Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws

TL;DR

This paper investigates the structure of conservation laws for second-order parabolic equations, showing they depend on at most second derivatives and identifying Monge-Ampère equations as unique with non-trivial conservation laws.

Contribution

It introduces a method to analyze conservation laws via linearized characteristic cohomology and characterizes equations with non-trivial conservation laws as Monge-Ampère type.

Findings

Conservation laws depend on at most second derivatives.

Only Monge-Ampère type equations have non-trivial conservation laws.

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.