Theoretical and numerical aspects for nonlocal equations of porous medium type
- Juan Luis Vázquez Doktorvater/Doktormutter
Universität der Verteidigung: Universidad Autónoma de Madrid
Fecha de defensa: 11 von Dezember von 2015
- Fernando Quirós Gracián Präsident/in
- Carlos Escudero Liébana Sekretär/in
- Bruno Volzone Vocal
- Gabriele Grillo Vocal
- Espen Robstad Jakobsen Vocal
Art: Dissertation
Zusammenfassung
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.