Stefan-type problems consist in finding two functions u (the “temperature”) and w (the “enthalpy”) satisfying \(w_ t+div(- k(u)\nabla u)=0\) and \(w\in \int^{u}_{u_ 0}c(\eta)d\eta +LH(u-u_ 0)\) where k and c are the conductivity and the specific heat and \(u_ 0\) and L are the melting point and the latent heat. The differential equation is to be satisfied in a suitable weak sense and the problem is completed by prescribing initial values for w and boundary conditions usually involving u and/or the thermal flux -k(u)\(\nabla u.\)
When convection with a prescribed velocity field \b{v} is taken into account the flux is \b{v}w-k(u)\(\nabla u\). If the problem is studied in a domain \(\Omega \subset {\mathbb{R}}^ n\), it is necessary to identify \(B\subset \partial \Omega\) such that the convection field is inwards at points of B. (If div \b{v} is assumed non negative (as in the paper) \(B\subset \subset \partial \Omega).\)
Of course, prescribing the temperature at points of B may not be sufficient to solve the problem uniquely: if we just say that material enters \(\Omega\) at temperature \(u_ 0\), this does not specify the energy exchange with the surroundings.
Existence and uniqueness are proved for the solution of the corresponding Cauchy problem for the abstract evolution equation. Then it is proved that this solution is a weak solution in the usual \(L^ 2\) sense. Since the weak solution is unique, its notion is appropriate for the problem.