Global well‐posedness of 3D incompressible inhomogeneous magnetohydrodynamic equations

Abstract

In this paper, we investigate the 3D inhomogeneous incompressible magnetohydrodynamic (MHD) system. By the assumption of the smallness of initial velocity and magnetic fluids in the critical Besov space, the local and global well‐posedness of 3D inhomogeneous incompressible equations is obtained. It improves some previous results of MHD equations by generalizing the range of exponent p$$ p $$ in Besov spaces trueB˙p,13false/p−1$$ {\dot{B}}_{p,1}&#x0005E;{3/p-1} $$ with 1<p<6$$ 1&lt;p&lt;6 $$. Besides, the initial density belongs to the critical Besov space Bq,13false/q$$ {B}_{q,1}&#x0005E;{3/q} $$ with 1<q<6$$ 1&lt;q&lt;6 $$, and it is removed the additional restriction of 1<q≤p$$ 1&lt;q\le p $$, which is an important condition in some previous results for both 3D inhomogeneous incompressible Navier–Stokes equations and MHD system.