Communications in Mathematical Sciences

Volume 17 (2019)

Number 7

Global stability of large steady-states for an isentropic Euler–Maxwell system in $\mathbb{R}^3$

Pages: 1841 – 1860

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n7.a4

Authors

Cunming Liu (School of Mathematical Sciences, Qufu Normal University, Shandong, China)

Zuji Guo (Department of Mathematics, Taiyuan University of Technology, Taiyuan, China)

Yue-Jun Peng (Laboratoire de Mathématiques Blaise Pascal, Université Clermont Auvergne, Clermont-Ferrand, France)

Abstract

This paper concerns the global existence and stability of smooth solutions near large steady-states for an isentropic Euler–Maxwell system in $\mathbb{R}^3$. This system describes the dynamics of electrons in magnetized plasmas where the ion density is a given smooth function with a positive lower bound, but without any restriction on the size. We establish the well-posedness of large steady-state solutions with zero velocity in $\mathbb{R}^3$. It is achieved through a study for a semilinear elliptic equation by using variational methods. For the initial data close to the steady-state solutions, we solve the stability problem by means of classical energy estimates and an anti-symmetric matrix technique together with an induction argument on the order of the derivatives of solutions with respect to the time and space variables.

Keywords

Euler–Maxwell system, global stability, large steady-state solution, energy estimate

2010 Mathematics Subject Classification

35B40, 35Q35, 35Q60

Full Text (PDF format)

The research of this work was supported in part by National Natural Science Foundation of China under grants 11401421, 11601363, 11671295 and Science Foundation of Shanxi Province under grants 201601D021011, 201701D121008.

Received 15 November 2018

Accepted 9 May 2019

Published 6 January 2020