Global Solutions of Nonlinear Magnetohydrodynamics with Large Initial Data

A free boundary problem for nonlinear magnetohydrodynamics with general large initial data is investigated. The existence, uniqueness, and regularity of global solutions are established with large initial data in H¹. It is shown that neither shock waves nor vacuum and concentration in the solutions are developed in a finite time, although there is a complex interaction between the hydrodynamic and magnetodynamic effects. An existence theorem of global solutions with large discontinuous initial data is also established.