Lezioni LEONARDESCHE
Martedi’ 7 giugno 2011, ore 16.30, in Aula Chisini, via Saldini 50,
il Professore JUAN LUIS VAZQUEZ Universidad Autonoma de Madrid
parlera’ su
NONLINEAR DIFFUSION, FROM POROUS MEDIA TO FRACTIONAL DIFFUSION. EVOLUTIONS, FREE BOUNDARIES AND ASYMPTOTICS
Abstract: In the talk we will make a presentation of the theory of Nonlinear Diffusion centered on one of the popular models, the porous medium equation and its close relative, the fast diffusion equation. The existence of free boundaries is one of the most peculiar properties of the former equation. Related models and literature will be mentioned. In the final section, I will present recent work that combines degenerate nonlinear diffusion with nonlocal operators of fractional Laplacian type. Apart from the unexpected existence of free boundaries, the model admits mass preserving self-similar solutions that are found by solving an elliptic fractional-Laplacian obstacle problem. We use entropy methods to show that the asymptotic behaviour is described after renormalization by these self-similar solutions. This is intended to give an idea of the variety of topics and techniques in the area.
ABOUT THE SPEAKER: Juan Luis Vazquez is a leading expert in the fields of nonlinear partial differential equations and their applications. He is well known for his numerous contributions such as strong maximum principles and blow up analysis for non linear elliptic and parabolic problems and the analysis of diffusion processes such as those and their applications, especially for porous media. Vazquez has recieved the Spanish National Prize for Research in Mathematics in 2003, has delived a plenary address at the 2006 International congress of Mathematicians, and has served as President of the Spanish Society for Applied Mathematics. Vazquez is currently Full Professor of Mathematics at Universidad Autonoma de Madrid in Spain.