$L^p$-$L^{q}$-$L^{r}$ estimates and minimal decay regularity for compressible Euler-Maxwell equations

TL;DR

This paper introduces the concept of minimal decay regularity for dissipative systems with regularity-loss, establishing new $L^p$-$L^q$-$L^r$ decay estimates and applying them to compressible Euler-Maxwell equations to determine the lowest regularity for optimal decay.

Contribution

It proposes the notion of minimal decay regularity, develops new decay estimates in Fourier space, and applies these results to Euler-Maxwell equations with weaker dissipation.

Findings

Minimal decay regularity matches the critical regularity for global solutions.

New $L^p$-$L^q$-$L^r$ decay estimates are established.

Extended decay properties to symmetric hyperbolic systems with non-symmetric dissipation.

Abstract

Due to the dissipative structure of \textit{regularity-loss}, extra higher regularity than that for the global-in-time existence is usually imposed to obtain the optimal decay rates of classical solutions to dissipative systems. The aim of this paper is to seek the lowest regularity index for the optimal decay rate of $L^{1}(\mathbb{R}^n)$-$L^2(\mathbb{R}^n)$. Consequently, a notion of minimal decay regularity for dissipative systems of regularity-loss is firstly proposed. To do this, we develop a new time-decay estimate of $L^p(\mathbb{R}^n)$-$L^{q}(\mathbb{R}^n)$-$L^{r}(\mathbb{R}^n)$ type by using the low frequency and high-frequency analysis in Fourier spaces. As an application, for compressible Euler-Maxwell equations with the weaker dissipative mechanism, it is shown that the minimal decay regularity coincides with the critical regularity for global classical solutions. Moreover, the recent decay property for symmetric hyperbolic systems with non-symmetric dissipation is also extended to be the $L^p$-version.