Near-invariance under dynamic scaling for the Navier-Stokes equations in critical spaces: a probabilistic approach to regularity problems

We make a detailed comparison between the Navier-Stokes equations and their dynamically-scaled counterpart, the so-called Leray equations. The Navier-Stokes equations are invariant under static scaling transforms, but are not generally invariant under dynamic scaling transforms. We will study how close they can be brought together using the critical dependent variables and discuss the implications on the regularity problems. Assuming that the Navier-Stokes equations written in the vector potential have a solution that blows up at t = 1, we derive the Leray equations by dynamic scaling. We observe: (1) The Leray equations have only one term extra on top of those of the Navier-Stokes equations. (2) We can recast the Navier-Stokes equations as a Wiener path integral and the Leray equations as another Ornstein-Uhlenbeck path integral. By the Maruyama-Girsanov theorem, both equations take the identical form modulo the Maruyama-Girsanov density, which is valid up to t = 2√ 2 by the Novikov condition. (3) The global solution of the Leray equations is given by a finite-dimensional projection R of a functional of an Ornstein-Uhlenbeck process and a probability measure. If R remains smooth beyond t = 1 under an absolute continuous change of the probability measure, we can rule out finite-time blowup by contradiction. There are two cases: (A) R given by a finite number of Wiener integrals, and (B) otherwise. Ruling out blowup in (A) is straightforward. For (B), a condition based on a limit passage in the Picard iterations is identified for such a contradiction to come out. The whole argument equally holds in R d for any d ≥ 2.