Optimal regularity for phase transition problems with convection
In this paper, the author considers a stationary phase transition problem where the liquid phase is in motion. For given convection \({\mathbf v}\), the problem consists of determining the temperature \(T\) from the equation \[ \Delta T =\mathrm{div}\left({\mathbf v}\beta(T-T_S)\right)+f, \] where \(\beta(s)=as+lH(s)\) is the enthalpy, \(a\) is the specific heat constant, \(l\) is the latent heat constant, \(H\) is the Heaviside function, \(T_S\) is the solidification temperature and \(f\) is a function that accounts for heat sources or sinks. This is also known as Stefan problem with convection and portrays various phase transition models. The main results of this paper state that weak solutions of this problem are locally Lipschitz continuous, provided that \({\mathbf v}=D\xi\), where \(\xi\) is a harmonic function, and the Lipschitz free boundary is \(C^1\) regular when \({\mathbf v}\) is a constant vector (that is, in the case of continuous casting).
scientific article; zbMATH DE number 6476997
scientific article