Journal of Differential Equations最新文献

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On the splash singularity for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic equations in 3D 关于三维粘性和非阻力不可压缩磁流体动力学方程自由边界问题的飞溅奇点

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-11-28 DOI: 10.1016/j.jde.2024.11.026

Guangyi Hong , Tao Luo , Zhonghao Zhao

In this paper, the existence of finite-time splash singularity is proved for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic (MHD) equations in

R^{3}

, based on a construction of a sequence of initial data alongside delicate estimates of the solutions. The result and analysis in this paper generalize those by Coutand and Shkoller in [14, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2019] from the viscous surface waves to the viscous conducting fluids with magnetic effects for which non-trivial magnetic fields may present on the free boundary. The arguments in this paper also hold for any space dimension

d \geq 2

本文基于初始数据序列的构造以及解的微妙估计，证明了 R3 中粘性和非阻性不可压缩磁流体动力学（MHD）方程的自由边界问题存在有限时间飞溅奇点。本文的结果和分析概括了 Coutand 和 Shkoller 在[14, Ann. Inst. H. Poincaré C Anal. Non Linéaire，2019]中从粘性表面波到具有磁效应的粘性导电流体的结果和分析，其中自由边界上可能存在非三维磁场。本文的论点在空间维数 d≥2 时也同样成立。

引用次数: 0

Global analyticity and the lower bounds of analytic radius for the Chaplygin gas equations with source terms 带有源项的查普利金气体方程的全局解析性和解析半径下限

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-11-28 DOI: 10.1016/j.jde.2024.11.027

Zhengyan Liu , Xinglong Wu , Boling Guo

This paper is devoted to studying the global existence and the analytic radius of analytic solutions to the Chaplygin gas equations with source terms. If the initial data belongs to Gevrey spaces and it is sufficiently small, we show the solution has the global persistent property in Gevrey spaces. In particular, we obtain uniform lower bounds on the spatial analytic radius which is given by

C e^{- C t}

, for some constant

C > 0

, this tells us that the decay rate of the analytic radius is at most a single exponential decay, which is the slowest decay rate of lower bounds on the analytic radius compared with the double and triple exponential decay of analytic radius derived by Levermore, Bardos, et al. (see Remark 1.2). Our method is based on the Fourier transformation and Gevrey-class regularity.

本文致力于研究带有源项的查普利金气体方程解析解的全局存在性和解析半径。如果初始数据属于 Gevrey 空间且足够小，我们证明解在 Gevrey 空间中具有全局持久性。特别是，我们得到了空间解析半径的均匀下界，即在某个常数 C>0 下，解析半径由 Ce-Ct 给定，这告诉我们解析半径的衰减率最多是单指数衰减，与 Levermore、Bardos 等人推导的解析半径的双倍和三倍指数衰减相比，这是解析半径下界的最慢衰减率（见备注 1.2）。我们的方法基于傅里叶变换和 Gevrey 级正则性。

引用次数: 0

A note on the log-perturbed Brézis-Nirenberg problem on the hyperbolic space

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-11-28 DOI: 10.1016/j.jde.2024.11.025

Monideep Ghosh, Anumol Joseph, Debabrata Karmakar

We consider the log-perturbed Brézis-Nirenberg problem on the hyperbolic space

Δ_{B^{N}} u + λ u + | u |^{p - 1} u + θ u \ln u^{2} = 0, u \in H^{1} (B^{N}), u > 0 in B^{N},

and study the existence vs non-existence results. We show that whenever

θ > 0

, there exists an

H^{1}

-solution, while for

θ < 0

, there does not exist a positive solution in a reasonably general class. Since the perturbation

u \ln u^{2}

changes sign, Pohozaev type identities do not yield any non-existence results. The main contribution of this article is obtaining an “almost” precise lower asymptotic decay estimate on the positive solutions for

θ < 0

, culminating in proving their non-existence assertion.

引用次数: 0

Time splitting method for nonlinear Schrödinger equation with rough initial data in L2 具有 L2 中粗糙初始数据的非线性薛定谔方程的时间分割法

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-11-26 DOI: 10.1016/j.jde.2024.11.018

Hyung Jun Choi , Seonghak Kim , Youngwoo Koh

We establish convergence results related to the operator splitting scheme on the Cauchy problem for the nonlinear Schrödinger equation with rough initial data in

L^{2}

{\begin{matrix} i \partial_{t} u + Δ u = λ | u |^{p} u, & (x, t) \in R^{d} \times R_{+}, \\ u (x, 0) = ϕ (x), & x \in R^{d}, \end{matrix}

where

λ \in {- 1, 1}

and

p > 0

. While the Lie approximation

Z_{L}

is known to converge to the solution u when the initial datum ϕ is sufficiently smooth, the convergence result for rough initial data is open to question. In this paper, for rough initial data

ϕ \in L^{2} (R^{d})

, we prove the

L^{2}

convergence of the filtered Lie approximation

Z_{f l t}

to the solution u in the mass-subcritical range,

0 < p < \frac{4}{d}

. Furthermore, we provide a precise convergence result for radial initial data

ϕ \in L^{2} (R^{d})

我们建立了非线性薛定谔方程 Cauchy 问题的算子分裂方案的收敛结果，该问题的初始数据为 L2 中的粗糙数据，{i∂tu+Δu=λ|u|pu,(x,t)∈Rd×R+,u(x,0)=j(x),x∈Rd，其中λ∈{-1,1}和 p>0。众所周知，当初始数据 j 足够光滑时，Lie 近似值 ZL 会收敛于解 u，但粗糙初始数据的收敛结果却有待商榷。在本文中，对于粗糙初始数据ϕ∈L2(Rd)，我们证明了滤波Lie近似Zflt在质量次临界范围（0<p<4d）内对解u的L2收敛性。此外，我们还提供了径向初始数据 j∈L2(Rd) 的精确收敛结果。

引用次数: 0

The extended second Painlevé hierarchy: Auto-Bäcklund transformations and special integrals 扩展的第二潘列维层次：自贝克伦德变换和特殊积分

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-11-26 DOI: 10.1016/j.jde.2024.11.017

P.R. Gordoa, A. Pickering

We return to our study of the extended second Painlevé hierarchy presented in a previous paper. For this hierarchy we give a new local auto-BT. We also give an extensive discussion of the iterative construction of solutions and special integrals using auto-BTs. Furthermore, we show that Lax pairs can be provided for special integrals. Even though this will, in fact, be the case quite generally, it seems that Lax pairs for special integrals have not been given previously. Amongst the equations for which we present Lax pairs are examples due to Cosgrove and, in classical Painlevé classification results, Chazy.

我们回到前一篇论文中对扩展的第二潘列维层次结构的研究。对于这个层次，我们给出了一个新的局部自BT。我们还广泛讨论了使用自BT迭代构造解和特殊积分的问题。此外，我们还证明了可以为特殊积分提供 Lax 对。尽管这实际上是一种很普遍的情况，但以前似乎还没有给出过特殊积分的 Lax 对。在我们给出 Lax 对的方程中，有科斯格罗夫（Cosgrove）的例子，也有查兹（Chazy）的经典 Painlevé 分类结果。

引用次数: 0

Asymptotically additive families of functions and a physical equivalence problem for flows 函数的渐近相加族和流动的物理等价问题

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-11-26 DOI: 10.1016/j.jde.2024.11.023

Carllos Eduardo Holanda

We show that additive and asymptotically additive families of continuous functions with respect to suspension flows are physically equivalent. In particular, the equivalence result holds for hyperbolic flows. We also obtain an equivalence relation for expansive flows. Moreover, we show how this equivalence result can be used to extend the nonadditive thermodynamic formalism and multifractal analysis for flows.

我们证明，相对于悬浮流的连续函数的可加族和渐近可加族在物理上是等价的。特别是，对于双曲流，等价结果成立。我们还得到了膨胀流的等价关系。此外，我们还展示了如何利用这一等价结果来扩展流动的非加性热力学形式主义和多分形分析。

引用次数: 0

Global well-posedness of the three-dimensional free boundary problem for viscoelastic fluids without surface tension 无表面张力粘弹性流体三维自由边界问题的全局拟合性

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-11-26 DOI: 10.1016/j.jde.2024.11.020

Jingchi Huang, Zheng-an Yao, Xiangyu You

In this paper, we consider the three-dimensional free boundary problem of incompressible and compressible neo-Hookean viscoelastic fluid equations in an infinite strip without surface tension, provided that the initial data is sufficiently close to the equilibrium state. By reformulating the problems in Lagrangian coordinates, we can get the stabilizing effect of elasticity. In both cases, we utilize the elliptic estimates to improve the estimates. Moreover, for the compressible case, we find there is an extra ODE structure that can improve the regularity of the free boundary, thus we can have the global well-posedness. To prove the global well-posedness for the incompressible case, we employ two-tier energy method introduced in [11][12][13] to compensate for the inferior structure.

本文考虑了无表面张力的无限条带中不可压缩和可压缩新胡克粘弹性流体方程的三维自由边界问题，前提是初始数据足够接近平衡状态。通过在拉格朗日坐标中重新表述问题，我们可以获得弹性的稳定效应。在这两种情况下，我们都利用椭圆估计来改进估计。此外，在可压缩情况下，我们发现有一个额外的 ODE 结构可以改善自由边界的正则性，因此我们可以得到全局好拟性。为了证明不可压缩情况下的全局最优性，我们采用了[11][12][13]中介绍的两层能量法来补偿劣化结构。

引用次数: 0

Poincaré-Perron problem for high order differential equations in the class of almost periodic type functions 几乎周期型函数类高阶微分方程的 Poincaré-Perron 问题

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-11-26 DOI: 10.1016/j.jde.2024.11.016

H. Bustos , P. Figueroa , M. Pinto

We address the Poincaré-Perron's classical problem of approximation for high order linear differential equations in the class of almost periodic type functions, extending the results for a second order linear differential equation in [23]. We obtain explicit formulae for solutions of these equations, for any fixed order

n \geq 3

, by studying a Riccati type equation associated with the logarithmic derivative of a solution. Moreover, we provide sufficient conditions to ensure the existence of a fundamental system of solutions. The fixed point Banach argument allows us to find almost periodic and asymptotically almost periodic solutions to this Riccati type equation. A decomposition property of the perturbations induces a decomposition on the Riccati type equation and its solutions. In particular, by using this decomposition we obtain asymptotically almost periodic and also p-almost periodic solutions to the Riccati type equation. We illustrate our results for a third order linear differential equation.

我们解决了 Poincaré-Perron 提出的近似几乎周期型函数类高阶线性微分方程的经典问题，扩展了 [23] 中二阶线性微分方程的结果。通过研究与解的对数导数相关的里卡提式方程，我们得到了这些方程在任何固定阶数 n≥3 时的解的明确公式。此外，我们还提供了充分条件，以确保基本解系的存在。通过定点巴拿赫论证，我们可以找到这个里卡提式方程的近周期解和渐近近周期解。扰动的分解特性诱导出里卡蒂方程及其解的分解。特别是，通过使用这种分解，我们得到了里卡提式方程的渐近近周期解和 p 近似周期解。我们用一个三阶线性微分方程来说明我们的结果。

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引用次数: 0

Explicit approximation for stochastic nonlinear Schrödinger equation 随机非线性薛定谔方程的显式近似值

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-11-22 DOI: 10.1016/j.jde.2024.11.022

Jianbo Cui

In this paper, we study explicit approximations of stochastic nonlinear Schrödinger equations (SNLSEs). We first prove that the classical explicit numerical approximations are divergent for SNLSEs with polynomial nonlinearities. To enhance the stability, we propose a kind of explicit numerical approximations, and establish the regularity analysis and strong convergence rate of the proposed approximations for SNLSEs. There are two key ingredients in our approach. One ingredient is constructing a logarithmic auxiliary functional and exploiting the Bourgain space to prove new regularity estimates of SNLSEs. Another one is providing a dedicated error decomposition formula and presenting the tail estimates of underlying stochastic processes. In particular, our result answers the strong convergence problem of numerical approximation for 2D SNLSEs.

本文研究随机非线性薛定谔方程（SNLSE）的显式近似。我们首先证明了经典的显式数值近似对于具有多项式非线性的 SNLSE 是发散的。为了提高稳定性，我们提出了一种显式数值近似方法，并建立了针对 SNLSE 的近似方法的正则性分析和强收敛率。我们的方法有两个关键要素。一个是构建对数辅助函数，利用布尔干空间证明 SNLSE 的新正则性估计。另一个是提供专门的误差分解公式，并给出底层随机过程的尾估计值。特别是，我们的结果回答了二维 SNLSE 数值逼近的强收敛问题。

引用次数: 0

Corrigendum to “Differential inclusions in Wasserstein spaces: The Cauchy-Lipschitz framework” [J. Differ. Equ. 271 (2021) 594–637]

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-11-22 DOI: 10.1016/j.jde.2024.10.045

Benoît Bonnet-Weill , Hélène Frankowska

This corrigendum is concerned with the technical preliminary [1, Lemma 1]. Unfortunately, its proof contains a mistake which ultimately renders its conclusion erroneous. In this note, we provide a corrected version of the latter, and show that this modification has no impact on the other results of [1] while incurring very benign changes in none but two series of computations throughout the manuscript.

引用次数: 0

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