Entropies, geometric structures, and interactions for systems of conservation laws

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The Principal Investigator and his collaborators study hyperbolic conservation laws, a class of partial differential equations that covers many of the fundamental equations in applied math, and in particular in fluid dynamics. The main lines of current research are: (A) a systematic approach to the study of one-dimensional systems of conservation laws based on geometric properties of their eigen-frames; (B) employing vanishing viscosity approximations to obtain interaction estimates for conservation laws in several space dimensions; (C) radially symmetric solutions to systems of conservation laws, and in particular solutions to the compressible Euler system.All of the projects are motivated by the need to provide rigorous statements about the properties of physical models that are used by scientists and engineers. The results assess the range of validity of various physical models and will be of use in delimiting and refining existing models. Particular emphasis is put on understanding solutions of complex, nonlinear equations that are used in simulations of e.g. high speed flight, traffic flow, combustion and detonations.