Weak Solutions to Stefan Problems with Prescribed Convection
Jim Rulla · SIAM Journal on Mathematical Analysis · 1987
This paper deals with the equation \[ w_t + {\operatorname{div}}({\bf v}w - abla \alpha (w)) = g,\] which is to hold in a smooth, bounded domain $G \subset \mathbb{R}^n $. The function $\alpha :\mathbb{R} \to \mathbb{R}$ is uniformly Lipschitz and ondecreasing, but may be identically zero. For certain smooth functions ${\bf v}:\bar G \to \mathbb{R}^n $ satisfying ${\operatorname{div}}({\bf v}) \geqq 0$, there are integral solutions to the Cauchy problem associated with this equation, provided the boundary conditions on w are chosen appropriately. We prove that these integral solutions are weak solutions in the usual sense. Moreover, these weak solutions are unique; hence the notion of a weak solution is adequate for problems of this type.