Hyperbolic Conservation Laws in Continuum Physics

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Proposal #0202888PI: Constantine M. DafermosInstitution: Brown UniversityTitle: Hyperbolic Conservation Laws in Continuum PhysicsABSTRACTThe proposed research program lies on the interface between continuum physics and the theory of hyperbolic systems of conservation laws. It involves the following projects, which are motivated by the recent breakthrough of Bianchini and Bressan on the vanishing-viscosity method: (a) The study of balance laws with dissipative source terms through the vanishing-viscosity method; (b) An attempt to extend the layering method from scalar conservation laws to systems thereof; and (c) The study of the classical Riemann problem with the help of the new estimates of Bianchini and Bressan.Hyperbolic conservation laws belong to a class of nonlinear partial differential equations that govern the dynamics of materials with nonlinear elastic response, including solids like rubber and gases like air. The special feature of these equations is that the nonlinearity induces the development of discontinuities, akin to the familiar phenomenon of the breaking of waves on the beach. The discontinuities then propagate as shock waves. The proposed research program is to investigate a number of methods for constructing solutions with shocks. The objective is to settle questions of a theoretical nature, such as the issue of existence of solutions, by an approach that brings out the intimate connection between mathematics and physics, while suggesting, at the same time, algorithms for the actual computation of such solutions.