ON THE STATIONARY BOUSSINESQ-STEFAN PROBLEM WITH CONSTITUTIVE POWER-LAWS

We discuss the existence of weak solutions to a steady-state coupled s ystem between a two-phase Stefan problem, with convection and non-Four ier heat diffusion, and an elliptic variational inequality traducing t he non-Newtonian flow only in the liquid phase. In the Stefan problem for the p-Laplacian equation the main restriction comes from the requi rement that the liquid zone is at least an open subset, a fact that le ads us to search for a continuous temperature field. Through the heat convection coupling term, this depends on the q-integrability of the v elocity gradient and the imbedding theorems of Sobolev. We show that t he appropriate condition for the continuity to hold, combining these t wo powers, is pq > n. This remarkably simple condition, together with q > 3n/(n + 2), that assures the compactness of the convection term, i s sufficient to obtain weak solvability results for the interesting sp ace dimension cases n = 2 and n = 3. (C) 1997 Elsevier Science Ltd.