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

Content Provider	Springer Nature Link
Author	Nielsen Clelland, J.
Copyright Year	1997
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 formA (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.
Starting Page	1
Ending Page	77
Page Count	77
File Format	PDF
ISSN	10221824
Journal	Selecta Mathematica
Volume Number	3
Issue Number	1
e-ISSN	14209020
Language	English
Publisher	Birkhäuser-Verlag
Publisher Date	1997-04-01
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Conservation laws Parabolic PDE Exterior differential systems Cartan's method Mathematics
Content Type	Text
Resource Type	Article
Subject	Physics and Astronomy Mathematics

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