Scalar parabolic PDEs and braids
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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
- MR Author ID: 346210
- 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
- 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 [R. Ghrist, Braids and differential equations, Proc. International Congress of Mathematicians, vol. III, 2006, 1–26] - © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 2755-2788
- MSC (2000): Primary 35K90, 37B30
- DOI: https://doi.org/10.1090/S0002-9947-08-04823-X
- MathSciNet review: 2471938