On the Classical Solvability of the Stefan Problem in a Viscous Incompressible Fluid Flow

This paper is devoted to the study of a solidification/melting process in the case where the fluid is flowing. Such a phenomenon is described by the Stefan problem with transport terms in the equation of the temperature distribution and the initial-boundary value problem for the incompressible Navier--Stokes equations. The existence of the classical solution is proved locally in time.