A remark on the time-decay estimates for the compressible magnetohydrodynamic system

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 ${\dot{B}}_{2,\mathrm{\infty }}^{-s_1}$ with $s_1\in (l1-\frac{d}{2},s_0]$ ($s_0\triangleq \frac{2d}{p}-\frac{d}{2}$), then the $L^p$ norm (the slightly stronger ${\dot{B}}_{p,1}^0$ norm in fact) of strong solutions has the optimal decay $t^{-\frac{s_1}{2}-\frac{d}{2}(\frac{1}{2}-\frac{1}{p})}$ ($t^{-\frac{d}{2p}}$ if $s_1=s_0$) for $t\to \mathrm{\infty }$, 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 ${\dot{B}}_{2,\mathrm{\infty }}^{-s_1}$-$L^r$ type are also captured in the critical framework.

Keywords:

• Compressible MHD equations
• time-decay rates
• critical Besov space

2010 MSC:

• 76W05
• 35Q35
• 35L65
• 35K65

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Note that for technical reasons, we need a small overlap between low and high frequencies.