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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Scalar parabolic PDEs and braids


Authors: R. W. Ghrist and R. C. Vandervorst
Journal: Trans. Amer. Math. Soc. 361 (2009), 2755-2788
MSC (2000): Primary 35K90, 37B30
Published electronically: December 17, 2008
MathSciNet review: 2471938
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Abstract: The comparison principle for scalar second order parabolic PDEs on functions $ u(t,x)$ admits a topological interpretation: pairs of solutions, $ u^1(t,\cdot)$ and $ u^2(t,\cdot)$, evolve so as to not increase the intersection number of their graphs. We generalize to the case of multiple solutions $ \{u^\alpha(t,\cdot)\}_{\alpha=1}^n$. By lifting the graphs to Legendrian braids, we give a global version of the comparison principle: the curves $ u^\alpha(t,\cdot)$ evolve so as to (weakly) decrease the algebraic length of the braid.

We define a Morse-type theory on Legendrian braids which we demonstrate is useful for detecting stationary and periodic solutions to scalar parabolic PDEs. This is done via discretization to a finite-dimensional system and a suitable Conley index for discrete braids.

The result is a toolbox of purely topological methods for finding invariant sets of scalar parabolic PDEs. We give several examples of spatially inhomogeneous systems possessing infinite collections of intricate stationary and time-periodic solutions.


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Additional Information

R. W. Ghrist
Affiliation: Departments of Mathematics and Electrical & Systems Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Email: ghrist@math.upenn.edu

R. C. Vandervorst
Affiliation: Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081 HV, Amsterdam, The Netherlands
Email: vdvorst@few.vu.nl

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04823-X
PII: S 0002-9947(08)04823-X
Received by editor(s): November 11, 2004
Received by editor(s) in revised form: December 14, 2007
Published electronically: December 17, 2008
Additional Notes: The first author was supported in part by NSF PECASE grant DMS-0337713.
The second author was supported by NWO VIDI grant 639.032.202
These results were announced in \cite{G}
Article copyright: © Copyright 2008 American Mathematical Society