The Cauchy problem for the non-isentropic 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 non-isentropic compressible
magnetohydrodynamic (MHD) system. First, motivated by Pu–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 data
should be bounded in L1-norm and is small in
H2-norm. Finally, based on Fourier theory and the idea
of cancellation 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 non-isentropic MHD
fluids. Our result is the first one concerning with the optimal decay
estimates of the highest-order derivatives of the non-isentropic MHD
system.