Global low-energy weak solutions of the equations of three-dimensional compressible magnetohydrodynamics

We prove the global-in-time existence of weak solutions of the equations of compressible magnetohydrodynamics in three space dimensions with initial data small in L² and initial density positive and essentially bounded. A great deal of information concerning partial regularity is obtained: velocity, vorticity, and magnetic field become relatively smooth in positive time (H¹ but not H²) and singularities in the pressure cancel those in a certain multiple of the divergence of the velocity, thus giving concrete expression to conclusions obtained formally from the Rankine–Hugoniot conditions. Copyright © 2012 Springer-Verlag.