17
views
0
recommends
+1 Recommend
0 collections
    0
    shares
      • Record: found
      • Abstract: found
      • Article: found
      Is Open Access

      Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws

      Preprint

      Read this article at

      Bookmark
          There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

          Abstract

          This is the second of two papers, the first of which developed geometric invariants for the class of scalar parabolic PDE, up to a generalized change of variable. The results developed in Part I are used here to study the characteristic cohomology of parabolic exterior differential systems, which directly translates into results on the conservation laws of parabolic equations. I find several results: First, I calculate the auxilliary differential equation whose solutions are in bijection with conservation laws of a parabolic equation. This result holds for fully general non-linear scalar parabolic equations, including ones not of evolutionary form. Second, any conservation law of an evolutionary parabolic equation is defined by a Jacobian function that depends on at most the second derivatives of solutions (as opposed to arbitrarily many derivatives in the general case). Finally, any evolutionary parabolic system with at least one non-trivial conservation law is neccessarily a Monge-Amp\`ere equation.

          Related collections

          Most cited references2

          • Record: found
          • Abstract: not found
          • Article: not found

          Characteristic cohomology of differential systems II: Conservation laws for a class of parabolic equations

            Bookmark
            • Record: found
            • Abstract: not found
            • Article: not found

            Characteristic Cohomology of Differential Systems (I) General Theory

              Bookmark

              Author and article information

              Journal
              04 October 2018
              Article
              1810.02346
              71dd8fb9-098d-40b9-a322-f961fbc3f23c

              http://arxiv.org/licenses/nonexclusive-distrib/1.0/

              History
              Custom metadata
              Primary 35L65, 58A15, Secondary 35K10, 35K55, 35K96
              22 pages
              math.AP math.DG

              Analysis,Geometry & Topology
              Analysis, Geometry & Topology

              Comments

              Comment on this article