We investigate the general Porous Medium Equations with drift and source terms that model tumor growth. Incompressible limit of such models has been well-studied, where convergence of the density and pressure variables are established, while it remains unclear whether the free boundaries of the solutions exhibit convergence as well. In this talk, we shall present an affirmative result by showing that the free boundaries converge in the Hausdorff distance in the incompressible limit. It relies on quantifying the relation between the free boundary motion and spatial average of the pressure, and establishing a uniform-in-m strict expansion property of the pressure supports. We derive upper bounds for the Hausdorff dimensions of the free boundaries. This is based on a joint work with Yuming Paul Zhang.

学术海报.pdf