The Stefan problem with mushy regions: Differentiability of the interfaces (Q1342819)

The Stefan problem is one of the most frequently studied free boundary problems. The present paper is concerned with the one-dimensional problem with a heat source: \[ u_t=\varphi (u)_{xx} +f(u), \] where \(u(x,t)\) and \(\varphi (u(x,t))\) denote respectively the enthalpy and temperature of a material which is in the liquid phase if \(\varphi (u)>1\) and in the solid phase if \(\varphi (u)<1\). The set \[ M= \biggl\{(x,t) \mid\varphi \bigl(u(x,t)\bigr) =1\biggr\} \] has a non-empty interior which is referred to as the mushy region. The main aim of the paper is a study of regularity of the solid-mush, mush-liquid and solid-liquid interfaces. The paper is carefully written and will certainly appeal to specialists in free boundary problems.