The frequency-localization technique and minimal decay-regularity for Euler-Maxwell equations

TL;DR

This paper introduces a new frequency-localization technique to improve decay estimates for dissipative hyperbolic systems, specifically applying it to Euler-Maxwell equations to achieve optimal decay rates without requiring extra regularity.

Contribution

The paper develops a novel frequency-localization time-decay method that reduces the minimal regularity needed for decay estimates in dissipative systems, exemplified by Euler-Maxwell equations.

Findings

Achieved optimal decay rate for Euler-Maxwell equations at critical regularity s_c=5/2.

Reduced the regularity requirement for decay estimates in dissipative hyperbolic systems.

Established a new approach that overcomes technical difficulties in decay analysis.

Abstract

Dissipative hyperbolic systems of \textit{regularity-loss} have been recently received increasing attention. Usually, extra higher regularity is assumed to obtain the optimal decay estimates, in comparison with that for the global-in-time existence of solutions. In this paper, we develop a new frequency-localization time-decay property, which enables us to overcome the technical difficulty and improve the minimal decay-regularity for dissipative systems. As an application, it is shown that the optimal decay rate of $L^1(\mathbb{R}^3)$-$L^2(\mathbb{R}^3)$ is available for Euler-Maxwell equations with the critical regularity $s_{c}=5/2$, that is, the extra higher regularity is not needed.