Theoretical and numerical aspects for nonlocal equations of porous medium type
- Autores: Félix del Teso Méndez
- Directores de la Tesis: Juan Luis Vázquez (dir. tes.)
- Lectura: En la Universidad Autónoma de Madrid ( España ) en 2015
- Idioma: inglés
- Tribunal Calificador de la Tesis: Fernando Quirós Gracián (presid.) , Carlos Escudero Liébana (secret.) , Gabriele Grillo (voc.) , Espen Robstad Jakobsen (voc.) , Bruno Volzone (voc.)
- Enlaces
- Tesis en acceso abierto en: Biblos-e Archivo
- Resumen
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.
