Global existence and decay estimates of strong solutions for compressible non-isentropic magnetohydrodynamic flows with vacuum

Abstract

We study the Cauchy problem of three-dimensional compressible non-isentropic magnetohydrodynamic (MHD) fluids with both interior and far field vacuum states. Applying delicate energy estimates, initial layer analysis, and continuation theory, we establish the global existence and uniqueness of strong solutions, which may be of possibly large oscillations, provided that the initial data are of small total energy. Furthermore, we also derive algebraic decay estimates of the solution. This improves our previous results (Z. Angew. Math. Phys. 71: Paper No. 188, 2020; J. Differential Equations 336: 456--478, 2022) where viscosity coefficients should additionally satisfy $3\mu>\lambda$. To our best knowledge, this is the first result on time-decay rates of solutions to the Cauchy problem of multi-dimensional compressible non-isentropic MHD equations with vacuum.