Decay-in-time of the highest-order derivatives of solutions for the compressible isentropic MHD equations

Abstract

In this paper, we consider the large time behavior of the Cauchy problem for the three-dimensional isentropic compressible magnetohydrodynamic (MHD) equations. The global existence of smooth solutions for the 3D compressible MHD equations has been proved by Chen–Tan [6], under the condition that the initial data are close to the constant equilibrium state in the Sobolev space H3. However, to our best knowledge, the decay estimate of the highest-order spatial derivatives of the solution to the compressible MHD equations has not been solved. The main goal in this paper is to give a positive answer to this problem. Exactly, under the assumption that the initial perturbation is small in Hl(R3)∩B˙2,∞−s(R3) with l⩾3, s∈[0,52], combining the spectral analysis on the semigroup generated by the linear system at the constant state and the energy method to the compressible MHD equations, then we get the optimal convergence rates of any order spatial derivatives (including the highest-order derivatives) of the solution. This result concern with the optimal time decay rates of the solutions, which extends the work obtained by Chen [5] for the compressible Navier–Stokes equations in R3 to the 3D compressible MHD equations. Moreover, we have expanded the range of s from (0,3/2] to [0,5/2], compared with the previous results on optimal decay rates of global strong solutions for the 3D isentropic compressible MHD system.