Optimal time decay estimation for large-solution about 3D compressible MHD equations

Abstract

This paper mainly focus on optimal time decay estimation for large-solution about compressible magnetohydrodynamic equations in 3D whole space, provided that $(\sigma_{0}-1,u_{0},M_{0})\in L^1\cap H^2$. In [2](Chen et al.,2019), they proved time decay estimation of $\|(\sigma-1,u,M)\|_{H^1}$ being $(1+t)^{-\frac{3}{4}}$. Based on it, we obtained that of $\|\nabla(\sigma-1,u,M)\|_{H^1}$ being $(1+t)^{-\frac{5}{4}}$ in [24]. Therefore, we are committed to improving that of $\|\nabla^2 (\sigma-1,u,M)\|_{L^2}$ in this paper. Thanks to the method adopted in [25] (Wang and Wen, 2021), we get the optimal time decay estimation to the highest-order derivative for space of solution, which means that time decay estimation of $\|\nabla^2 (\sigma-1,u,M)\|_{L^2}$ is $(1+t)^{-\frac{7}{4}}$