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Symmetry, Integrability and Geometry: Methods and Applications, 2021, Volume 17, 047, 24 pp.
DOI: https://doi.org/10.3842/SIGMA.2021.047 (Mi sigma1730) 		 
Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws

Benjamin B. Mcmillan

University of Adelaide, Adelaide, South Australia
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DOI: https://doi.org/10.3842/SIGMA.2021.047
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.
Keywords: conservation laws, parabolic symbol PDEs, Monge–Ampère equations, characteristic cohomology of exterior differential systems.
Funding agency Grant number
National Science Foundation DGE-1106400
74341.2010
Australian Research Council DP190102360
This material is based upon work supported by the National Science Foundation under Grants No. DGE-1106400 and 74341.2010, as well as the Australian Research Council, Discovery Program DP190102360.
Received: March 17, 2020; in final form April 27, 2021; Published online May 11, 2021
Bibliographic databases:
Document Type: Article
MSC: 35L65, 58A15, 35K10, 35K55, 35K96
Language: English
Citation: Benjamin B. Mcmillan, “Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws”, SIGMA, 17 (2021), 047, 24 pp.
Citation in format AMSBIB
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