Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 047, 24 pages      arXiv:1810.02346      https://doi.org/10.3842/SIGMA.2021.047

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

Benjamin B. McMillan
University of Adelaide, Adelaide, South Australia

Received March 17, 2020, in final form April 27, 2021; Published online May 11, 2021

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-Ampère type.

Key words: conservation laws; parabolic symbol PDEs; Monge-Ampère equations; characteristic cohomology of exterior differential systems.

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