Optimal decay for the compressible MHD equations in the critical regularity framework

Abstract

In this paper, we study the large time behavior of solutions to the compressible magnetohydrodynamic equations in the $L^p$-type critical Besov spaces. Precisely, we show that if the initial data in the low frequencies additionally belong to some Besov space ${\dot{B}}_{2,\infty }^{-{\sigma }_1}$ with ${\sigma }_1\in (1-N/2,2N/p-N/2]$, then the ${\dot{B}}_{p,1}^0$ norm of the critical global solutions presents the optimal decay $t^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{{\sigma }_1}{2}}$ for $t\to +\infty$. The pure energy argument without the spectral analysis is performed, which allows us to remove the usual smallness assumption of low frequencies.

arXiv admin note: text overlap with arXiv:1812.11714 by other authors