Large-time behavior of strong solutions to the compressible magnetohydrodynamic system in the critical framework

We study the compressible viscous magnetohydrodynamic (MHD) system and investigate the large-time behavior of strong solutions near constant equilibrium (away from vacuum). In the 80s, Umeda et al. considered the dissipative mechanisms for a rather general class of symmetric hyperbolic–parabolic systems, which is given by [Formula: see text] Here, [Formula: see text] denotes the characteristic root of linearized equations. From the point of view of dissipativity, Kawashima in his doctoral dissertation established the optimal time-decay estimates of [Formula: see text]-[Formula: see text]) type for solutions to the MHD system. Now, by using Fourier analysis techniques, we present more precise description for the large-time asymptotic behavior of solutions, not only in extra Lebesgue spaces but also in a full family of Besov norms with the negative regularity index. Precisely, we show that the [Formula: see text] norm (the slightly stronger [Formula: see text] norm in fact) of global solutions with the critical regularity, decays like [Formula: see text] as [Formula: see text]. Our decay results hold in case of large highly oscillating initial velocity and magnetic fields, which improve Kawashima’s classical efforts.