Global Strong solution with vacuum to the 2D nonhomogeneous incompressible MHD system

In this paper, we first prove the unique global strong solution with vacuum to the two dimensional nonhomogeneous incompressible MHD system, as long as the initial data satisfies some compatibility condition. As a corollary, the global existence of strong solution with vacuum to the 2D nonhomogeneous incompressible Navier-Stokes equations is also established. Our main result improves all the previous results where the initial density need to be strictly positive. The key idea is to use some critical Sobolev inequality of logarithmic type, which is originally due to Brezis-Wainger.Comment: 16 page

Global Strong Solutions to the incompressible Magnetohydrodynamic Equations with Density-Dependent Viscosity and Vacuum in 3D Exterior Domains

The nonhomogeneous incompressible Magnetohydrodynamic Equations with density-dependent viscosity is studied in three-dimensional (3D) exterior domains with slip boundary conditions. The key is the constraint of an additional initial value condition $B_0\in L^p (1\leqslant p<12/7)$, which increase decay-in-time rates of the solutions, thus we obtain the global existence of strong solutions provided the gradient of the initial velocity and initial magnetic field is suitably small. In particular, the initial density is allowed to contain vacuum states and large oscillations. Moreover, the large-time behavior of the solution is also shown.Comment: arXiv admin note: text overlap with arXiv:2205.05925, arXiv:1709.05608, arXiv:1506.03884, arXiv:2112.08111 by other author