn-Hilliard equation for suitable subsequences. Moreover, we show existence of weak solutions for the nonlocal equation under a condition, which guarantees existence of weak solutions for suitably localized or singular kernels.

15. Functional-Analytic Justification of the Time-Domain Foldy-Lax Approximation for Dispersive Acoustic Media: A Feynman-Diagram Viewpoint

色散声学介质中时域Foldy-Lax近似的函数分析依据：一种费曼图观点

作者: Arpan Mukherjee, Mourad Sini

类别: math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01532v1

链接: http://arxiv.org/abs/2512.01532v1

摘要: This work provides a rigorous functional-analytic justification for a time-domain Foldy-Lax framework that describes multiple acoustic scattering by a cluster of dispersive resonators (modeling gas-filled bubbles), explicitly incorporating dispersion via the Minnaert resonance. The model is formulated as a delayed-coupled hyperbolic system for bubble amplitude interactions. We combine time-domain integral equations, Laplace transforms, and Hardy-Sobolev space techniques to analyze this system, establishing its unique solvability in anisotropic Hilbert spaces, with solutions expressed as convergent Neumann series of convolution operators. We derive geometric decay of truncation errors for resonant incident waves and quantify the contribution of $N$-th order multi-scattering, showing it scales with $\varepsilon^{N(1-p)+1}$ (relating bubble radius $\varepsilon$ and inter-bubble distance scaling as $\varepsilon^p$, $p<1$). This dominates the measurement errors, which are of order $\varepsilon^2$, thereby allowing us to capture fields generated by inter-bubble interactions of order $N<\frac{1}{1-p}$. This provides a quantitative relation between the spectra band width of the source field, the closeness distance between the bubbles and the order $N$ of the relevant interactions between the bubbles. Furthermore, a novel connection to Feynman diagrams maps multi-scattering paths to diagrammatic vertices and propagators, simplifying the interpretation of higher-order interactions and kinematic constraints. This framework advances accurate transient wave prediction in dispersive media, with implications for cavitation therapy, seismic imaging, and metamaterial engineering.

16. Velocity Averaging Lemmas: Classical, Quantum and Semi-Classical

速度平均引理：经典、量子与半经典情形

作者: François Golse, Norbert J. Mauser, Jakob Möller

类别: math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01529v1

链接: http://arxiv.org/abs/2512.01529v1

摘要: Averaging lemmas were introduced as a tool of the mathematical analysis of kinetic equations, i.e. PDEs for functions in phase space $(x,v)$ containing a transport (“advection”) term. By integrating over $v$ in velocity space $\mathbb{R}_v^d$ (velocity averaging), one gains regularity for the density in position space $\mathbb{R}_x^d$. The concept was invented independently by V.I. Agoshkov and by F. Golse, B. Perthame, R. Sentis and P.-L. Lions, and successfully applied to the analysis of Vlasov or Boltzmann equations in “classical kinetic theory”. In “quantum kinetic theory”, the Schrödinger equation for the complex-valued “wave function” in the physical space is converted into the Wigner equation for the real-valued Wigner function in phase space (which can take negative values). The Wigner (“Quantum Vlasov”) equation contains the transport term of classical kinetic equations plus a pseudo-differential operator containing the potential. We give answers to the long standing question of whether and to which extent averaging lemmas apply to the “quantum” case of the Wigner equation. The hard part are the “semi-classical” averaging lemmas, where one considers the asymptotics of vanishing Planck constant towards the non-negative Wigner measure. In that context “pure vs. mixed states” play a crucial role, as well as the connection between the Schrödinger equation and Quantum Hydrodynamics (QHD). We present the results for the classical and quantum cases and sketch the “semi-classical” case which is worked out in full detail in a follow-up article.

17. Mountain pass for the Ginzburg-Landau energy in a strip: solitons and solitonic vortices

带状区域中Ginzburg-Landau能量的山路解：孤子与孤子型涡旋

作者: Amandine Aftalion, Luc Nguyen

类别: math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01506v1

链接: http://arxiv.org/abs/2512.01506v1

摘要: Motivated by recent experiments, we study critical points of the Ginzburg-Landau energy in an infinite strip where phase imprinting is applied to half of the domain. We prove that there is a critical width of the cross section below which the soliton solution is a mountain pass solution and the minimizer within the subspace of odd functions. Above the critical width, we find that the mountain pass solution is a vortex with a solitonic behaviour in the infinite direction, called a solitonic vortex. Moreover, depending on the width, we prove that the minimizer in a space with some symmetries can display one or several solitonic vortices. While the problem shares some similarities with the analysis of stability and minimality of the Ginzburg-Landau vortex of degree one in a disk or the whole plane, the change in geometry introduces subtle analytical differences. Extensions to the case of an infinite cylinder in 3D are also given.

18. A Generalization of Caffarelli’s Contraction Theorem to Nearly Spherical Manifolds

一种将Caffarelli收缩定理推广至近球面流形的方法

作者: Yuxin Ge, Jordan Serres

类别: math.AP, math.DG, math.FA, math.PR

发布日期: 2025-12-01

arXiv ID: 2512.01496v1

链接: http://arxiv.org/abs/2512.01496v1

摘要: We show that every nearly spherical manifold can be realized as the volume-preserving image of a round sphere, via the Brenier-McCann optimal transport map. This theorem extends Caffarelli’s contraction theorem to nearly spherical manifolds and yields, as a corollary, a proof of a perturbative form of Milman’s conjecture. The proof is based on a novel stability result for optimal transport maps on the sphere.

19. Existence of two thresholds in a bistable equation with nonlocal competition

具有非局部竞争的双稳方程中两个阈值的存在性

作者: Matthieu Alfaro, Cédric Chane Ki Chune, Lionel Roques

类别: math.AP, q-bio.PE

发布日期: 2025-12-01

arXiv ID: 2512.01435v1

链接: http://arxiv.org/abs/2512.01435v1

摘要: We consider a nonlocal bistable reaction-diffusion equation, which serves as a model for a population structured by a phenotypic trait, subject to mutation, trait-dependent fitness, and nonlocal competition. Within this replicator-mutator framework, we further incorporate a ‘‘pseudo-Allee effect’’ so that the long time behavior (extinction vs. survival) depends on the size of the initial data. After proving the well-posedness of the associated Cauchy problem, we investigate its long-time behavior. We first show that small initial data lead to extinction. More surprisingly, we then prove that that extinction may also occur for too large initial data, in particular when selection is not strong enough. Finally, we exhibit situations where intermediate initial data lead to persistence, thereby revealing the existence of (at least) two thresholds. These results stand in sharp contrast with the behavior observed in local bistable equations.

20. Asymptotic stability of solitary waves for the b-family of equations

b族方程孤波的渐近稳定性

作者: Jun Wu, Yue Liu, Zhong Wang

类别: math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01225v1

链接: http://arxiv.org/abs/2512.01225v1

摘要: We establish the asymptotic stability of lefton solutions-exponentially localized stationary solitary waves-for the $b$-family of equations with positive momentum density in the regime $b < -1$. Unlike the completely integrable Camassa-Holm $(b=2)$ and Degasperis-Procesi $(b=3)$ cases, this parameter range lies outside integrability and exhibits distinct nonlinear dynamics. Our analysis adapts the Martel-Merle framework for generalized KdV equations to the nonlocal, non-integrable structure of the $b$-family of equations. The proof combines a nonlinear Liouville property for solutions localized near leftons with a refined spectral analysis of the associated linearized operator. These results provide the first rigorous asymptotic stability theory for leftons in the non-integrable $b$-family of equations.

21. Data assimilation and discrepancy modeling with shallow recurrent decoders

基于浅层循环解码器的数据同化与差异建模

作者: Yuxuan Bao, J. Nathan Kutz

类别: cs.LG, cs.AI, math.AP, nlin.CD

发布日期: 2025-12-01

arXiv ID: 2512.01170v1

链接: http://arxiv.org/abs/2512.01170v1

摘要: The requirements of modern sensing are rapidly evolving, driven by increasing demands for data efficiency, real-time processing, and deployment under limited sensing coverage. Complex physical systems are often characterized through the integration of a limited number of point sensors in combination with scientific computations which approximate the dominant, full-state dynamics. Simulation models, however, inevitably neglect small-scale or hidden processes, are sensitive to perturbations, or oversimplify parameter correlations, leading to reconstructions that often diverge from the reality measured by sensors. This creates a critical need for data assimilation, the process of integrating observational data with predictive simulation models to produce coherent and accurate estimates of the full state of complex physical systems. We propose a machine learning framework for Data Assimilation with a SHallow REcurrent Decoder (DA-SHRED) which bridges the simulation-to-real (SIM2REAL) gap between computational modeling and experimental sensor data. For real-world physics systems modeling high-dimensional spatiotemporal fields, where the full state cannot be directly observed and must be inferred from sparse sensor measurements, we leverage the latent space learned from a reduced simulation model via SHRED, and update these latent variables using real sensor data to accurately reconstruct the full system state. Furthermore, our algorithm incorporates a sparse identification of nonlinear dynamics based regression model in the latent space to identify functionals corresponding to missing dynamics in the simulation model. We demonstrate that DA-SHRED successfully closes the SIM2REAL gap and additionally recovers missing dynamics in highly complex systems, demonstrating that the combination of efficient temporal encoding and physics-informed correction enables robust data assimilation.

22. On solvability of parabolic equations with singular coefficients in odd mixed-norm Morrey-Sobolev spaces

关于奇混合范数Morrey-Sobolev空间中具奇异系数抛物方程的可解性研究

作者: N. V. Krylov

类别: math.AP

发布日期: 2025-12-01

arXiv ID: 2512.01168v1

链接: http://arxiv.org/abs/2512.01168v1

摘要: We prove an existence and uniqueness theorem for second-order parabolic equations in the whole space with constant zeroth-order coefficient in mixed-norm Morrey-Sobolev spaces. The main coefficient $a$ is assumed to be measurable in $t$ and BMO in $x$ and the first-order coefficients $b$ are in an appropriate mixed-norm Morrey classes (thus admitting rather rough singularities). The mixed-norm Morrey-Sobolev spaces are ``odd’’ in the sense that the interior integration in the formula defining the norm is performed with respect to $t$ and not to $x$ as is customary.

23. $\mathcal{C}^α$-regularity for nonlinear non-diagonal parabolic systems

非线性非对角抛物型系统的 $\mathcal{C}^α$ 正则性研究

作者: Miroslav Bulíček, Jens Frehse

类别: math.AP

发布日期: 2025-11-30

arXiv ID: 2512.01122v1

链接: http://arxiv.org/abs/2512.01122v1

摘要: In the elliptic theory for $p$-Laplacian-like problems, the Hölder continuity of solutions has been proven for problems arising as Euler–Lagrange equations of a convex potential with $p$-growth that additionally satisfies the splitting condition. In this article, we extend these results to the parabolic setting. We investigate nonlinear parabolic systems whose structure parallels the elliptic case but incorporates time dependence. Assuming suitable space-time regularity of $F$ and natural structural conditions analogous to the stationary theory, we establish $\mathcal{C}^α$-regularity of weak solutions in space and time whenever the growth parameter $p>d/2$. This extends the classical result for parabolic systems, which is valid only for $p>d-2$. This is the only regularity result for systems that are far from the radial (Uhlenbeck) structure.

24. Homogenization of a thin linear elastic plate reinforced with a periodic mosaic of small rigid plates

周期性镶嵌小刚性板增强的细线弹性薄板的均匀化

作者: Amartya Chakrabortty, Georges Griso, Julia Orlik

类别: math.AP

发布日期: 2025-11-30

arXiv ID: 2512.01042v1

链接: http://arxiv.org/abs/2512.01042v1

摘要: In the framework of linearized elasticity, we study thin elastic composite plates with thickness $δ$. The plates contain small, rigid rectangular plates distributed periodically along $\varepsilon$. Between two neighboring rigid plates is an elastic beam with thickness $δ< \varepsilon/3 < 1$. Through a simultaneous process of homogenization and dimension reduction, we obtain the limit model. Our analysis yields Korn-type inequalities adapted to the rigid-elastic geometry of the structure and provides a precise characterization of the limit deformation and displacement fields. In the $2$D limit problem, the bending is the sum of two functions, each depending on only one variable. This is due to the fact that the mixed derivatives of the outer-plane displacement vanish. Finally, the limiting 2D problem is two decoupled plates or strips, each one with just three degrees of freedom: shear along the strip axis, the cross-contraction (-extension), and the cross-bending. The corresponding correctors are defined in the same way in the periodicity cell. In the linearized setting, all the correctors are decomposed.

25. On the smoothness of solutions of fully nonlinear second order equations in the plane

关于平面上完全非线性二阶方程解的光滑性

作者: Alessandro Goffi

类别: math.AP

发布日期: 2025-11-30

arXiv ID: 2512.00951v1

链接: http://arxiv.org/abs/2512.00951v1

摘要: We study interior $C^{2,α}$ regularity estimates for solutions of fully nonlinear uniformly elliptic equations of the general form $F(D^2u)=0$ in two independent variables and without any geometric condition on $F$. By means of the theory of divergence form equations we prove that $C^2$ solutions of the previous equation are $C^{2,\barα(λ/Λ)}$ in the interior of the domain, where $0<λ\leqΛ$ are the ellipticity constants. We finally exploit the theory of nondivergence equations in the plane to obtain $C^{2,\tildeα}$ regularity for an explicit exponent $\tildeα=\tildeα(λ/Λ)>λ/Λ$.

26. Multiplicity of normalized solutions to the upper critical fractional Choquard equation with $L^2$-supercritical perturbation

具有 $L^2$-超临界扰动的上临界分数阶Choquard方程归一化解的多重性

作者: Yergen Aikyn, Yongpeng Chen, Michael Ruzhansky, Zhipeng Yang

类别: math.AP

发布日期: 2025-11-30

arXiv ID: 2512.00922v1

链接: http://arxiv.org/abs/2512.00922v1

摘要: We investigate normalized solutions with prescribed $L^2$-norm for the upper critical fractional Choquard equation

$$ (-Δ)^s u+V(\varepsilon x)u=λu+\big(I_α*|u|^{p}\big)|u|^{p-2}u+\big(I_α*|u|^{q}\big)|u|^{q-2}u\quad\text{in }\mathbb{R}^N, $$

where $N>2s$, $0<s<1$, $(N-4s)^+<α<N$, and the nonlocal exponents satisfy

$$ \frac{N+2s+α}{N}< q< p=\frac{N+α}{N-2s}, $$

so that both nonlinearities are $L^2$-supercritical and the $p$ term has upper critical growth of Hartree type. Under standard assumptions on the slowly varying potential $V$, we develop a constrained variational approach on the $L^2$-sphere, based on a truncation-penalization of the critical term in the energy functional, to overcome the lack of compactness. We prove that, for all sufficiently small $\varepsilon>0$, the problem admits at least $\mathrm{cat}_{M_δ}(M)$ distinct normalized solutions, where $M$ is the set of global minima of $V$ and these solutions concentrate near $M$ as $\varepsilon\to0$.

27. Nonlinear instability of rolls in the 2-dimensional generalized Swift-Hohenberg equation

二维广义Swift-Hohenberg方程中 roll解的非线性不稳定性

作者: Myeongju Chae, Soyeun Jung

类别: math.AP

发布日期: 2025-11-30

arXiv ID: 2512.00764v1

链接: http://arxiv.org/abs/2512.00764v1

摘要: Within the framework developed in \cite{Gr, JLL, RT1}, we rigorously establish the nonlinear instability of roll solutions to the two-dimensional generalized Swift-Hohenberg equation (gSHE). Our analysis is based on spectral information near the maximally unstable Bloch mode, combined with precise semigroup estimates. We construct a certain class of small initial perturbations that grow in time and cause the solution to deviate from the underlying roll solution within a finite time. This result provides a clear transition from spectral to nonlinear instability in a genuinely two-dimensional setting, where the Bloch parameter $σ$ ranges over an unbounded domain.

28. Non-Euclidean elasticity for rods and almost isometric embeddings of geodesic tubes

非欧几里得弹性理论中的杆模型与测地管的几乎等距嵌入

作者: Milan Kroemer, Stefan Müller

类别: math.AP

发布日期: 2025-11-29

arXiv ID: 2512.00643v1

链接: http://arxiv.org/abs/2512.00643v1

摘要: We consider a geodesic $γ$ of length $2L$ in an oriented Riemannian manifold $(\mathcal M, g)$ and a thin tube $Ω^*_h$ around $γ$ of radius $h$. We study an ’elastic’ energy per unit volume $E_h(u)$ of maps $u$ from $Ω^*_h$ into another oriented Riemannian manifold $(\tilde {\mathcal M},\tilde g)$. The energy $E_h$ is based on the squared distance of the differentials $du$ from the set of orientation preserving linear maps between the corresponding tangent spaces. We prove a compactness result for sequences of maps $u_h$ for which $h^{-4} E_h(u_h)$ remains bounded and we study the $Γ$-Limit of $h^{-4} E_h(u_h)$ as $h \to 0$ with respect to a suitable notion of convergence for $u_h$ that involves certain blow-ups in the radial direction. This $Γ$-convergence result ge-ne-ra-lizes work by Mora and Müller on the limiting energy of thin rods in the Euclidean setting. We also obtain an expression for the minimum of the limiting energy as a specific quadratic functional in the difference of the pullbacks of the curvature tensors of $\mathcal M$ and $\tilde{\mathcal M}$ along the curves $γ$ and $u \circ γ$, respectively, thus answering a question by Maor and Shachar, J. Elasticity 134 (2019), pp. 149–173.

29. Turing pattern induced by cross-diffusion in cancer immunotherapy model

交叉扩散在癌症免疫治疗模型中诱导的图灵模式研究

作者: C. F. Munafò, S. Bonfiglio, P. Rogolino

类别: math.DS, math-ph, math.AP

发布日期: 2025-11-29

arXiv ID: 2512.00599v1

链接: http://arxiv.org/abs/2512.00599v1

摘要: In this paper, we investigate a mathematical model describing the interactions between effector cells (E), cancer cells (T), and the IL-2 compound (IL). The model considered here is a generalization, taking into account some cross-diffusion effects, of a spatial cancer immunotherapy model proposed by S. Suddin et al in 2021. These modifications allow us to describe two biologically relevant scenarios: a patient treated with Adoptive Cell Immunotherapy (ACI) and a patient not receiving any treatment/therapy. Cross-diffusion effects are particularly relevant in the interactions between tumor cells and the immune system, in fact they play a key role in immune response dynamics and cannot be neglected. We analyze the equilibrium points of the homogeneous system, along with their stability and bifurcation mechanisms. Furthermore, adopting the Turing approach for reaction-diffusion systems, we investigate the diffusion-driven instability and the emergence of spatial regular structures (stationary in time), i.e. the patterns. Finally, numerical simulations based on the Finite Difference Method (FDM) are presented for the two previously mentioned scenarios.

30. Differential and Variational Approach to First Order Mean Field Games in a Generalized Form

广义形式下的一阶平均场博弈的微分与变分方法

作者: Antonio Siconolfi

类别: math.AP

发布日期: 2025-11-29

arXiv ID: 2512.00587v1

链接: http://arxiv.org/abs/2512.00587v1

摘要: We investigate time dependent, first order Mean Field Games on the torus comparing, in a broad and general framework, the classical differential formulation , given by a Hamilton Jacobi equation coupled with a continuity equation, with a variational approach based on fixed points of a multivalued map acting on probability measures over trajectories. We prove existence of fixed points for very general Hamiltonians. When the Hamiltonian is differentiable with respect to the momentum, we show that the evaluation curve of any such fixed point solves a continuity equation driven by a vector field associated with the final condition in the Hamilton Jacobi equation. This field is defined without requiring additional regularity conditions on the value function solving the Hamilton–Jacobi equation. The field coincides with the classical vector field of Mean Field Systems at space–differentiability points of the value function lying in the space–time regions where optimal trajectories concentrate. Our analysis therefore provides a unified framework that bridges the differential and variational viewpoints in Mean Field Games, showing how aggregate optimality conditions naturally lead to continuity-equation descriptions under the sole assumption of differentiability of the Hamiltonian in the momentum variable.

31. Parabolic problems with slightly superlinear convection terms

具有略强于线性对流项的抛物型问题

作者: Fessel Achhoud

类别: math.AP

发布日期: 2025-11-29

arXiv ID: 2512.00495v1

链接: http://arxiv.org/abs/2512.00495v1

摘要: In this paper we deal with a non-linear parabolic problem which involving a convection term with super–linear growth, whose model is

$$ \frac{\partial u}{\partial t}-÷(\mathcal{M}(x,t)\nabla u)= -÷(u\log (e+|u|)E(x,t))+f(x,t), $$

where $\mathcal{M}$ is a bounded measurable matrix, the vector field $E$ and the function $f$ belong to suitable Lebesgue spaces. We prove the existence of a unique bounded and unbounded weak solution.

32. Anisotropic elliptic equations involving unbounded coefficients and singular nonlinearities

具有无界系数和奇异非线性的各向异性椭圆方程

作者: Fessel achhoud, Hichem Khelifi

类别: math.AP

发布日期: 2025-11-29

arXiv ID: 2512.00485v1

链接: http://arxiv.org/abs/2512.00485v1

摘要: In this paper, we study the existence and regularity of solutions for a class of nonlinear singular elliptic equations involving unbounded coefficients and a singular right-hand side. Specifically, we are interested to problem whose simplest model is \begin{equation*} -\sum_{j=1}^N\partial_{j}\left([1+u^{q}]\vert \partial_{j} u \vert^{p_{j}-2} \partial_{j} u\right)= \frac{f}{u^γ}\text{ in $\mathcal{D},$}\quad u>0 \text{ in $\mathcal{D},$} \quad u=0 \hbox{ on};; \partial\mathcal{D}, \end{equation*} where $\mathcal{D}$ is a bounded open subset of $\mathbb{R}^{N}$ with $N>2$, $ γ\geq0$, $q >0 $, $p_{j}>2$ for all $j=1,...,N$ and the source term $f$ belongs to $L^1(\mathcal{D})$, with $f \geq 0$ and $f \not\equiv 0$.

33. Existence, Stability and Controllability of the parabolic-parabolic thermistor model

抛物-抛物型热敏电阻模型的存在性、稳定性和可控性

作者: Miguel R. Nuñez-Chávez, Luis P. Yapu, Juan Límaco

类别: math.AP

发布日期: 2025-11-29

arXiv ID: 2512.00478v1

链接: http://arxiv.org/abs/2512.00478v1

摘要: In this article we establish the well-posedness, energy estimates, stability, and local null controllability for the thermistor system modeled by a parabolic-parabolic system using a control force acting on just one equation of the system. The proof of the controllability is based on appropriate Carleman estimates and Liusternik’s inverse function theorem to obtain the local controllability of the nonlinear system. The coupling of the system happens both in the terms of order zero and one, which requires the use of a special Carleman estimate for the system.

34. Existence and bounds of nonlinear singularity-free cosmological solutions in a string-inspired gravity

一类弦理论启发引力模型中非线性无奇异性宇宙解的存在性与界限

作者: Chihang He, Chao Liu

类别: math.AP, gr-qc

发布日期: 2025-11-29

arXiv ID: 2512.00455v1

链接: http://arxiv.org/abs/2512.00455v1

摘要: We provide a rigorous proof for the existence of homogeneous, isotropic and globally singularity-free cosmological solutions in Einstein-dilaton-Gauss-Bonnet (EdGB) gravity with exponential coupling. While numerical studies suggested such solutions exist, a formal proof remained elusive. By employing a novel ``power identity method’’ and overcoming significant challenges posed by the strong nonlinearities of the exponential coupling, which are not present in the quadratic coupling analyzed in our companion paper \cite{he2025proofssingularityfreesolutionsscalarization}, we establish a FLRW solution valid for all time $t\in(-\infty,+\infty)$, where the Hubble parameter remains positive and vanishes asymptotically, while the scalar field evolves monotonically. This result align with numerical simulations and offer a firm mathematical foundation for singularity-free cosmology in a string-inspired setting.

35. Controllability of a Semilinear System of Parabolic Equations with Nonlocal Terms

具有非局部项的半线性抛物型方程组的可控性

作者: Juan Limaco, Rafael Martins Lobosco, Luis P. Yapu

类别: math.AP

发布日期: 2025-11-29

arXiv ID: 2512.00445v1

链接: http://arxiv.org/abs/2512.00445v1

摘要: This paper extends our previous controllability results for a class of coupled linear parabolic systems with nonlocal interactions, motivated by applications in finance such as generalized Black–Scholes models. We establish local null controllability at a fixed time T>0 for a class of semilinear, nonlocally coupled systems driven by a single internal control acting on one component. The proof combines Kakutani’s fixed-point theorem with a controllability/observability estimate for the associated linearized dynamics. In addition, we obtain controllability for a broader class of linear systems than those considered in the first article. The paper concludes with remarks on boundary controllability within the same nonlocal framework and with perspectives for future research.

36. The weak maximum principle for solutions of degenerate elliptic equations with lower order terms

具有低阶项的退化椭圆方程解的弱极值原理

作者: David Cruz-Uribe, Scott Rodney

类别: math.AP

发布日期: 2025-11-28

arXiv ID: 2512.00244v1

链接: http://arxiv.org/abs/2512.00244v1

摘要: We prove a weak maximum principle for subsolutions of a degenerate, linear, second order elliptic operator with lower order terms, building on the existence results recently proved by the authors and Çetin, Dal and Zeren.

37. Convergence and front position for an FKPP-type free boundary problem

一类FKPP型自由边界问题的收敛性与前沿位置

作者: Julien Berestycki, Sarah Penington, Oliver Tough

类别: math.AP, math.PR

发布日期: 2025-11-28

arXiv ID: 2511.23457v1

链接: http://arxiv.org/abs/2511.23457v1

摘要: The free boundary problem

$$ \begin{cases} \partial_tu=\frac{1}{2}Δu+u,\quad &t>0, \, x>L_t,\\ u(t,x)=0,\quad &t>0,\, x\le L_t,\\ \int_{L_t}^{\infty}u(t,y)dy=1,\quad &t> 0,\\ u(t,x)dx \to u_0(dx)&\text{weakly as }t\to 0, \end{cases} $$

has long been conjectured to be in the universality class of the so-called FKPP reaction-diffusion equation. It appears naturally as the hydrodynamic limit of a branching-selection particle system, the $N$-BBM. In the present work, we show that for any initial condition $u_0(dx)$ that decays fast enough as $x\to\infty$, the solution of the free boundary problem converges to the minimal travelling wave solution. We further show how the decay of the initial condition precisely determines the position of the free boundary $L_t$ at large times $t$, mirroring the celebrated results of Bramson \cite{Bramson1983} in the context of the FKPP equation. Our conditions for convergence to the minimal travelling wave, and for $L_t$ to have the Bramson asymptotics

$$ L_t=\sqrt{2}t-\frac{3}{2\sqrt{2}}\log t+c+o(1)\quad\text{as }t\to\infty, $$

are necessary and sufficient. We also apply our results to a more general free boundary problem that depends on a parameter $β$, where we see a transition from \emph{pulled} to \emph{pushed} behaviour (with \emph{pushmi-pullyu} behaviour at the critical value of $β$). We obtain analogous sharp conditions for convergence to the minimal travelling wave, along with precise asymptotics for the front position, in each of these regimes. To our knowledge, such necessary and sufficient conditions had not previously been established in the pushmi-pullyu or pushed regimes, even for classical monostable reaction-diffusion equations. Our results prove and extend non-rigorous predictions in the physics literature of the first author, Brunet and Derrida.