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

TL;DR

This paper establishes the optimal decay rates over time for second derivatives of solutions to 3D compressible MHD equations with large initial data, improving previous decay estimates for these derivatives.

Contribution

It provides the first optimal decay estimate for the highest-order derivatives of solutions to 3D compressible MHD equations.

Findings

Decay rate of nabla^2 (sigma-1,u,M) in L^2 norm is (1+t)^{-7/4}

Improves previous decay estimates for first derivatives

Confirms optimality of the decay rate for high-order derivatives

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