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, 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, 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 $(1+t)^{-\frac{7}{4}}$.