Diminishing functionals for nonclassical entropy solutions selected by kinetic relations

Abstract

We consider nonclassical entropy solutions to hyperbolic conservation laws with concave-convex flux functions, whose undercompressive shocks are selected by a kinetic function ${\phi }^{♭}$. Extending earlier work of Baiti, LeFloch and Piccoli, we reinterpret their construction of the (generalized) strength of classical and nonclassical shocks, allowing us to simplify it, highlight its true nature and identify new degrees of freedom. Relying mainly upon the natural assumption that the composite function ${\phi }^{♭}◦{\phi }^{♭}$ is uniformly contracting, we establish that the generalized total variation of front-tracking approximations is non-increasing in time, and we conclude with the existence of nonclassical solutions to the initial value problem. We also propose a definition of a generalized interaction potential, and investigate its monotonicity properties. In particular, we established that the interaction functional is globally non-increasing along a splitting-merging interaction pattern.