# 2006 DiPerna Lectures

*The 2006 DiPerna Memorial Lecture will be delivered by Alberto Bressan, Penn State, on January 26.*

Department of Mathematics, University of California, Berkeley, presents

The DiPerna Memorial Lecture 2006

**Alberto Bressan**

Penn State

Thursday, January 26, 4 p.m.

60 Evans

**Solutions to hyperbolic conservation laws and their approximations**

The talk will present a survey of basic techniques and recent results in the theory of hyperbolic systems of conservation laws.

Even for smooth initial data, solutions to nonlinear conservation laws can develop shocks in finite time. This lack of regularity prevents the use of many classical tools of analysis. Two main approaches are currently available. One due to Glimm, based on the control of the total variation by means of wave-interaction funtionals, and one introduced by Di Perna, relying on weak convergence and compensated compactness arguments.

Recent analysis has shown that front tracking, vanishing viscosity and semidiscrete approximations preserve a uniform bound on the total variation of the solutions. All these approximations converge to a unique limit, depending continuously on the initial data in the L

^{1}norm.

On the other hand, fully discrete numerical schemes can generate an arbitrary large amount of oscillations. For the 2 x 2 system of isentropic gas dynamics, convergence has been established by a compensated compactness method. However, for general hyperbolic systems, the convergence of these numerical schemes remains an open problem.

*Refreshments will be served in 1015 Evans, 5:15-6:00pm*