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

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## 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 form*A (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.