Non-uniqueness for the compressible Euler–Maxwell equations

IF 2.1 2区数学 Q1 MATHEMATICS Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-20 DOI:10.1007/s00526-024-02798-2

Shunkai Mao, Peng Qu

引用次数: 0

Abstract

We consider the Cauchy problem for the isentropic compressible Euler–Maxwell equations under general pressure laws in a three-dimensional periodic domain. For any smooth initial electron density away from the vacuum and smooth equilibrium-charged ion density, we could construct infinitely many $\alpha $-Hölder continuous entropy solutions emanating from the same initial data for $\alpha <\frac{1}{7}$. Especially, the electromagnetic field belongs to the Hölder class $C^{1,\alpha }$. Furthermore, we provide a continuous entropy solution satisfying the entropy inequality strictly. The proof relies on the convex integration scheme. Due to the constrain of the Maxwell equations, we propose a method of Mikado potential and construct new building blocks.

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可压缩欧拉-麦克斯韦方程的非唯一性

我们考虑的是三维周期域中一般压力定律下等熵可压缩欧拉-麦克斯韦方程的考奇问题。对于任何远离真空的光滑初始电子密度和光滑平衡带电离子密度，我们都可以构造出无限多的（\α \）-霍尔德连续熵解，这些解都来自于相同的初始数据（\α <\frac{1}{7}\）。特别是，电磁场属于霍尔德类（C^{1,\alpha }\ ）。此外，我们还提供了严格满足熵不等式的连续熵解。证明依赖于凸积分方案。由于麦克斯韦方程的约束，我们提出了一种 Mikado 势的方法，并构建了新的构件。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.

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