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

TL;DR

This paper explores the geometric structure of second-order parabolic equations, identifying invariants and conditions for evolutionary form, with implications for conservation laws and classification of Monge-Ampère type equations.

Contribution

It introduces a geometric framework using Cartan techniques to classify parabolic equations and characterizes when they can be transformed into evolutionary form.

Findings

A family of invariants characterizes Monge-Ampère type equations.

Conservation laws depend on at most second derivatives of solutions.

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

Abstract

I consider the geometry of the general class of scalar 2nd-order differential equations with parabolic symbol, including non-linear and non-evolutionary parabolic equations. After defining the appropriate $G$-structure to model parabolic equations, I apply Cartan techniques to determine local geometric invariants (quantities invariant up to a generalized change of variables). One family of invariants gives a geometric characterization for parabolic equation of Monge-Amp\`ere type. A second family of invariants determines when a parabolic equation has a local choice of coordinates putting it in evolutionary form. In addition to their intrinsic interest, these results are applied in a follow up paper on the conservation laws of parabolic equations. It is shown there that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, the only evolutionary parabolic equations with at least one non-trivial conservation law are of Monge-Amp\`ere type.