On the Stefan type problems

The melting ice in a cup of water presents a typical example of phase transition of a material. An intermediate state of the material naturally arises at the melting temperature. Some mathematical models regard it as a moving surface between the two phases, while others introduce a mushy regiona mixture of ice and water. The underlying physics consists of the heat diffusion in the two phases and the exchange of energy at the intermediate state. A material can also have several states, for instance, ice-water-vapor, and give arise to an example of multi-phase transition. In fact, at temperature 273.16 kelvins and a partial vapor pressure of 611.657 pascals, ice, water and vapor coexist in a stable equilibrium. Therefore, two interfaces can be observed. Another motivation stems from the petroleum geologythe saturation of two immiscible fluids in a porous medium. In nature, subsurface rocks were initially wet and the pores among them were saturated with water. It is important to understand how the oil in a reservoir eventually filled up these pores that were once occupied by the water. The displacement of the water by the oil is driven by the so-called capillary pressure that exists on the interface of the two immiscible fluids. The capillary pressure increases as the oil saturation increases, and meanwhile the water saturation is forced to decrease. Such a process continues until all water at the center of the pores is displaced, and the only water left is the layer adherent to the rock grains. In such a case, the remaining water becomes immobile, no matter how high the capillary pressure is exerted. This limiting saturation of water is called the connate water saturation. These natural phenomena abide by certain physical laws, which are described by partial differential equations under our study. Our central goal is to establish mathematical evidences supporting that the physical models are sufficiently complete descriptions within their framework. To establish this latter point is the real significance behind the mathematical effort expanded, though they are of intrinsic interest in their own right.