一小时邀请报告简介之十：扩散

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一小时邀请报告简介之十：扩散
（转自2006年国际数学家大会官方网站）

Plenary Session: Juan Luis Vázquez
Difusión with Limits
Heat propagation, contamination, the nuclear mushroom, population studies, stellar dynamics… All these topics arose during a conversation about nonlinear diffusion, the subject being studied by the mathematician Juan Luis Vázquez, who will give one of the plenary lectures at the 2006 International Congress of Mathematicians in Madrid.  According to Vázquez, the common denominator of his studies is “the mathematics of entities that diffuse in groups (in heaps, says Vázquez); they occupy a zone in space and have a certain density”. This accounts for their application to particles, fluids and populations; he is looking for the answers to questions such these: Where are they? How many are there? Where are they going? Do they tend to stabilize?
The model that immediately comes to mind is the one used on probability and equation courses illustrating how a substance that diffuses in time eventually assumes a bell-shaped distribution, a phenomenon known as the Gauss bell, with tails that tend to the infinite. This is a classical pattern in mathematics and one that is followed by many models in nature, although it by no means occurs on every occasion.
Specifically, it is not fulfilled in ‘nonlinear diffusion phenomena with limited support”, which are those whose pattern of diffusion has a bound or limit. This Spanish mathematician will base his lecture on these types of phenomena. For example, if a viscous substance is poured onto a table and allowed to spread, its distribution will be propagated like a miniature mountain with finite support, with tails that tend to the infinite. Neither is the Gaussian model adequate for the cloud that forms after a nuclear explosion, which has marked boundary and does not propagate infinitely. The same can be said for the distribution of pollutants in the ground, a problem related to the so-called “equations in porous media”, which constitute one of Vázquez’s most notable scientific contributions.  
Despite the many applications deriving from his work, Juan Luis Vázquez points out that, “We are concerned with mathematics that are directly inspired by problems of fluid mechanics, stellar dynamics, population studies, chemical reactions, etc.. But we’re really specialists in the concepts and calculations lying behind these subjects; nonlinear differential equations, functional spaces, the density of mass or energy, measurements and probabilities, and asymptotic methods”.
While this research involves a high degree of abstraction and appears to have no immediate practical interest, and while the results obtained are not aimed at solving concrete problems in industry, they nevertheless provide a theoretical framework for a great variety of scientific problems. It involves painstaking work which can mean concentrating on the same integral throughout the course of a whole year. Despite this exclusive dedication to one particular element, the field is concerned neither with a specific equation nor functional analysis; as indicated by the title of his lecture – “Nonlinear Diffusion, from Analysis to Physics and Geometry” - Juan Luis Vázquez’s work is multidisciplinary. The lecture will cover subjects such as functional analysis, probability, physics, and geometry.
Juan Luis Vázquez was born in Oviedo (Spain) in 1946. Although from 1964 to 1969 he studied Telecommunications Engineering, he graduated in Mathematics at the Complutense Univesity of Madrid in 1973, where he also obtained his doctorate in 1979. Since 1986 he has been a professor of mathematics at the Autonomous University of Madrid. He was chairperson of the Spanish Society of Applied Mathematics from 1996 to 1998 and participated in the organization of the World Year of Mathematics in 2000. He recently received recognition as a “Highly Cited Scientist” from the ISI in 2003, and in the same year was awarded the Julio Rey Pastor National Prize for Mathematical Research.
Speaker: Juan Luis Vázquez
Title: Nonlinear Diffusion, from Analysis to Physics and Geometry
Date: Wednesday, August 30th: 9:00-10:00
ICM2006 Scientific Programme: http://www.icm2006.org/scientificprogram/plenarylectures/
（转自2006年国际数学家大会官方网站）

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