Diminishing Functionals for Nonclassical Entropy Solutions Selected by Kinetic Relations

We consider nonclassical entropy solutions to scalar conservation laws with concaveconvex flux functions, whose set of leftand right-hand admissible states ul, ur across undercompressive shocks is selected by a kinetic function ur = φ(ul). We introduce a new definition for the (generalized) strength of classical and nonclassical shocks, allowing us to propose a generalized notion of total variation functional. Relying only upon the natural assumption that the composite function φ ♭ ◦ φ ♭ is uniformly contracting, we prove 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 two definitions of generalized interaction potentials which are adapted to handle nonclassical entropy solutions and we investigate their monotonicity properties. In particular, we exhibit an interaction functional which is globally non-increasing along a splitting-merging interaction pattern.