Optimal Decay Rates of Classical Solutions for the Full Compressible MHD Equations

TL;DR

This paper establishes optimal decay rates for higher order derivatives of solutions to the full compressible MHD equations in three dimensions, improving previous results by applying Fourier splitting methods under small initial perturbations.

Contribution

It provides the first optimal decay rate results for higher order derivatives of classical solutions to the full compressible MHD equations in 3D, using Fourier splitting techniques.

Findings

Optimal decay rates for second order spatial derivatives of solutions.

Optimal decay rates for third order spatial derivatives of magnetic field.

Improved decay estimates over previous work by Pu and Guo.

Abstract

In this paper, we are concerned with optimal decay rates for higher order spatial derivatives of classical solutions to the full compressible MHD equations in three dimensional whole space. If the initial perturbation are small in $H^3$-norm and bounded in $L^q(q\in \left[1, \frac{6}{5}\right))$-norm, we apply the Fourier splitting method by Schonbek[Arch. Rational Mech. Anal. 88 (1985)] to establish optimal decay rates for the second order spatial derivatives of solutions and the third order spatial derivatives of magnetic field in $L^2$-norm. These results improve the work of Pu and Guo [Z. Angew. Math. Phys. 64 (2013) 519-538].