Conservation Laws of Multidimensional Diffusion-Convection Equations

TL;DR

This paper systematically classifies all local conservation laws for multidimensional diffusion-convection equations using symmetry and variational methods, providing explicit formulas and analyzing symmetry actions.

Contribution

It introduces a comprehensive classification of conservation laws for these equations, employing the direct method and variational principles, with explicit closed-form results.

Findings

Complete set of conservation laws derived

Explicit formulas for conservation laws provided

Symmetry group actions on conservation laws analyzed

Abstract

All possible linearly independent local conservation laws for $n$-dimensional diffusion--convection equations $u_t=(A(u))_{ii}+(B^i(u))_i$ were constructed using the direct method and the composite variational principle. Application of the method of classification of conservation laws with respect to the group of point transformations [R.O. Popovych, N.M. Ivanova, J. Math. Phys., 2005, V.46, 043502 (math-ph/0407008)] allows us to formulate the result in explicit closed form. Action of the symmetry groups on the conservation laws of diffusion equations is investigated and generating sets of conservation laws are constructed.