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 L1 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