Series: Mathematics Colloquium

Date: Friday, April 4, 2003

Time: 3:30 - 4:30 PM

Place: 102 McAllister Building

Host: Yuxi Zheng

Refreshments: 3:00 - 3:30 PM, in 212 McAllister

Speaker: Wen Shen, SISSA, Trieste, Italy

Title: Systems of Conservation Laws with Relaxation

Abstract:

In this talk I shall give an overview of the recent results on
hyperbolic systems of conservation laws with relaxation terms.
Conservation laws describe physical phenomena where some quantities
are conserved.  They can be used to model traffic flow, gas dynamics,
flow in porous media, etc.  A brief introduction to conservation laws
will be given, showing the difficulties caused by the nonlinearity.
Relaxation problems arise when the conservation equations also contain
stiff source terms.  They appear in several physical models, and also
in numerical schemes for computational purposes. In the scalar case,
under suitable assumptions one can prove the existence of a weak
entropy solution for the relaxation equations.  The solutions have
uniform BV bound, which gives the convergence to a limit by a
compactness argument.  Error estimates can also be proved, both in one
and in several space dimensions.  For systems with relaxation,
however, uniform BV bounds are very difficult to obtain.  At present,
stability and convergence results are known only for the special class
of "Temple systems".  A probabilistic technique for deriving these BV
bounds will be described.