Hierarchy of Conservation Laws of Diffusion--Convection Equations

TL;DR

This paper develops a systematic approach to classify conservation laws of diffusion-convection equations, including local and nonlocal potentials, using symmetry groups and equivalence transformations.

Contribution

It introduces new notions of equivalence for conservation laws, revises linear dependence concepts, and provides a complete classification of potential conservation laws for diffusion-convection equations.

Findings

Complete classification of potential conservation laws for diffusion-convection equations.

Development of an effective direct method for constructing conservation laws.

Generalization of the iteration procedure for finding nonlocal conservation laws.

Abstract

We introduce notions of equivalence of conservation laws with respect to Lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems. We also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials. To construct conservation laws, we develop and apply the most direct method which is effective to use in the case of two independent variables. Admitting possibility of dependence of conserved vectors on a number of potentials, we generalize the iteration procedure proposed by Bluman and Doran-Wu for finding nonlocal (potential) conservation laws. As an example, we completely classify potential conservation laws (including arbitrary order local ones) of diffusion--convection equations with respect to the equivalence group and construct an exhaustive list of locally inequivalent potential systems corresponding to these equations.