The Vanishing Viscosity Method in Two-Phase Stefan Problems with Nonlinear Flux Conditions

Abstract

This chapter discusses the vanishing viscosity method in two-phase Stefan problems with nonlinear flux conditions. Two-phase Stefan problems with non-standard conditions imposed on the fixed boundary often create substantial difficulties at their analysis. Of such problems, those with maximal monotone (multivalued) operators in boundary conditions are especially interesting because of their relevance in continuum physics. In particular, problems with time-dependent unilateral conditions of Signorini type fall into this category. After all the uniqueness of solutions are quite difficult for this class of problems and require, as a rule, the use of rather advanced tools. Neither classical nor standard variational approaches turn out fruitful. One of the possible ways of solving the problems is based on introducing the so-called viscosity solutions and discussing their behavior as viscosity parameter vanishes.