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Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws (1810.02346v3)
Published 4 Oct 2018 in math.AP and math.DG
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.