Geometry of conservation laws for a class of parabolic partial differential equations
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)].
scientific article; zbMATH DE number 1036225