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

Benjamin B. McMillan

Published 2018-10-04Version 1

This is the second of two papers, the first of which developed geometric invariants for the class of scalar parabolic PDE, up to a generalized change of variable. The results developed in Part I are used here to study the characteristic cohomology of parabolic exterior differential systems, which directly translates into results on the conservation laws of parabolic equations. I find several results: First, I calculate the auxilliary differential equation whose solutions are in bijection with conservation laws of a parabolic equation. This result holds for fully general non-linear scalar parabolic equations, including ones not of evolutionary form. Second, any conservation law of an evolutionary parabolic equation is defined by a Jacobian function that depends on at most the second derivatives of solutions (as opposed to arbitrarily many derivatives in the general case). Finally, any evolutionary parabolic system with at least one non-trivial conservation law is neccessarily a Monge-Amp\`ere equation.