Geometry of conservation laws for a class of parabolic partial differential equations

Abstract

I consider the problem of computing the space of conservation laws for a second-order parabolic partial differential equation for one function of three independent variables. The PDE is formulated as an exterior differential system ${\cal I}$ on a 12-manifold M, and its conservation laws are identified with the vector space of closed 3-forms in the infinite prolongation of ${\cal I}$ modulo the so-called "trivial" conservation laws. I use the tools of exterior differential systems and Cartan's method of equivalence to study the structure of the space of conservation laws. My main result is: Theorem. Any conservation law for a second-order parabolic PDE for one function of three independent variables can be represented by a closed 3-form in the differential ideal ${\cal I}$ on the original 12-manifold M. I show that if a nontrivial conservation law exists, then ${\cal I}$ has a deprolongation to an equivalent system ${\cal J}$ on a 7-manifold N, and any conservation law for ${\cal I}$ can be expressed as a closed 3-form on N that lies in ${\cal J}$. Furthermore, any such system in the real analytic category is locally equivalent to a system generated by a (parabolic) equation of the formA (u _xx u _yy -u ² _xy )+Bu _xx +2Cu _xy +Du _yy +E = 0 where A, B, C, D, E are functions of x, y, t, u, u _x , u _y , u _t . I compute the space of conservation laws for several examples, and I begin the process of analyzing the general case using Cartan's method of equivalence. I show that the non-linearizable equation $u_t = {1 \over 2} e^{-u} (u_{xx}+u_{yy})$ has an infinite-dimensional space of conservation laws. This stands in contrast to the two-variable case, for which Bryant and Griffiths showed that any equation whose space of conservation laws has dimension 4 or more is locally equivalent to a linear equation, i.e., is linearizable.