The asymptotic behavior of globally smooth solutions of non-isentropic Euler–Maxwell equations for plasmas

Abstract

In this paper we study the asymptotic behavior of globally smooth solutions of the Cauchy problem for the non-isentropic Euler–Maxwell equations arising in plasmas. We prove that smooth solutions (close to equilibrium) of the problem converge to a stationary solution as $t\to +\infty$.

Introduction

We investigate the asymptotic behavior of globally smooth solutions for the (rescaled) non-isentropic Euler–Maxwell systems, which takes the following (non-conservative) form [1], [3], [5], [6], [15]:

$$\left\{\begin{array}{cc}\hfill & {\partial }_{t}n+\nabla ·(\mathit{nu})=0\text{,}\hfill \\ \hfill & {\partial }_{t}u+(u·\nabla )u+\nabla \theta +\theta \nabla \ln n+u=-(E+u\times B)\text{,}\hfill \\ \hfill & {\partial }_t\theta +u·\nabla \theta +\frac{2}{3}\theta \nabla ·u=\frac{1}{3}|u{|}^2-(\theta -1)\text{,}\hfill \\ \hfill & {\partial }_{t}E-\nabla \times B=\mathit{nu}\text{,}\nabla ·E=1-n\text{,}\hfill \\ \hfill & {\partial }_{t}B+\nabla \times E=0\text{,}\nabla ·B=0\text{,}\hfill \end{array}\right.$$

for $(t\text{,}x)\in (0\text{,}\infty )\times \mathbb{T}$. Here, $n\text{,}u\text{,}\theta$ denote the scaled macroscopic density, mean velocity vector and temperature of the electrons and $E\text{,}B$ the scaled electric field and magnetic field. They are functions of a three-dimensional position vector $x\in \mathbb{T}$ and of the time $t>0$, where $\mathbb{T}={(\mathbb{R}/2\pi )}^3$ is the three-dimensional torus. The fields E and B are coupled to the particles through the Maxwell equations and act on the particles via the Lorentz force $E+u\times B$.

In this paper, we are interested in the asymptotics and global existence of smooth solutions of system (1.1) with the initial conditions:

$$t=0:(n\text{,}u\text{,}\theta \text{,}E\text{,}B)=(n_0\text{,}{u}_0\text{,}{\theta }_0\text{,}{E}_0\text{,}{B}_0)x\in \mathbb{T}\text{.}$$

The Euler–Maxwell system (1.1) is a symmetrizable hyperbolic system for $n\text{,}\theta >0$. Then the periodic problem (1.1), (1.2) has a local smooth solution when the initial data are smooth. In a simplified one dimensional Euler–Maxwell system, the global existence of entropy solutions has been given in [2] by the compensated compactness method. For the three dimensional Euler–Maxwell system, the existence of global smooth solutions with small amplitude to the Cauchy problem in the whole space and to the periodic problem in the torus is established by Peng et al. in [13] and Ueda et al. in [14] respectively, and the decay rate of the smooth solution when t goes to infinity is obtained by Duan in [4]. For asymptotic limits with small parameters, see [11], [12] and references therein. For the three dimensional non-isentropic Euler–Maxwell system, the diffusive relaxation limit is investigated by Yang et al. in [15]. So far there is no results about asymptotics and global existence for the non-isentropic Euler–Maxwell system.

The goal of the present paper is to establish the global existence of smooth solutions around a constant state $(1\text{,}0\text{,}1\text{,}0\text{,}\bar{B})$, which is a equilibrium solution of system (1.1), where $\bar{B}\in {\mathbb{R}}^3$ is any given constant. As a consequence, we obtain the asymptotic stability for $(n\text{,}u\text{,}\theta )$, i.e. $(n-1\text{,}u\text{,}\theta -1)⟶0$ as $t\to +\infty$ by using the elaborate energy method and the symmetric structure of the system.