The Global Solvability of 3-D Inhomogeneous Viscous Incompressible Magnetohydrodynamic Equations with Bounded Density

Global well‐posedness of 3D incompressible inhomogeneous magnetohydrodynamic equations

“…One of an interesting topic in inhomogeneous incompressible fluids is to get the global well‐posedness with weaker smallness conditions on the initial density and velocity with critical regularity, which also attracts much attention for 3D inhomogeneous incompressible MHD equations, such as Chen et al in their work 8 established the global existence and uniqueness of the solution with the initial velocity and magnetic field are sufficiently small in $$$ {H}^s\left({\mathbb{R}}^3\right)\left(1/2<s\le 1\right) $$$. Then, Xu et al in their work 9 reduce the regularities on the initial data stated in Chen et al 8 to the critical framework, so that the result of global existence of solutions to the corresponding systems still holds in critical Besov spaces. However, without any smallness assumption on initial density and the initial data in critical Besov spaces, Ai and Li 10 first proved the global well‐posedness of () with initial velocity and magnetic field being sufficiently small in $$$ {\mathcal{B}}_2^{\frac{1}{2}}\left({\mathbb{R}}^3\right) $$$ and the initial inhomogeneity in $$$ {\mathcal{B}}_2^{\frac{3}{2}}\left({\mathbb{R}}^3\right) $$$.…”