Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence and non-convergence phenomena in Euler-Maxwell to MHD transitions

Dong-ha Kim, Junha Kim, Jihoon Lee

Abstract

In this work, we investigate the difference estimate for a class of Euler-Maxwell system and those of magnetohydrodynamics (in short, MHD) systems in three dimensions. We decompose the Euler-Maxwell system into three parts, namely the MHD system, auxiliary linear system and error part system. As a result, we obtain the convergence of the velocity of the fluid $u$, electric fields $E$ and magnetic fields $B$ from the Euler-Maxwell system toward the MHD system in $L^{p}_{t}L^{2}_{x}$ as the speed of light $c$ approaches infinity for $p\in[1,\infty]$. We also derived non-convergence results of electric current $j$ or $cE$, and these results are classified by a certain threshold for $p$. Finally, we investigate how the $L^2$-energy flow of Euler-Maxwell system evolves as c tends to infinity, leading to the vanishing of Ampère's equation in the Euler-Maxwell system.

Convergence and non-convergence phenomena in Euler-Maxwell to MHD transitions

Abstract

In this work, we investigate the difference estimate for a class of Euler-Maxwell system and those of magnetohydrodynamics (in short, MHD) systems in three dimensions. We decompose the Euler-Maxwell system into three parts, namely the MHD system, auxiliary linear system and error part system. As a result, we obtain the convergence of the velocity of the fluid , electric fields and magnetic fields from the Euler-Maxwell system toward the MHD system in as the speed of light approaches infinity for . We also derived non-convergence results of electric current or , and these results are classified by a certain threshold for . Finally, we investigate how the -energy flow of Euler-Maxwell system evolves as c tends to infinity, leading to the vanishing of Ampère's equation in the Euler-Maxwell system.

Paper Structure

This paper contains 18 sections, 13 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Motivation of the work
  3. Main results
  4. Strategy of the proof
  5. Outline of the paper
  6. Local existence of solutions
  7. Existence of local-in-time solutions
  8. Improvement of the convergence
  9. Auxiliary linear system
  10. Preliminaries
  11. Linear parts estimates
  12. Proof of the main result
  13. Error part estimates
  14. Proof of the main results
  15. Appendix
  16. ...and 3 more sections

Key Result

Theorem 1.2

We assume that there exists $T>0$ satisfying the following: For each $c \geq c_0$, let $(u^c,B^c,E^c) \in C([0,T];H^m(\mathbb R^3))$ be the solution of Euler-Maxwell system main_eqn satisfying Condition cond_initial. By picking $(u_0,B_0)$ up in Condition cond_initial, let $(\overline{u},\overline{B and Below, recalling $\overline{E}$ in def_Ebar, we set Then, for any $p \in [1,+\infty]$, there

Theorems & Definitions (29)

  • Theorem 1.2: Difference estimates
  • Remark 1.3
  • Corollary 1.4: Convergence of solutions $u^c,B^c$ and $E^c$
  • Corollary 1.5: Convergence and Non-convergence of $cE^c$ and $j^c$
  • Remark 1.6
  • Corollary 1.7
  • Theorem 2.1: Euler-Maxwell system
  • Lemma 2.2: a priori estimate for Euler-Maxwell
  • proof : Proof of Theorem \ref{['thm_ext']}
  • Theorem 2.3: MHD system
  • ...and 19 more