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Characteristic cohomology of differential systems. II: Conservation laws for a class of parabolic equations. (English) Zbl 0853.58005

In Part I of this series of papers [J. Am. Math. Soc. 8, No. 3, 507-596 (1995; 845.58004)] the authors introduced the notion of the characteristic cohomology of an exterior differential system (EDS) and showed that in the local involutive case the characteristic cohomology \(\overline{H}^q\) vanishes if \(0 < q < n - \ell\), where \(\ell\) is some geometric invariant of the system. The group \(\overline{H}^{n - \ell}\) was defined to be the space \(\mathcal C\) of conservation laws associated to the EDS. The present paper (second in the series) is devoted to the analysis of conservation laws for a class of parabolic systems, i.e. exterior differential systems on 7-dimensional manifolds which are locally equivalent to EDS generated by a second-order parabolic equation for one unknown function of two independent variables. In general, the space of local conservation laws is isomorphic to a certain space \(\mathcal C\) of closed 2-forms in the infinitely prolonged differential ideal. In the parabolic case, these 2-forms turn out to be well defined on the original 7-manifold.
Parabolic equations of Goursat type (i.e. equations for which the Goursat invariant vanishes) can be integrated by using only PDE techniques, so most of the paper is devoted to the non-Goursat case. In this case, it is described a general form of any conservation law. Reducing the problem to the parabolic Monge-Ampere system, the authors give a local normal form for all systems with \(\dim {\mathcal C} = 1, 2, 3,\geq 4\). In particular, a parabolic system has at least four independent conservation laws iff it is locally equivalent to a linear PDE system. The authors also describe an effective algorithm of determining of conservation laws for parabolic systems with \(\dim {\mathcal C} = 1,2,3,\) or more. This algorithm is used in the analysis of many examples, especially in the case of parabolic evolution equations. The paper ends with introducing the concept of an integrable extension of an EDS. Conservation laws may be used for generating local integrable extensions. A theorem classifying integrable extensions is proved and illustrated.

MSC:

58A15 Exterior differential systems (Cartan theory)
35S35 Topological aspects for pseudodifferential operators in context of PDEs: intersection cohomology, stratified sets, etc.
35A30 Geometric theory, characteristics, transformations in context of PDEs
35K10 Second-order parabolic equations
35K55 Nonlinear parabolic equations
58J35 Heat and other parabolic equation methods for PDEs on manifolds
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations

Citations:

Zbl 0845.58004
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References:

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