Optimal regularity for phase transition problems with convection

In this paper we consider a steady state phase transition problem with given convection v. We prove, among other things, that the weak solution is locally Lipschitz continuous provided that v=Dξv=Dξ and ξ is a harmonic function. Moreover, for continuous casting problem, i.e. when v is constant vector, we show that Lipschitz free boundaries are C1C1 regular surfaces.