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

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 H2$$ {H}^2 $$‐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 L1$$ {L}^1 $$‐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.