Global well-posedness and large-time behavior for a special $2\frac{1}{2}$D full compressible viscous non-resistive MHD system

Abstract: In this paper, we consider the full compressible, viscous, non-resistive MHD system under the assumption that the fluids move on a plane while the magnetic field is oriented vertically. Within the framework of Besov spaces, by introducing several new unknown quantities and exploiting the intrinsic structure of the system, we prove the global well-posedness of strong solutions for initial data close to a constant equilibrium state. Furthermore, under some suitable additional conditions involving only the low-frequency part of the initial perturbation, we develop a Lyapunov-type energy argument, which yields the optimal time-decay rates of the global solution. To the best of our knowledge, our result is the first one on global solvability to the full compressible, viscous, non-resistive MHD system in multi-dimensional whole space.