The phase-field transition system with non-homogeneous Cauchy-Stefan-Boltzmann and homogeneous Neumann boundary conditions and non-constant thermal conductivity (Q393208)

In this article, the author studies the phase-field transition system with applications in metallurgy. The model consists of a nonlinear parabolic system with non-homogeneous and nonlinear Cauchy-Stefan-Boltzmann conditions and homogeneous Neumann boundary conditions. The unknowns are the reduced temperature distribution and the phase function. The main results of the paper are the existence, the regularity, and the uniqueness of the solution. Basic techniques rely on the Leray-Schauder degree theory, the standard theory of linear and quasi-linear parabolic equations, as well as the Lions-Peetre embeddings. Also, necessary a priori estimates are obtained by choosing appropriate test-functions in the weak formulation. The author claims that, in comparison with other similar models, the one presented is able to describe the real physical phenomena, for example the phenomenon of phase-separation.