In this work we study the inhomogeneous porous medium equation (PME). In particular, we introduce a deterministic particle method that approximates the PME in chapter 2 and how the PME is used to model tumor growth and study the incompressible limit in chapter 3. Particle methods aim to discretize a PDE into a system of ODEs for particles. In the absence of diffusion, particle methods can approximate solutions while maintaining the Wasserstein gradient flow structure. Diffusion smooths things out so that particles do not remain particles. This issue can be dealt with via regularization of the diffusion term. Much has been developed for convex and semi-convex energies. We would like to extend this particle method (blob method) to an energy with more general convexity (omega-convexity) containing nonlinear porous medium diffusion (m=2). We connect this pde to chemotaxis via a Keller-Segel model for the Newtonian kernel and the Bessel kernel. Then, we perform numerical simulations. The porous medium equation can be used to model tumor growth.We study the incompressible limit of an inhomogeneous porous medium equation (PME) with a cell division term that directly depends on space, time, and the pressure. The incompressible limit connects the PME to a Hele-Shaw free boundary problem (FBP). This relation is known as the complementarity condition. We first achieve convergence to the limiting problem along with uniqueness. Then, we gain enough compactness using L^3 bound of the pressure gradient and L^3 AB-estimate to get the complementarity condition. To finish the connection to the FBP, the velocity law to the boundary of the tumor is found. In particular, a novel inhomogeneous velocity of the free boundary is obtained.
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