Considering the one-phase Stefan problem, we present an account of some recent mathematical results within the framework of variational inequalities. We discuss several situations corresponding to different boundary conditions and different geometries, like the exterior problem, the continuous casting model, and the degenerate case of the quasi-steady model. We develop a few continuous-dependen...

We prove strong convergence to singular limits for a linearized fully inhomogeneous Stefan problem subject to surface tension and kinetic undercooling effects. Different combinations of σ → σ0 and δ → δ0, where σ, σ0 ≥ 0 and δ, δ0 ≥ 0 denote surface tension and kinetic undercooling coefficients respectively, altogether lead to five different types of singular limits. Their strong convergence is...

We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mus...

Temperature and phase evolution in phase transitions are represented here by coupling the energy balance equation with a multivalued constitutive relation between the density of internal energy and the temperature, and with a nonlinear conduction law. This doubly nonlinear problem generalizes the classical Stefan model. Existence of a weak solution is proved via time discretization, a priori es...

The exact solution is found for the Stefan problem of inwards freezing of a nonhomogeneous cylinder whose specific heat and latent heat depend upon the inverse square of the radial distance. The freezing time for the corresponding annular cylinder is found exactly. The validity of the pseudo-steady-state approximation for the solvable cylinder problem is verified explicitly. It is argued that t...

A simple one-dimensional model for crystal dissolution and precipitation is presented. The model equations resemble a one-phase Stefan problem and involve nonlinear and multi-valued exchange rates at the free boundary. The original equations are formulated on a variable domain. By transforming the model to a fixed domain and applying a regularization, we prove the existence and uniqueness of a ...