It is well known how to construct a Laplacian on a fractal using the Dirichlet forms (see e.g. the survey article by Strichartz). This implies, in particular, that a fractal can be "heated", i.e. one can write (and solve) the heat equation on the fractal.
The question is, can one run a fluid flow through a fractal set? In other words, is there a proper way to write the Navier-Stokes equations on a fractal? In order to do this, it seems that we need a "correct" notion of divergence at least.
More generally, is there a "correct" way to define a differential form on a fractal?