Stabilité des chocs non classiques pour des lois de conservation non convexes
First, we study the existence and the properties of shock waves realized as the limit of a linear diffusive-dispersive regularization for the class of hyperbolic conservation laws with generic non-convex flux having an arbitrary number of inflection points. We introduce the construction of a corresponding family of kinetic functions, based on a generalization ofMaxwell’s equal area rule. Our main result is a characterization (via a kinetic function) of all admissible shocks : for each left-hand state and each shock speed, we prove that there exists a nonclassical shock with arbitrary many oscillations. Moreover, we state conditions under which a discontinuity is entropy dissipative and we describe the family of all entropy dissipative solutions to the Riemann problem. This work is an extension of the work done by Bedjaoui, Chalons, Coquel and LeFloch on the Van der Walls isentropic gases covering the case of two inflection points. Next, we are interested in the existence of a class of nonclassical entropy solutions to the initial value problem for nonlinear hyperbolic systems of conservation laws in one space variable. The solutions under consideration exhibit a splitting/merging pattern made of classical and nonclassical waves that interact together. Our analysis is based on a new non-classical Riemann solver which is motivated by a kinetic relation for the propagation of nonclassical shocks and by a nucleation criterion that selects between classical and nonclassical behavior. All solutions satisfy a single entropy inequality associated with a given entropy pair. Our existence proof of nonlinear stability builds on the introduction of a new total variation functional which suitably measure the total variation and wave interaction of nonclassical and classical waves. This work is a generalization of the work done by Laforest-LeFloch who have already established the existence of nonclassical entropy solutions for scalar equations.
Hyperbolic conservation law Nonconvex flux Diffusive-dispersive regularization Nonclassical shock Kinetic function