Perspectives in nonlinear diffusion: between analysis, physics and geometry

• Autores: Juan Luis Vázquez
• Localización: Proceedings oh the International Congress of Mathematicians: Madrid, August 22-30,2006 : invited lectures / coord. por Marta Sanz Solé, Javier Soria de Diego, Juan Luis Varona Malumbres, Joan Verdera, Vol. 1, 2006, ISBN 978-3-03719-022-7, págs. 609-634
• Idioma: inglés
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• Resumen

  • We review some topics in the mathematical theory of nonlinear diffusion. Attention is focused on the porous medium equation and the fast diffusion equation, including logarithmic diffusion. Special features are the existence of free boundaries, the limited regularity of the solutions and the peculiar asymptotic laws for porous medium flows, while for fast diffusions we find the phenomena of finite-time extinction, delayed regularization, nonuniqueness and instantaneous extinction. Logarithmic diffusion with its strong geometrical flavor is also discussed.

    Connections with functional analysis, semigroup theory, physics of continuous media, probability and differential geometry are underlined.