Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on
  topologically one-dimensional fractals   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.

12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga
34
The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
1
The categories of this paper are: cs
19
The categories of this paper are: eess
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc