  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.
