4 February, 2014 inmath.AP,math.CA,paper| Tags:dyadic models,finite time blowup,Navier-Stokes equations| byTerence Tao
I’ve just uploaded to the arXiv the paper “Finite time blowup for an averaged three-dimensional Navier-Stokes equation“, submitted to J. Amer. Math. Soc.. The main purpose of this paper is to formalise the “supercriticality barrier” for the global regularity problem for the Navier-Stokes equation, which roughly speaking asserts that it is not possible to establish global regularity by any “abstract” approach which only uses upper bound function space estimates on the nonlinear part of the equation, combined with the energy identity. This is done by constructing a modification of the Navier-Stokes equations with a nonlinearity that obeys essentially all of the function space estimates that the true Navier-Stokes nonlinearity does, and which also obeys the energy identity, but for which one can construct solutions that blow up in finite time. Results of this type had been previously established byMontgomery-Smith,Gallagher-Paicu, andLi-Sinaifor variants of the Navier-Stokes equation without the energy identity, and byKatz-Pavlovicand byCheskidovfor dyadic analogues of the Navier-Stokes equations in five and higher dimensions that obeyed the energy identity (see also the workof Plechacand Sverakand ofHou and Leithat also suggest blowup for other Navier-Stokes type models obeying the energy identity in five and higher dimensions), but to my knowledge this is the first blowup result for a Navier-Stokes type equation in three dimensions that also obeys the energy identity. Intriguingly, the method of proof in fact hints at a possible route to establishing blowup for the true Navier-Stokes equations, which I am now increasingly inclined to believe is the case (albeit for a very small set of initial data).
To state the results more precisely, recall that the Navier-Stokes equations can be written in the form
for a divergence-free velocity fieldand a pressure field, whereis the viscosity, which we will normalise to be one. We will work in the non-periodic setting, so the spatial domain is, and for sake of exposition I will not discuss matters of regularity or decay of the solution (but we will always be working with strong notions of solution here rather than weak ones). Applying the Leray projectionto divergence-free vector fields to this equation, we can eliminate the pressure, and obtain an evolution equation
purely for the velocity field, whereis a certain bilinear operator on divergence-free vector fields (specifically,. The global regularity problem for Navier-Stokes is then equivalent to the global regularity problem for the evolution equation(1).
An important feature of the bilinear operatorappearing in(1)is the cancellation law
(using theinner product on divergence-free vector fields), which leads in particular to the fundamental energy identity
This identity (and its consequences) provide essentially the only knowna prioribound on solutions to the Navier-Stokes equations from large data and arbitrary times. Unfortunately, as discussed inthis previous post, the quantities controlled by the energy identity are supercritical with respect to scaling, which is the fundamental obstacle that has defeated all attempts to solve the global regularity problem for Navier-Stokes without any additional assumptions on the data or solution (e.g. perturbative hypotheses, ora prioricontrol on a critical norm such as thenorm).
Our main result is then (slightly informally stated) as follows
Theorem 1There exists anaveragedversionof the bilinear operator, of the form
for some probability space, some spatial rotation operatorsfor, and some Fourier multipliersof order, for which one still has the cancellation law
and for which the averaged Navier-Stokes equation