The Cauchy problem for the nonisentropic compressible MHD fluids: Optimal time‐decay rates

Abstract

This paper is concerned with the time‐decay rates of the strong solutions of the three‐dimensional nonisentropic compressible magnetohydrodynamic (MHD) system. First, motivated by Pu and Guo's result [Z. Angew. Math. Phys. 64 (2013) 519–538], we establish the existence result of a unique local‐in‐time strong solution for the MHD system. Then, we derive a priori estimates and use the continuity argument to obtain the global‐in‐time solution, where the initial perturbation is small in ‐norm. Finally, based on Fourier theory and the idea of cancelation of a low‐medium frequent part as in [Sci. China Math. 65 (2022) 1199–1228], we get the optimal time‐decay rates (including highest‐order derivatives) of strong solutions for nonisentropic MHD fluids when the boundedness of ‐norm of the initial perturbation is required. Our result is the first one concerning with the optimal decay estimates of the highest‐order derivatives of the nonisentropic MHD system.