Global well-posedness for the three-dimensional full compressible viscous non-resistive MHD system

Abstract: In this paper, we consider the three-dimensional full compressible viscous non-resistive MHD system. Global well-posedness is proved for an initial-boundary value problem around a strong background magnetic field. It is also shown that the unique solution converges to the steady state at an almost exponential rate as time tends to infinity. The proof is based on the celebrated two-tier energy method, due to Guo and Tice [\emph{Arch. Ration. Mech. Anal.}, 207 (2013), pp. 459--531; \emph{Anal. PDE.}, 6 (2013), pp. 287--369], reformulated in Lagrangian coordinates. The obtained result may be viewed as an extension of Tan and Wang [\emph{SIAM J. Math. Anal.}, 50 (2018), pp. 1432--1470] to the context of heat-conductive fluids. This in particular verifies the stabilization effects of vertical magnetic field in the full compressible non-resistive fluids.