Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws
Analysis of PDEs
2021-05-12 v3 Differential Geometry
Abstract
I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservation laws of a given parabolic equation. This is used to show that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, it is shown that the only evolutionary parabolic equations with at least one non-trivial conservation law are of Monge-Amp\`ere type.
Keywords
Cite
@article{arxiv.1810.02346,
title = {Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws},
author = {Benjamin B. McMillan},
journal= {arXiv preprint arXiv:1810.02346},
year = {2021}
}