The stabilizing effect of magnetic field on the two-dimensional partial dissipation MHD equations with large initial data

This paper investigates the stability of perturbations around a prescribed background magnetic field for the two-dimensional incompressible MHD equations with vertical velocity dissipation and horizontal magnetic diffusion. The absence of horizontal dissipation in the Navier–Stokes equations remains a significant open mathematical problem, in particular regarding the construction of global solutions in the whole space R². From a mathematical perspective, we demonstrate that the background magnetic field effectively enhances horizontal dissipation, thereby establishing the global-in-time stability of strong solutions to the MHD system with general initial data. Remarkably, both the initial velocity and magnetic field are allowed to be arbitrarily large in suitable Sobolev spaces. This significantly extends the result of [H. Lin, R. Ji, J. Wu, L. Yan, J. Funct. Anal., 108519, 2020], where global stability result was proved only for small initial perturbations, and the result of [C. Cao, J. Wu, Adv. Math., 1803–1822, 2011], where the corresponding norms grow exponentially in time. Furthermore, we also characterize the large-time behavior of the solutions.