| 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 which 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$$A(usb{xx}usb{yy}-usbsp{xy}{2}) + Bsb1usb{xx}+2Bsb2usb{xy} +Bsb3usb{yy}+C=0cr$$where $A, Bsb{i}, C$ are functions of x, y, t, u, $usb{x}, usb{y}, usb{t}$.;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$$usb{t} = {1over2}esp{-u}(usb{xx}+usb{yy})$$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. |
