Global Fujita–Kato Solutions of the Incompressible Inhomogeneous Magnetohydrodynamic Equations

We investigate the incompressible inhomogeneous magnetohydrodynamic (MHD) equations in , under the assumptions that the initial density is only bounded, and the initial velocity and magnetic field exhibit critical regularities. In particular, the density is allowed to be piecewise constant with jumps. First, we establish the global‐in‐time well‐posedness and large‐time behavior of solutions to the Cauchy problem in the case that has small variations, and and are sufficiently small in the critical Besov space with . Moreover, the small variation assumption on is no longer required in the case . Then, we construct a unique global Fujita–Kato solution under the weaker condition that and are small in but may be large in . Additionally, we show a general uniqueness result with only bounded and nonnegative density, without assuming the regularity of the velocity.