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The porous medium equation is a well-known nonlinear diffusion model describing the evolution of the density of a gas flowing through a porous medium. However, it also appears in many other situations, for instance as a simplified model of a tumor growth, whenever the diffusion coefficient degenerates as the density becomes zero. An important feature of this partial differential equation is the development of a free boundary that propagates with a finite speed. The pressure of the gas is assumed to be proportional to some power of the density. In the stiff-pressure limit, that is, the limit of an incompressible gas, the pressure converges to a solution of a related model, the Hele-Shaw problem. The limit pressure is harmonic in its positive set, and the boundary of this set moves with a normal velocity that is proportional to the normal derivative of the pressure. This can be easily seen formally, but the rigorous justification is rather delicate. In particular, an initial layer develops if the initial data is not matched to the Hele-Shaw problem. Furthermore, the initial density distribution influences the limit solution by increasing the speed of the free boundary in a nontrivial way. In this talk, I will discuss two approaches to establishing the asymptotic limit in a model of tumor growth: a variational approach, which is based on the recent work of Mellet, Perthame, Quirós and Vázquez, and a viscosity solutions approach, which we have developed in a joint work with Inwon Kim (UCLA). |