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

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 harmonicity for finite energy 1-forms. As as 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.

References in corpus (5)

• Dirac and magnetic Schrödinger operators on fractals
• Metrics and spectral triples for Dirichlet and resistance forms
• Upper estimate of martingale dimension for self-similar fractals
• Measurable Riemannian structures associated with strong local Dirichlet forms
• Geometry and Analysis of Dirichlet forms

Cited by in corpus (11)

• Dirac and magnetic Schrödinger operators on fractals
• Metrics and spectral triples for Dirichlet and resistance forms
• Fractal snowflake domain diffusion with boundary and interior drifts
• Long Time Decay of Leray Solution of 3D-NSE With Exponential Damping
• Finite energy coordinates and vector analysis on fractals
• Differential forms on Dirichlet spaces and Bakry-Émery estimates on metric graphs
• Closability, regularity, and approximation by graphs for separable bilinear forms
• Canonical diffusions on the pattern spaces of aperiodic Delone sets
• From non-symmetric particle systems to non-linear PDEs on fractals
• Differential forms for fractal subspaces and finite energy coordinates
• Gaps in the spectrum of the Laplacian on $3N$-Gaskets