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

TL;DR

This paper establishes global convergence rates for solutions of relaxed hyperbolic systems, specifically the compressible Euler and Euler-Maxwell systems, towards simpler models like the porous medium and drift-diffusion equations, using new asymptotic methods.

Contribution

It introduces a novel asymptotic expansion technique combined with stream function methods to obtain uniform-in-time error estimates for the convergence of these systems.

Findings

Convergence rate of ( ext{( ext{)} for the Euler system with ill-prepared data.

Enhanced convergence rate of ( ext{^2}) in well-prepared settings.

Global strong convergence of Euler-Maxwell solutions to the drift-diffusion model in (3).

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.