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We consider nonclassical entropy solutions to scalar conservation laws withconcave-convex flux functions, whose set of left- and right-hand admissiblestates across undercompressive shocks is selected by a kinetic function \phi.We introduce a new definition for the (generalized) strength of classical andnonclassical shocks, allowing us to propose a generalized notion of totalvariation functional. Relying only upon the natural assumption that thecomposite function \phi o \phi is uniformly contracting, we prove that thegeneralized total variation of front-tracking approximations is non-increasingin time, and we conclude with the existence of nonclassical solutions to theinitial-value problem. We also propose two definitions of generalizedinteraction potentials which are adapted to handle nonclassical entropysolutions and we investigate their monotonicity properties. In particular, weexhibit an interaction functional which is globally non-increasing along asplitting-merging interaction pattern.

Diminishing functionals for nonclassical entropy solutions selected by kinetic relations