Diminishing functionals for nonclassical entropy solutions selected by kinetic relations

TL;DR

This paper introduces a new framework for analyzing nonclassical entropy solutions to scalar conservation laws, proving their stability via a generalized total variation functional and exploring interaction potentials.

Contribution

It proposes a novel definition of shock strength and total variation for nonclassical solutions, and demonstrates their non-increasing behavior under certain conditions.

Findings

Generalized total variation is non-increasing over time.

Existence of nonclassical solutions is established.

Interaction functionals can be globally non-increasing.

Abstract

We consider nonclassical entropy solutions to scalar conservation laws with concave-convex flux functions, whose set of left- and right-hand admissible states across undercompressive shocks is selected by a kinetic function \phi. 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 \phi o \phi 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.