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In this paper, we are concerned with the compressible viscous magnetohydrodynamic system (MHD) and investigate the large time behavior of strong solutions near constant equilibrium (away from vacuum). In the eighties, Umeda, Kawashima & Shizuta [35] initiated the dissipative mechanism for a rather general class of symmetric hyperbolic-parabolic systems, which isHere λ = λ (iξ) is the characteristic root of linearized equations. From the point of view of dissipativity, Kawashima [25] in his doctoral dissertation established the optimal time-decay estimates of L q -L 2 (q ∈ [1, 2)) type for solutions to the MHD system. Here, by using Fourier analysis techniques, we shall improve Kawashima's efforts in [25] and give 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 L p norm (the slightly strongerḂ 0 p,1 norm in fact) of global solutions with the critical regularity, decays like tfor t → ∞. In particular, taking p = 2 and d = 3 goes back to the classical time decay t − 3 4 . We derive new estimates which are used to deal with the strong coupling between the magnetic field and fluid dynamics. 1991 Mathematics Subject Classification. 76W15, 35Q30, 35L65, 35K65.

The present work is dedicated to the well‐posedness issue of strong solutions (away from vacuum) to the compressible viscous magnetohydrodynamic (MHD) system in Rd (d  ≥  2). We aim at extending those results in previous studies to more general Lp critical framework. Precisely, by recasting the whole system in Lagrangian coordinates, we prove the local existence and uniqueness of solutions by means of Banach fixed‐point theorem. Furthermore, with the aid of effective velocity, we employ the energy argument to establish global a priori estimates, which lead to the unique global solution near constant equilibrium. Our results hold in case of small data but large highly oscillating initial velocity and magnetic field.

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