Summer Program

Nonlinear Conservation Laws and Applications

Description:

For hyperbolic systems of conservation laws in one space dimension, the method of local decomposition of solutions along traveling wave profiles has recently paved the way to understanding the convergence of various types of approximations: vanishing viscosity, relaxations, semi-discrete schemes. On the other hand, other approximations, such as physical viscosity, or fully discrete numerical schemes, are still poorly understood. The global existence or the finite time blow-up of solutions with large BV data is another major problem for investigation.

The theoretical analysis of hyperbolic conservation laws in several space dimensions remains a grand challenge. In the past few years, new techniques have been introduced, resulting in specific advances. Refined measure-theoretical tools have been developed for the study of scalar conservation laws, and for transport equation with rough coefficients. Intriguing counterexamples have been constructed by means of a newly developed Baire category technique. Moreover, major breakthroughs have recently been achieved in the understanding of shock reflection past a wedge in the equation of gas dynamics, and in the existence theory for global weak solutions with large data to the compressible Navier-Stokes equations in both the isentropic and non-isentropic cases.

Progress in theoretical understanding has been paralleled by an expansion in the applications of hyperbolic conservation laws. Traditional areas of applications in mathematical physics, such as fluid dynamics, magneto-hydrodynamics, nonlinear elasticity, combustion models, oil recovery, etc, have experienced sustained growth. In addition, new directions are emerging: continuum models based on conservation laws are increasingly used in the analysis of blood flow and of cell motion, in the modelling of traffic flow and of large scale supply-chains in economic and industrial applications. A novel aspect of these models is that the flow is not only studied on a single road, or pipeline, but on a network. These network models apply, in particular, also to the flow of information packets through the Internet.

The first week of the program will be largely devoted to tutorial sessions and general survey lectures. These will be targeted at young researchers and Ph.D. students who know only some basic facts on the subject and want to get familiar with the main lines of current research.

The following two weeks will contain more specialized research talks with one or two themes each day. These talks will be intermixed with discussions, pointing out the most promising areas for further research. We plan to overlap themes in several areas to stimulate interaction between different groups. The second week will focus more on the theoretical aspects, and the third week on applications and numerical methods.

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