Series: Mathematics Colloquium

Date: Thursday, April 10, 2003

Time: 4:30 - 5:30 PM

Place: 102 McAllister Building

Host: Yuxi Zheng

Refreshments: 4:00 - 4:30 PM, in 212 McAllister

Speaker: Alberto Bressan, SISSA, Trieste, Italy

Title: Recent Trends in the Theory of Conservation Laws

Abstract:

The talk will provide a survey of recent results and open problems in
the theory of hyperbolic systems of conservation laws.  Until
recently, most of the analysis of weak solutions was based on accurate
estimates on the "Riemann problem", where the initial data is
piecewise constant.  The vanishing viscosity approach now provides a
new perspective, based on the local decomposition of a viscous
solution in terms of viscous travelling waves.  This yields the
well-posedness of the Cauchy problem for a general class of hyperbolic
systems in one space dimension, with initial data having small total
variation.  A number of open problems will be discussed, including
finite time blow up for large initial data, existence of weak
solutions in several space dimensions and applications to differential
games.