The 2D magnetohydrodynamic system with Euler-like velocity equation and partial magnetic diffusion

When the velocity equation of the incompressible 2D magnetohydrodynamic (MHD) system is inviscid, the global well-posedness and stability problem in the whole space ${\mathbb{R}}^2$ case remains an extremely challenging open problem. Broadman, Lin, and Wu (SIAM J. Math. Anal. 52(5) (2020): 5001-5035) were able to establish the global well-posedness and stability near a background magnetic field when there is damping in one velocity component. Their work exploited the stabilizing effect of the background magnetic field. This paper presents new progress. We are able to prove the global well-posedness and stability even when the magnetic diffusion is degenerate and only in the vertical direction. The velocity equation is still inviscid and has damping only in the vertical component. The proof of this new result overcomes two main difficulties, the potential rapid growth of the velocity due to the lack of dissipation or horizontal damping and the control of nonlinearity associated with the magnetic field. By discovering the key hidden smoothing effects and incorporating them in the construction of a two-layered energy function, we are able to obtain uniform bounds on the solution in the H³-norm when the initial perturbation is near the background magnetic field. In addition, we prove that certain Lebesgue and Sobolev norms of the solution approach zero as time approaches infinity.