Nonlocal approximation of nonlinear diffusion equations
Nonlinear diffusion equations are ubiquitous in several real world applications. They were introduced to analyse gas expansion in a porous medium, groundwater infiltration, and heat conduction in plasmas, to name a few applications in physics. In this talk, I will present recent joint work with José A. Carrillo and Antonio Esposito concerning a nonlocal approximation inspired by the theory of gradient flows for a general family of equations closely related to the porous medium equation with m>1. Our approximation is motivated by recent ideas to use (nonlocal) interaction equations to approximate (local) diffusion equations. We prove under very general assumptions that weak solutions to our nonlocal approximation converge to weak solutions of the original local equation. One byproduct of our analysis is the development of a deterministic particle method for numerically approximating solutions to nonlinear diffusion equations.