Journal of Differential Equations最新文献

英文中文

Steklov eigenvalues of nearly circular area-normalized domains 近圆面积归一化域的Steklov特征值

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-10 DOI: 10.1016/j.jde.2025.114031

Lucas Alland , Robert Viator

We consider Steklov eigenvalues of nearly circular domains in

R^{2}

of fixed unitary area. In [19], the authors treated such domains as perturbations of the disk, and they computed the first-order term of the asymptotic expansions of the Steklov eigenvalues for reflection-symmetric perturbations; here, we expand these first-order results beyond reflection-symmetry. We also recover the second-order asymptotic expansions, which enable us to prove that no Steklov eigenvalue beyond the first positive one is locally shape-optimized by the disk.

研究了固定酉域R2上近圆域的Steklov特征值。在[19]中，作者将这些区域视为圆盘的扰动，并计算了反射对称扰动的Steklov特征值渐近展开式的一阶项；在这里，我们将这些一阶结果扩展到反射对称之外。我们还恢复了二阶渐近展开式，这使我们能够证明没有超出第一个正的Steklov特征值被磁盘局部形状优化。

引用次数: 0

Infinitely many self-similar blow-up profiles for the Keller-Segel system in dimensions 3 to 9 3 ~ 9维Keller-Segel系统的无穷多自相似爆破剖面

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-10 DOI: 10.1016/j.jde.2025.114033

Van Tien Nguyen , Zhi-An Wang , Kaiqiang Zhang

Based on the method of matched asymptotic expansions and Banach fixed point theorem, we rigorously construct infinitely many self-similar blow-up profiles for the parabolic-elliptic Keller-Segel system

{\begin{matrix} \partial_{t} u = Δ u - \nabla \cdot (u \nabla Φ_{u}), \\ 0 = Δ Φ_{u} + u, \\ u (\cdot, 0) = u_{0} \geq 0 \end{matrix} in R^{d},

where

d \in {3, \dots, 9}

. Our findings demonstrate that the infinitely many backward self-similar profiles approximate the rescaling radial steady-state near the origin (i.e.

0 < | x | ≪ 1

) and

\frac{2 (d - 2)}{| x |^{2}}

at spatial infinity (i.e.

| x | ≫ 1

). We also establish the convergence of the self-similar blow-up solutions as time tends to the blow-up time

T > 0

. Our results can give a refined description of backward self-similar profiles for all

| x | \geq 0

rather than for

0 < | x | ≪ 1

| x | ≫ 1

, indicating that the blow-up point is the origin and

u (x, t) \sim \frac{1}{| x |^{2}}, x \neq 0, as t \to T .

基于匹配渐近展开方法和Banach不动点定理，我们对抛物-椭圆型Keller-Segel系统{∂tu=Δu−∇⋅(u∇Φu),0=ΔΦu+u,u(⋅,0)=u0≥0inRd严格构造了无限多个自相似爆破轮廓，其中d∈{3，⋯,9}。我们的研究结果表明，无限多的后向自相似曲线近似于原点附近（即0<；|x|≪1）和空间无限大处（即|x|≠1）的重新标度径向稳态（即2(d−2)|x|2）。我们还建立了自相似爆破解随时间趋于爆破时间T>；0时的收敛性。我们的结果可以对所有|x|≥0而不是0<；|x|《1》或|x|《1》的后向自相似分布图进行精细描述，表明爆炸点是原点，并且u(x,t) ~ 1|x|2,x≠0,ast→t。

引用次数: 0

Symplectic techniques for stochastic differential equations on reductive Lie groups with applications to Langevin diffusions 约化李群上随机微分方程的辛技术及其在朗之万扩散中的应用

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-10 DOI: 10.1016/j.jde.2025.114034

Erwin Luesink , Oliver D. Street

We show how Langevin diffusions can be interpreted in the context of stochastic Hamiltonian systems with structure-preserving noise and dissipation on reductive Lie groups. Reductive Lie groups provide the setting in which the Lie group structure is compatible with Riemannian structures, via the existence of bi-invariant metrics. This structure allows for the explicit construction of Riemannian Brownian motion via symplectic techniques, which permits the study of Langevin diffusions with noise in the position coordinate as well as Langevin diffusions with noise in both momentum and position.

我们展示了如何在具有结构保持噪声和约化李群上耗散的随机哈密顿系统的背景下解释朗之万扩散。约化李群通过双不变度量的存在提供了李群结构与黎曼结构相容的条件。这种结构允许通过辛技术明确地构造黎曼布朗运动，从而允许在位置坐标中研究带噪声的朗之万扩散，以及在动量和位置中研究带噪声的朗之万扩散。

引用次数: 0

Strichartz estimates for the generalized Zakharov-Kuznetsov equation on R×T and applications R×T上广义Zakharov-Kuznetsov方程的Strichartz估计及其应用

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-09 DOI: 10.1016/j.jde.2025.113951

Jakob Nowicki-Koth

In this article, we address the Cauchy problem associated with the k-generalized Zakharov-Kuznetsov equation posed on

R \times T

. By establishing an almost optimal linear

L^{4}

-estimate, along with a family of bilinear refinements, we significantly lower the well-posedness threshold for all

k \geq 2

. In particular, we show that the modified Zakharov-Kuznetsov equation is locally well-posed in

H^{s} (R \times T)

for all

s > \frac{11}{24}

在本文中，我们讨论了与R×T上的k-广义Zakharov-Kuznetsov方程相关的柯西问题。通过建立一个几乎最优的线性l4估计，以及一系列双线性改进，我们显著降低了所有k≥2的适定性阈值。特别地，我们证明了修正的Zakharov-Kuznetsov方程在Hs（R×T）中对所有s>；1124都是局部适定的。

引用次数: 0

Asymptotic stability of composite waves of shock profile and rarefaction for the Navier–Stokes–Poisson system Navier-Stokes-Poisson系统激波剖面和稀疏复合波的渐近稳定性

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-09 DOI: 10.1016/j.jde.2025.114032

Wanyong Shim

We study the stability of composite waves consisting of a shock profile and a rarefaction wave for the one-dimensional isothermal Navier–Stokes–Poisson (NSP) system, which describes the ion dynamics in a collision-dominated plasma. More precisely, we prove that if the initial data are sufficiently close in the

H^{2}

norm to the Riemann data corresponding to a solution consisting of a shock and a rarefaction wave of the associated quasi-neutral Euler system, then the solution to the Cauchy problem for the NSP system converges, up to a dynamical shift, to a superposition of the corresponding shock profile and the rarefaction wave as time tends to infinity. Our proof is based on the method of a-contraction with shifts, which has recently been applied to the Navier–Stokes equations to establish the asymptotic stability of composite waves. To adapt this method to the NSP system, we employ a modulated relative functional introduced in our previous work on the stability of single shock profiles.

本文研究了一维等温Navier-Stokes-Poisson （NSP）系统中由激波剖面和稀疏波组成的复合波的稳定性，该系统描述了碰撞主导等离子体中的离子动力学。更准确地说，我们证明了如果初始数据在H2范数上与相关准中性欧拉系统的激波和稀疏波解对应的Riemann数据足够接近，那么NSP系统的Cauchy问题的解在时间趋于无穷时收敛到相应的激波和稀疏波的叠加，直到动态位移。我们的证明是基于a-带位移的收缩方法，该方法最近被应用于Navier-Stokes方程，以建立复合波的渐近稳定性。为了使这种方法适用于NSP系统，我们采用了在以前关于单激波剖面稳定性的工作中引入的调制相对泛函。

引用次数: 0

Nonlinear stability of periodic wave trains in the FitzHugh-Nagumo system against fully nonlocalized perturbations FitzHugh-Nagumo系统周期波列在完全非局域扰动下的非线性稳定性

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-08 DOI: 10.1016/j.jde.2025.114013

Joannis Alexopoulos, Björn de Rijk

Recently, a nonlinear stability theory has been developed for wave trains in reaction-diffusion systems relying on pure

L^{\infty}

-estimates. In the absence of localization of perturbations, it exploits diffusive decay caused by smoothing together with spatio-temporal phase modulation. In this paper, we advance this theory beyond the parabolic setting and propose a scheme designed for general dissipative semilinear problems. We present our method in the context of the FitzHugh-Nagumo system. The lack of parabolicity and localization complicates mode filtration in

L^{\infty}

-spaces using the Floquet-Bloch transform. Instead, we employ the inverse Laplace representation of the semigroup generated by the linearization to uncover high-frequency damping, while leveraging a link to the Floquet-Bloch representation for the smoothing low-frequency part. Another challenge arises in controlling regularity in the quasilinear iteration scheme for the modulated perturbation. We address this by extending the method of nonlinear damping estimates to nonlocalized perturbations using uniformly local Sobolev norms.

近年来，基于纯L∞估计建立了反应扩散系统中波列的非线性稳定性理论。在没有局部扰动的情况下，它利用了由平滑和时空相位调制引起的扩散衰减。在本文中，我们将这一理论推广到抛物型环境之外，并提出了一个设计用于一般耗散半线性问题的格式。我们在FitzHugh-Nagumo系统的背景下提出了我们的方法。利用Floquet-Bloch变换在L∞空间中进行模式滤波时，由于缺乏抛物性和局域性，使得滤波变得复杂。相反，我们采用由线性化产生的半群的逆变拉普拉斯表示来揭示高频阻尼，同时利用到Floquet-Bloch表示的链接来平滑低频部分。另一个挑战是如何控制调制扰动的拟线性迭代方案的正则性。我们通过使用一致局部Sobolev范数将非线性阻尼估计方法推广到非局部扰动来解决这个问题。

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引用次数: 0

On well-posedness of a mildly dissipative family of active scalar equations in borderline Sobolev spaces 边界Sobolev空间中一类微耗散有源标量方程的适定性

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-08 DOI: 10.1016/j.jde.2025.113939

Anuj Kumar , Vincent R. Martinez

This paper considers a family of active scalar equations which modify the generalized surface quasi-geostrophic (gSQG) equations through its constitutive law and a dissipative perturbation. These modifications are characteristically mild in the sense that they are logarithmic. The problem of well posedness, in the sense of Hadamard, is then studied in a borderline setting of regularity in analogy to the scaling-critical spaces of the gSQG equations. A novelty of the system considered is the nuanced form of smoothing provided by the proposed mild form of dissipation, which is able to support global well-posedness at the Euler endpoint, but in a setting where the inviscid counterpart is known to be ill-posed. A novelty of the analysis lies in the simultaneous treatment of modifications in the constitutive law, dissipative mechanism, and functional setting, which the existing literature has typically treated separately. A putatively sharp relation is identified between each of the distinct system-modifiers that is consistent with previous studies that considered these modifications in isolation. This unified perspective is afforded by the introduction of a linear model equation, referred to as the protean system, that successfully incorporates the more delicate commutator structure collectively possessed by the gSQG family and upon which each facet of well-posedness can effectively be reduced to its study.

本文研究了一类有源标量方程，它们通过本构律和耗散摄动对广义曲面拟地转方程进行了修正。这些修改的特点是温和的，因为它们是对数的。在Hadamard意义上的适定性问题，然后在正则性的边界设置中，类比于gSQG方程的标度临界空间，研究了适定性问题。所考虑的系统的一个新颖之处是所提出的温和耗散形式提供的细微平滑形式，它能够在欧拉端点支持全局适定性，但在已知无粘对应物是不适定的情况下。该分析的新颖之处在于，它同时处理了现有文献通常分别处理的本构法、耗散机制和功能设置的变化。在每个不同的系统修饰剂之间确定了假定的尖锐关系，这与先前孤立考虑这些修饰剂的研究是一致的。这种统一的视角是通过引入线性模型方程提供的，被称为protean系统，它成功地结合了gSQG家族共同拥有的更精致的换向器结构，并且在此基础上，适位性的每个方面都可以有效地减少到它的研究。

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引用次数: 0

A singular perturbation analysis for the Brusselator 布鲁塞尔子的奇异摄动分析

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-05 DOI: 10.1016/j.jde.2025.113866

Maximilian Engel , Guillermo Olicón-Méndez

In this work we study the Brusselator – a prototypical model for chemical oscillations– under the assumption that the bifurcation parameter is of order

O (1 / ϵ)

for positive

ϵ ≪ 1

. The dynamics of this mathematical model exhibits a time scale separation visible via fast and slow regimes along its unique attracting limit cycle. This limit cycle accumulates at infinity as

ϵ \to 0

, so that appropriate coordinates

(w, z)

are used to analyse the dynamics near the line at infinity, corresponding to the set

{z = 0}

. This object becomes a nonhyperbolic invariant manifold for which we use a desingularising rescaling, in order to study the closeby dynamics. Further use of geometric singular perturbation techniques allows us to give a decomposition of the Brusselator limit cycle in terms of four different fully quantified time scales for small ϵ.

在这项工作中，我们研究了Brusselator——化学振荡的一种原型模型——假设分岔参数为O阶（1/ λ），且正λ≪1。这个数学模型的动力学表现出时间尺度的分离，通过沿着其独特的吸引极限环的快、慢状态可见。这个极限环在无穷远处累积为λ→0，因此使用适当的坐标（w,z）来分析无穷远处线附近的动力学，对应于集合{z=0}。该对象变成一个非双曲不变流形，我们对其使用去具体化的重标度，以便研究近距离动力学。进一步使用几何奇异摄动技术，我们可以用四种不同的完全量化的时间尺度来分解布鲁塞尔极限环。

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引用次数: 0

Probabilistic representation for viscosity solutions to double-obstacle quasi-variational inequalities 双障碍拟变分不等式粘度解的概率表示

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-05 DOI: 10.1016/j.jde.2025.114006

Magnus Perninge

We prove the existence and uniqueness of viscosity solutions to quasi-variational inequalities (QVIs) with both upper and lower obstacles. In contrast to most previous works, we allow all involved coefficients to depend on the state variable and do not assume any type of monotonicity. It is well known that double obstacle QVIs are related to zero-sum games of impulse control, and our existence result is derived by considering a sequence of such games. Full generality is obtained by allowing one player in the game to randomize their control. A by-product of our result is that the corresponding zero-sum game has a value.

Utilizing recent results for backward stochastic differential equations (BSDEs), we find that the unique viscosity solution to our QVI is related to optimal stopping of BSDEs with constrained jumps and, in particular, to the corresponding non-linear Snell envelope. This gives a new probabilistic representation for double obstacle QVIs.

证明了具有上下障碍的拟变分不等式（QVIs）的粘滞解的存在唯一性。与大多数以前的工作相反，我们允许所有相关系数依赖于状态变量，并且不假设任何类型的单调性。众所周知，双障碍QVIs与冲动控制的零和博弈有关，我们的存在性结果是通过考虑一系列此类博弈而得出的。充分的通用性是通过允许游戏中的一个玩家随机控制而获得的。我们的结果的副产品是，相应的零和博弈具有价值。利用倒向随机微分方程（BSDEs）的最新结果，我们发现QVI的唯一粘度解与具有约束跳跃的BSDEs的最优停止有关，特别是与相应的非线性Snell包络有关。给出了双障碍QVIs的一种新的概率表示。

引用次数: 0

Fractional Caffarelli-Kohn-Nirenberg type inequalities on the Heisenberg group Heisenberg群上的分数阶Caffarelli-Kohn-Nirenberg型不等式

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-05 DOI: 10.1016/j.jde.2025.114030

Rama Rawat, Haripada Roy, Prosenjit Roy

The aim of this work is to establish some cases of the Caffarelli-Kohn-Nirenberg inequalities on the Heisenberg group for fractional Sobolev spaces. Here, we work with the fractional Sobolev spaces introduced by Adimurthi and Mallick (2018) [1], which provide a general framework in the context of the Heisenberg group. Our inequalities, in particular, contain the fractional Hardy's inequality and Sobolev inequality established by them, and also extend the admissible range of indices for the fractional Hardy's inequality.

本文的目的是建立分数Sobolev空间的Heisenberg群上的Caffarelli-Kohn-Nirenberg不等式的一些情形。在这里，我们使用Adimurthi和Mallick(2018)[1]引入的分数Sobolev空间，它提供了海森堡群背景下的一般框架。特别地，我们的不等式包含了分数阶Hardy不等式和由它们建立的Sobolev不等式，并扩展了分数阶Hardy不等式的指标可容许范围。

引用次数: 0

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Journal of Differential Equations

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