Nonlinear stability of a background magnetic field for the 3D compressible MHD equations with anisotropic dissipation

We study the nonlinear stability of an equilibrium with a background magnetic field for the three-dimensional compressible magnetohydrodynamic (MHD) equations in the whole space $\mathbb{R}^3$, in the strongly anisotropic regime where the velocity is dissipated only in the horizontal directions and the magnetic field is diffused in a single direction.

We prove that for initial data sufficiently close to the equilibrium in a Sobolev space, the system admits a unique global-in-time solution that remains close to the equilibrium and enjoys quantitative dissipation estimates.

The proof overcomes the severe lack of dissipation through two mechanisms: the background magnetic field is shown to generate enhanced dissipation for the magnetic field and the density, while a nonlinear cancellation mechanism is devised to resolve the loss of vertical derivatives caused by the compressible coupling.