[1402.0290] Finite time blowup for an averaged three-dimensional Navier-Stokes equation

[1402.0290] Finite time blowup for an averaged three-dimensional Navier-Stokes equation:



The Navier-Stokes equation on the Euclidean space $\mathbf{R}^3$ can be expressed in the form $\partial_t u = \Delta u + B(u,u)$, where $B$ is a certain bilinear operator on divergence-free vector fields $u$ obeying the cancellation property $\langle B(u,u), u\rangle=0$ (which is equivalent to the energy identity for the Navier-Stokes equation). In this paper, we consider a modification $\partial_t u = \Delta u + \tilde B(u,u)$ of this equation, where $\tilde B$ is an averaged version of the bilinear operator $B$ (where the average involves rotations and Fourier multipliers of order zero), and which also obeys the cancellation condition $\langle \tilde B(u,u), u \rangle = 0$ (so that it obeys the usual energy identity). By analysing a system of ODE related to (but more complicated than) a dyadic Navier-Stokes model of Katz and Pavlovic, we construct an example of a smooth solution to such a averaged Navier-Stokes equation which blows up in finite time. This demonstrates that any attempt to positively resolve the Navier-Stokes global regularity problem in three dimensions has to use finer structure on the nonlinear portion $B(u,u)$ of the equation than is provided by harmonic analysis estimates and the energy identity. We also propose a program for adapting these blowup results to the true Navier-Stokes equations.

 This  blowup solution will have a signi ficant portion of its energy concentrating on smaller and smaller balls around the spatial origin x = 0; more precisely, there will be an increasing sequence of times tn 
converging exponentially fast to a fi nite limit  ,  such that a large fraction of the energy  (at least  for some small ,  > 0)  is concentrated in the ball  centred at the origin.

'via Blog this'