Geometry and conservation laws for a class of second-order parabolic equations ii: conservation laws
Date
2021
Authors
McMillan, B.B.
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Symmetry, Integrability and Geometry: Methods and Applications, 2021; 17:1-24
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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-Ampère type.
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