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Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals

Authors: Michael Hinz and Alexander Teplyaev
Journal: Trans. Amer. Math. Soc. 367 (2015), 1347-1380
MSC (2010): Primary 31E05, 60J45; Secondary 28A80, 31C25, 35J25, 35Q30, 46L87, 47F05, 58J65, 60J60, 81Q35
Published electronically: April 30, 2014
Corrigendum: Trans. Amer. Math. Soc. 369 (2017), 6777-6778.
MathSciNet review: 3280047
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Abstract: We consider finite energy and $ L^2$ differential forms associated with strongly local regular Dirichlet forms on compact connected topologically one-dimensional spaces. We introduce notions of local exactness and local harmonicity and prove the Hodge decomposition, which in our context says that the orthogonal complement to the space of all exact 1-forms coincides with the closed span of all locally harmonic 1-forms. Then we introduce a related Hodge Laplacian and define a notion of harmonicity for finite energy 1-forms. As a corollary, under a certain capacity-separation assumption, we prove that the space of harmonic 1-forms is nontrivial if and only if the classical Čech cohomology is nontrivial. In the examples of classical self-similar fractals these spaces typically are either trivial or infinitely dimensional. Finally, we study Navier-Stokes type models and prove that under our assumptions they have only steady state divergence free solutions. In particular, we solve the existence and uniqueness problem for the Navier-Stokes and Euler equations for a large class of fractals that are topologically one-dimensional but can have arbitrary Hausdorff and spectral dimensions.

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Additional Information

Michael Hinz
Affiliation: Department of Mathematics, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany

Alexander Teplyaev
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009

Received by editor(s): May 30, 2013
Published electronically: April 30, 2014
Additional Notes: The research of the first author was supported in part by the Alexander von Humboldt Foundation (Feodor Lynen Research Fellowship Program) and carried out during a stay at the Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
The research of the first and second authors was supported in part by NSF grant DMS-0505622
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