Geodesic
Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021)

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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.
Keywords: conservation laws, characteristic cohomology of exterior differential systems.
Mots-clés : parabolic symbol PDEs, Monge–Ampère equations 
Benjamin B. Mcmillan. Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a46/
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