Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws
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 satisfi...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2021 |
| Main Author: | |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2021
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211302 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws. Benjamin B. McMillan. SIGMA 17 (2021), 047, 24 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862569069128974336 |
|---|---|
| author | McMillan, Benjamin B. |
| author_facet | McMillan, Benjamin B. |
| citation_txt | Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws. Benjamin B. McMillan. SIGMA 17 (2021), 047, 24 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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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| first_indexed | 2026-03-13T11:23:54Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211302 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T11:23:54Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | McMillan, Benjamin B. 2025-12-29T11:06:08Z 2021 Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws. Benjamin B. McMillan. SIGMA 17 (2021), 047, 24 pages 1815-0659 2020 Mathematics Subject Classification: 35L65; 58A15; 35K10; 35K55; 35K96 arXiv:1810.02346 https://nasplib.isofts.kiev.ua/handle/123456789/211302 https://doi.org/10.3842/SIGMA.2021.047 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. This material is based upon work supported by the National Science Foundation under Grant No. DGE-1106400 and 74341.2010, as well as the Australian Research Council, Discovery Program DP190102360. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws Article published earlier |
| spellingShingle | Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws McMillan, Benjamin B. |
| title | Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws |
| title_full | Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws |
| title_fullStr | Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws |
| title_full_unstemmed | Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws |
| title_short | Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws |
| title_sort | geometry and conservation laws for a class of second-order parabolic equations ii: conservation laws |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211302 |
| work_keys_str_mv | AT mcmillanbenjaminb geometryandconservationlawsforaclassofsecondorderparabolicequationsiiconservationlaws |