Gibb's minimization principle for approximate solutions of scalar conservation laws

TL;DR

This paper explores how approximate solutions to scalar conservation laws can be derived from a BGK-type kinetic equation that enforces Gibbs' entropy minimization, linking variational principles with kinetic representations.

Contribution

It introduces a novel approach using Gibbs' minimization principle within a BGK framework to obtain approximate solutions of scalar conservation laws.

Findings

Approximate solutions satisfy a kinetic equation similar to Lions-Perthame-Tadmor representation.

Solutions exhibit small-scale non-equilibrium behavior.

Gibbs' entropy minimization guides the equilibrium densities in the model.

Abstract

In this work we study variational properties of approximate solutions of scalar conservation laws. Solutions of this type are described by a kinetic equation which is similar to the kinetic representation of admissible weak solutions due to Lions-Perthame-Tadmor, but also retain small scale non-equilibrium behavior. We show that approximate solutions can be obtained from a BGK-type equation with equilibrium densities satisfying Gibb's entropy minimization principle.