A counterexample to the smoothness of the solution to an equation
arising in fluid mechanics
We analyze the equation coming from the Eulerian-Lagrangian description
of fluids. We discuss a couple of ways to extend this notion to
viscous fluids. The main focus of this paper is to discuss
the first way, due to Constantin. We show that this description
can only work for short times, after which the ``back to coordinates map''
may have no smooth inverse. Then we briefly discuss a second way that
uses Brownian motion. We use this to provide a plausibility argument
for the global regularity for the Navier-Stokes equations.