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

TL;DR

This paper develops a Hodge theory framework for differential forms on topologically one-dimensional fractals, linking harmonic forms to cohomology, and analyzes Navier-Stokes equations in this fractal setting.

Contribution

It introduces a Hodge decomposition and Laplacian for differential forms on fractals, connecting harmonic forms to classical cohomology and studying fluid equations on these spaces.

Findings

Hodge decomposition holds for fractals under certain conditions.

Harmonic 1-forms are nontrivial iff classical cohomology is nontrivial.

Navier-Stokes equations have only steady divergence-free solutions on these 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 \v{C}ech 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.