Geometry of conservation laws for a class of parabolic partial differential equations
type: Item
instance of: scholarly article
title: Geometry of conservation laws for a class of parabolic partial differential equations
zbMATH Open document ID: 0879.35072
author: Jeanne N. Clelland
published in: Selecta Mathematica. New Series
publication date: 1997-12-22T00:00:00Z
review text: The author continues the research initiated by \textit{R. L. Bryant} and \textit{P. A. Griffiths} [Duke Math. J. 78, 531-676 (1995; Zbl 0853.58005)]. Tools of exterior differential systems and Cartan's method of equivalence are applied to the study of conservation laws of second-order nonlinear parabolic partial differential equations on one function of three variables, the main result being that all conservation laws are representable by a closed 3-form on second order jets. Examples collected are of independent interest. E.g., the equation \(u_t = \frac12 e^{-u} (u_{xx} + u_{yy})\) is shown to be intrinsically nonlinear and yet to possess an infinite set of conservations laws, parametrized by solutions \(A\) of the Laplace equation \(A_{xx} + A_{yy} = 0\). This contrasts with the result of Bryant and Griffiths that parabolic equations in the plane with at least four independent conservation laws are necessarily linearizable. Alternatively, one could compute the conservation laws from the earlier \textit{A. M. Vinogradov}'s theory [J. Math. Anal. Appl. 100, 41-129 (1984; Zbl 0548.58015)].
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 58A15
Mathematics Subject Classification ID: 35K10
zbMATH DE Number: 1036225
zbMATH Keywords: exterior differential systems
zbMATH Keywords: Cartan's method
reviewed by: Michal Marvan
MaRDI profile type: Publication
full work available at URL: s000290050005
OpenAlex ID: W2133326914
Wikidata QID: Q115389796
DOI: 10.1007/S000290050005
description: scientific article; zbMATH DE number 1036225