The dot B_{p,1}^0 norm of global solutions to compressible MHD decays at the optimal rate t to the power of minus N/2 times (1/2 minus 1/p) minus sigma_1 over 2, when low frequencies lie in dot B_{2,infty}^{-sigma_1} for sigma_1 in (1-N/2, 2N/p-N/2].
Hao, Well-posedness to the compressible viscous magnetohy drodynamic system, Nonlinear Anal- ysis: RWA , 12 (2011), 2962–2972
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Optimal decay for the compressible MHD equations in the critical regularity framework
The dot B_{p,1}^0 norm of global solutions to compressible MHD decays at the optimal rate t to the power of minus N/2 times (1/2 minus 1/p) minus sigma_1 over 2, when low frequencies lie in dot B_{2,infty}^{-sigma_1} for sigma_1 in (1-N/2, 2N/p-N/2].