Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations
I study the geometry and the conservation laws of second-order partial differential equations of parabolic type. The general strategy is to replace the differential equation with an exterior differ- ential system—a smooth manifold with extra geometric structure that keeps track of the solutions—and then use geometric methods.
I use Cartan’s method of equivalence to determine the essential geometric curvatures of parabolic equations. I then explain the geometric significance of these curvatures, including some normal form results. The study of these curvatures also leads me naturally to a nice class of equations, the parabolic Monge-Ampe`re equations.
In the second half, I study the relationship between the geometry and the conservation laws of parabolic equations. In particular, I prove that for a specific class of parabolic equations, the gen- erating function of any conservation law depends on at most second derivatives of solutions. This is in contrast to examples such as the KdV equation, which have conservation laws of arbitrarily high order.