It is well known how to construct a Laplacian on a fractal using the Dirichlet forms (see e.g.
[the survey article](http://www.ams.org/notices/199910/fea-strichartz.pdf) 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?*



How to define a differential form on a fractal?
Take a look at [Jenny Harrison, "Flux across nonsmooth boundaries and fractal Gauss/Green/Stokes' theorem,"](http://iopscience.iop.org/0305-4470/32/28/310/) which should at least answer your question about a "correct" notion of flux and divergence for a fractal domain -- here's the abstract:



> 
> By replacing the parametrization of a
>  domain with polyhedral approximations
>  we give optimal extensions of theorems
>  of Gauss, Green and Stokes'. Permitted
>  domains of integration range from
>  smooth submanifolds to structures that
>  may not be locally Euclidean and have
>  no tangent vectors defined anywhere.
>  One may still calculate divergence and
>  curl over a domain, and flux across
>  its boundary which itself may have no
>  normal vectors defined anywhere.
> 
> 
> 


