Hai-Liang Li

We consider the Cauchy problem of three-dimensional isentropic compressible magnetohydrodynamic equations in the present paper. For regular initial data with small energy but possibly large oscillations, we prove the global well-posedness of classical solution, where the flow density is allowed to contain vacuum states, and the large-time behavior of the solution is also shown.

We consider the long-time behavior and optimal decay rates of global strong solution to three-dimensional isentropic compressible Navier–Stokes (CNS) system in the present paper. When the regular initial data also belong to some Sobolev space with l⩾4 and s∈[0, 1], we show that the global solution to the CNS system converges to the equilibrium state at a faster decay rate in time. In particular, the density and momentum converge to the equilibrium state at the rates (1 + t)−3/4−s/2 in the L2-norm or (1 + t)−3/2−s/2 in the L∞-norm, respectively, which are shown to be optimal for the CNS system. Copyright © 2010 John Wiley & Sons, Ltd.