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arXiv · 1810.02346

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

Benjamin B. McMillan
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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.

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Benjamin B. McMillan. 2018-10-04. Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws. https://doi.org/10.3842/sigma.2021.047

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