The idea of using self-similarity -- essentially symmetry -- to prove lack of well-posedness for the Navier-Stokes equations has attracted some attention recently, in particular due to the work of Jia and Sverak. This idea has a long history, starting with Leray who attempted to use self-similar solutions to construct a blow-up at a prescribed time T. How important are the symmetry properties of the Navier-Stokes equations to the question of well-posedness? For example, uniqueness of self-similar (i.e. "symmetric") solutions combined with lack of uniquness in general would amount to "symmetry breaking" in the Navier-Stokes problem, and suggest that another mechanism, rather then symmetry drives the behavior of solutions.
We re-cast these uniqueness questions in terms of an explosion problem for the underlying stochastic processes and provide evidence against symmetry breaking in the Navier-Stokes equations.
Based on the joint work with Nick Michalowski, Enrique Thomann, and Ed Waymire.