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Global solutions of Euler-Maxwell equations with dissipation

Bernard Ducomet, Šárka Nečasová, John Sebastian H. Simon

Abstract

We consider the Cauchy problem for a damped Euler-Maxwell system with no ionic background. For smooth enough data satisfying suitable so-called dispersive conditions, we establish the global in time existence and uniqueness of a strong solution that decays uniformly in time. Our method is inspired by the works of D. Serre and M. Grassin dedicated to the compressible Euler system.

Global solutions of Euler-Maxwell equations with dissipation

Abstract

We consider the Cauchy problem for a damped Euler-Maxwell system with no ionic background. For smooth enough data satisfying suitable so-called dispersive conditions, we establish the global in time existence and uniqueness of a strong solution that decays uniformly in time. Our method is inspired by the works of D. Serre and M. Grassin dedicated to the compressible Euler system.
Paper Structure (8 sections, 6 theorems, 114 equations)

This paper contains 8 sections, 6 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Decay estimates in Sobolev spaces
  4. Decay estimates for the generalized Burgers equation
  5. Sobolev estimates for System \ref{['eq:MP']}
  6. Proving Theorem \ref{['isentro']}
  7. Existence
  8. Uniqueness

Key Result

Proposition 2.1

Let $v_0$ be in $E^s$ with $s>5/2$ and satisfy: where ${\rm Sp}\,A$ denotes the spectrum of the matrix $A$. Then auxiso supplemented with auxiso0 has a classical solution $v$ on $\mathbb{R}_+\times\mathbb{R}^3$ such that Moreover, $D v\in {\mathcal{C}}_b(\mathbb{R}_+\times\mathbb{R}^3)$ and we have for any $t\geq 0$ and $x\in \mathbb{R}^3,$ for some function $K\in{\mathcal{C}}_b(\mathbb{R}_+\ti

Theorems & Definitions (8)

  • Proposition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1