A remark on the time-decay estimates for the compressible magnetohydrodynamic system. Issue 11 (18th August 2021) - Shi, Weixuan; Zhang, Jianzhong
ABSTRACT: The present paper is dedicated to the optimal time-decay estimates of global strong solutions near constant equilibrium (away from vacuum) to the compressible magnetohydrodynamic (MHD) equations in the L p critical Besov spaces. In which we claim a new low-frequency assumption that plays a key role in the large-time behavior of solutions. Precisely, we exhibit that if the low frequencies of initial data belong to some Besov space B ˙ 2, ∞ − s 1 with s 1 ∈ ( l 1 − d 2, s 0 ] ( s 0 ≜ 2 d p − d 2 ), then the L p norm (the slightly stronger B ˙ p, 1 0 norm in fact) of strong solutions has the optimal decay t − s 1 2 − d 2 ( 1 2 − 1 p ) ( t − d 2 p if s 1 = s 0 ) for t → ∞, which improve the results of [Shi WX, Xu J. Large-time behavior of strong solutions to the compressible magnetohydrodynamic system in the critical framework. J Hyperbol Differ Equ. 2018;15:259–290]. The proof mainly depends on a sharp time-weighted energy estimates in light of low and high frequencies for the solutions. As a by-product, those optimal time-decay rates of B ˙ 2, ∞ − s 1 - L r type are also captured in the critical framework.