Variational inequalities in magneto-hydrodynamics

We study subdifferential initial boundary-value problems for the magneto-hydrodynamics (MHD) equations of a viscous incompressible liquid. We construct a solvability theory for an abstract evolution inequality in Hilbert space for operators with quadratic nonlinearity. The results obtained are applied to the study of MHD flows. For three-dimensional flows, we prove the existence of weak solutions of variational inequalities “globally” with respect to time, while, for two-dimensional flows, we establish the existence and uniqueness of strong solutions.