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Global convergence rates in the relaxation limits for the compressible Euler and Euler-Maxwell systems in Sobolev spaces

Timothée Crin-Barat, Yue-Jun Peng, Ling-Yun Shou

TL;DR

The paper analyzes global-in-time relaxation limits for two partially dissipative systems: the damped compressible Euler equations and the damped compressible Euler–Maxwell system. By developing a multi-dimensional stream-function framework and a systematic asymptotic expansion, the authors prove explicit convergence rates in Sobolev spaces for ill-prepared data and enhanced $O(^2)$ rates to first-order approximations under well-prepared data, with the Euler–Maxwell system converging to a drift-diffusion model. The work introduces initial-layer corrections and carefully manages nonlinear residuals to obtain uniform-in-time error bounds, avoiding frequency-localization techniques. These results deepen understanding of diffusion limits in high dimensions and establish quantitative rates relevant to plasma and fluid-EM coupling models, offering robust tools for further extensions to bounded domains or discrete settings.

Abstract

We study two relaxation problems in the class of partially dissipative hyperbolic systems: the compressible Euler system and the compressible Euler-Maxwell system. In classical Sobolev spaces, we derive a global convergence rate of $\mathcal{O}(\varepsilon)$ between strong solutions of the relaxed Euler system and the porous medium equation in $\mathbb{R}^d$ ($d\geq1$) for \emph{ill-prepared} initial data. In a well-prepared setting, we derive an enhanced convergence rate of order $\mathcal{O}(\varepsilon^2)$ between the solutions of the relaxed compressible Euler system and their first-order asymptotic approximation. Regarding the relaxed Euler-Maxwell system, we prove the global strong convergence of its solutions to the drift-diffusion model in $\mathbb{R}^3$ in an \emph{ill-prepared} setting. These results are achieved by developing a new asymptotic expansion approach that, combined with stream function techniques, ensures uniform-in-time error estimates.

Global convergence rates in the relaxation limits for the compressible Euler and Euler-Maxwell systems in Sobolev spaces

TL;DR

The paper analyzes global-in-time relaxation limits for two partially dissipative systems: the damped compressible Euler equations and the damped compressible Euler–Maxwell system. By developing a multi-dimensional stream-function framework and a systematic asymptotic expansion, the authors prove explicit convergence rates in Sobolev spaces for ill-prepared data and enhanced rates to first-order approximations under well-prepared data, with the Euler–Maxwell system converging to a drift-diffusion model. The work introduces initial-layer corrections and carefully manages nonlinear residuals to obtain uniform-in-time error bounds, avoiding frequency-localization techniques. These results deepen understanding of diffusion limits in high dimensions and establish quantitative rates relevant to plasma and fluid-EM coupling models, offering robust tools for further extensions to bounded domains or discrete settings.

Abstract

We study two relaxation problems in the class of partially dissipative hyperbolic systems: the compressible Euler system and the compressible Euler-Maxwell system. In classical Sobolev spaces, we derive a global convergence rate of between strong solutions of the relaxed Euler system and the porous medium equation in () for \emph{ill-prepared} initial data. In a well-prepared setting, we derive an enhanced convergence rate of order between the solutions of the relaxed compressible Euler system and their first-order asymptotic approximation. Regarding the relaxed Euler-Maxwell system, we prove the global strong convergence of its solutions to the drift-diffusion model in in an \emph{ill-prepared} setting. These results are achieved by developing a new asymptotic expansion approach that, combined with stream function techniques, ensures uniform-in-time error estimates.

Paper Structure

This paper contains 30 sections, 23 theorems, 293 equations.

Table of Contents

  1. Introduction
  2. The compressible Euler system with damping
  3. The compressible Euler-Maxwell system
  4. Literature on damped Euler equations and partially dissipative hyperbolic systems
  5. Literature on the Euler-Maxwell system
  6. Aims of the paper
  7. Main results
  8. The compressible Euler system
  9. Strategy of proof for the Euler system
  10. The compressible Euler-Maxwell system
  11. Strategy of proof for the Euler-Maxwell system
  12. Preliminaries and toolbox
  13. The Euler system (I): Proof of Theorem \ref{['T1.1']}
  14. The error equations
  15. Error estimates in $L^2( \mathbb{R}^+;L^2)$
  16. ...and 15 more sections

Key Result

Proposition 2.1

( CoulombelGoudonCoulombelLin) Let $d\geq1$, $m\geq [\frac{d}{2}]+2$ and F1.1 hold. There exists a positive constant $\delta$, independent of $\varepsilon$, such that, if $\mathcal{E}^\varepsilon_0\le\delta$, then the Cauchy problem F1.2-F1.3 admits a unique global classical solution $(\rho^\vareps for a generic constant $C>0$. Moreover, for any given $T>0$, as $\varepsilon\to 0$, up to subsequen

Theorems & Definitions (26)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.2
  • Proposition 2.3: Wasiolek2016
  • Proposition 2.4
  • Theorem 2.3
  • Remark 2.3
  • ...and 16 more