Large time behavior of strong solutions to the compressible magnetohydrodynamic system in the critical $L^{p}$ framework

TL;DR

This paper analyzes the long-term decay of strong solutions to the compressible magnetohydrodynamic system in a critical $L^{p}$ framework, extending previous results with more precise decay rates and Besov space estimates.

Contribution

It improves existing decay estimates for MHD solutions by providing detailed asymptotic behavior in Besov norms with negative regularity.

Findings

Solutions decay like $t^{-d(1/p - 1/4)}$ in $L^{p}$ norm as $t o rightarrow$

Established decay rates in Besov spaces with negative regularity

Extended classical decay results to a broader functional framework

Abstract

In this paper, we are concerned with the compressible viscous magnetohydrodynamic (MHD) system and investigate the large time behavior of strong solutions near constant equilibrium. In the eighties, Umeda, Kawashima and Shizuta initiated the dissipative mechanism for a rather general class of symmetric hyperbolic-parabolic systems. From the point of view of dissipativity, Kawashima in his doctoral dissertation established the optimal time-decay estimates of $L^{q}$-$L^{2}(q\in [1,2)$) type for solutions to the MHD system. Here, we shall improve Kawashima's efforts and give more precise description for the large time asymptotic behavior of solutions, not only in extra Lebesgue spaces but also in a full family of Besov norms with the negative regularity index. Precisely, we show that the $L^{p}$ norm (the slightly stronger $\dot{B}^{0}_{p,1}$ one in fact) of global solutions with the critical regularity, decays like $t^{-d(\frac{1}{p}-\frac{1}{4})}$ for $t\rightarrow\infty$. In particular, taking $p=2$ and $d=3$ goes back to the classical time decay $t^{-\frac{3}{4}}$. We derive new estimates which are used to deal with the strong coupling between the magnetic field and fluid dynamics.