CONDUCTION-CONVECTION WITH CHANGE OF PHASE E. Dibenedetto M. O’Leary Abstract. We discuss the conduction-convection system with change of phase, and review recent results. We give a generalization of the existence theorem proven in [13]. §1. Introduction. The purpose of this note is to review the results on conduction convec- tion systems and to give some generalizations of the results in [13]. We begin by considering the physics of the problem. Suppose we have a bounded region in R 3 in which we have water and ice. We assume that for every time t ≥ 0 and point x, the liquid and solid phases are determined by the value of the temperature u(x, t) at that point, with thawing and freezing occuring at u = 0. In the ice the temperature is determined by the heat equation, while in the water we have convective motions driven by temperature varia- tions that mix the fluid in an attempt to bring the system to thermal equilibrium. Such motion in a gravitational field is called free convection. The dynamic state of the liquid is determined by the velocity v and the pressure p, and we assume the fluid evolves via free convection. [21] Although we have described a water/ice system, this model applies to other systems of physical interest, for example in solidification of alloys in the continuous casting of metals, and in welding. [5,6,23] Partially supported by NSF grant DMS-9104088 Typeset by A M S-T E X 1