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Nonlinear Hyperbolic Waves in Phase Dynamics and Astrophysics

27th January 2003 to 11th July 2003

Organisers: Professor C M Dafermos (Brown), Professor P G LeFloch (CNRS), Professor E F Toro (Trento)

Programme Theme

Quasilinear hyperbolic systems in divergence form, commonly called 'hyperbolic conservation laws', govern a broad spectrum of physical phenomena in compressible fluid dynamics, nonlinear material science, etc. Such equations admit solutions that may exhibit various kinds of shocks and other nonlinear waves (propagating phase boundaries, fluid interfaces, etc.) which play a dominant role in multiple areas of physics; astrophysics, cosmology, dynamics of (solid-solid) material interfaces, multiphase (liquid-vapor) flows, combustion theory, etc.

In recent years, major progress has been made in both theoretical and the numerical aspects of the field, while the number of applications has skyrocketed. The time is ripe for building a bridge between the most recent developments in the general mathematical theory and the areas of applications that have developed most actively in the last few years. Hyperbolic problems in astrophysics and cosmology (relativistic compressible fluid models, the Einstein Field equations of general relativity) are particularly challenging for the applied mathematician; they are essential in order to uncover the structure and formation of the Universe. Hyperbolic problems arising in nonlinear material science (propagating phase boundaries in solids undergoing phase transitions of the austenite-martensite type, for instance) face serious mathematical difficulties (failure of strict hyperbolicity, elliptic regions in phase space); these models are an essential component for understanding the dynamics of shape memory alloys (also called smart materials). To describe the dynamics of phase boundaries, the notion of a kinetic relation was proposed and investigated by material scientists and applied mathematicians in recent years. Recent developments on the theory of systems of conservation laws will be covered; L1 well-posedness, singular limits based on diffusion-dispersion or relaxation, multidimensional hyperbolic problems, degenerate shock wave structure, failure of strict hyperbolicity, etc.

The programme will also have a computational component; numerical schemes for multiphase flows and fluids with non-convex equation of state, exact or approximate Riemann solvers for relativistic and non-relativistic flows, etc. Short-courses, theme weeks and conferences will provide a valuable opportunity for reviewing the state of the art on the physical models, the techniques of mathematical analysis, and the numerical analysis, and for discussing open problems among the participants.

Programme Outline

January to February:
Kinetic Relations for Nonclassical Shocks and Phase Boundaries
February to March:
General Theory of Hyperbolic Conservation Laws
March to April:
Shock Waves having Anomalous Structure and Multi-dimensional Problems
April to May:
Multiphase Flows in Fluids
June to July:
Hyperbolic Problems from Astrophysics and Cosmology.

Final Scientific Report: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons