Theoretical and numerical aspects for nonlocal equations of porous medium type

Abstract

In this thesis we consider three different models of nonlinear and nonlocal diffusion equations of porous medium type. The prototype is the classical Porous Medium Equation $$ \frac{\partial u}{\partial t} =\Delta u^m, $$ which models the flow of gasses through a porous media. The main topics of the thesis are the following: (1) Finite difference method for the Fractional Porous Medium Equation $u_t + (-\Delta)^s u^m=0$ with $m\geq1$ and $s\in(0,1)$. (2) Finite and infinite speed of propagation for the Porous Medium Equation with Fractional Pressure $u_t=\nabla\cdot(u^{m-1}\nabla (-\Delta)^{-s}u)$ with $m\geq1$ and $s\in (0,1)$. (3) Transformations of self-similar solutions for porous medium equations of fractional type. (4) Uniqueness and properties of distributional solutions of the more general nonlocal porous medium equation $\dell_t u -\mathcal{L}^\mu[\varphi(u)]=0$ \em where $\mathcal{L}^\mu$ is a very general nonlocal operator and the nonlinearity $\varphi$ is continuous and nondecreasing scalar function.