REGULARITY RESULTS FOR 2-DIMENSIONAL FLOWS OF MULTIPHASE VISCOUS FLUIDS

Global regularity results for weak solutions of the Navier-Stokes equa tions for two-dimensional multiphase incompressible fluids are proved under suitable conditions on the viscosity without assuming positive l ower bounds on the initial density. As an application, we deduce regul arity properties for the integral curves of the corresponding velocity field. Finally, we prove regularity results ''in the small'' for stro ng solutions.