The phenomenon of concentration in solutions to hyperbolic systems of conservation laws and related recent analytical techniques for studying such solutions are discussed. In particular, for the Euler equations for both isentropic and nonisentropic fluids, the phenomenon of concentration in solutions is fundamental as the pressure vanishes, which occurs not only in the multidimensional situations, but also even naturally in the one-dimensional case. From the point of view of hyperbolic conservation laws, since the limit system loses hyperbolicity, the phenomenon of concentration in the process of vanishing pressure limit can be regarded as a phenomenon of resonance among the characteristic fields. This phenomenon indicates that the flux-function limit can be very singular: the limit functions of solutions are no longer in the spaces of functions, $BV$ or $L^\infty$, yet the space of Radon measures, for which the divergences of certain entropy and entropy flux fields are also Radon measures, is a natural space to deal with such a limit in general. In this regard, a theory of divergence-measure fields developed recently is also presented. This theory especially includes normal traces, a generalized Gauss-Green theorem, and product rules, among others. Some applications of this theory to solving various nonlinear problems in conservation laws and related areas are also discussed.

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Author AddressGui-Qiang Chen Department of Mathematics Northwestern University Evanston, IL 60208-2730 USA gqchen@math.northwestern.edu