On Le Jan-Sznitman's stochastic approach to the Navier-Stokes equations

TL;DR

This paper investigates the connection between Navier-Stokes equations and stochastic cascades, introducing a simplified stochastic construction that reveals insights into solution existence, uniqueness, blowup, and convergence properties.

Contribution

It presents a new stochastic approach to Navier-Stokes equations that simplifies solution construction and provides new results on blowup, non-uniqueness, and global well-posedness.

Findings

Simpler stochastic construction for Navier-Stokes solutions

Demonstration of finite-time blowup and non-uniqueness

Proof of global well-posedness for small initial data

Abstract

The paper explores the symbiotic relation between the Navier-Stokes equations and the associated stochastic cascades. Specifically, we examine how some well-known existence and uniqueness results for the Navier-Stokes equations can inform about the probabilistic features of the associated stochastic cascades, and how some probabilistic features of the stochastic cascades can, in turn, inform about the existence and uniqueness (or the lack thereof) of solutions. Our method of incorporating the stochastic explosion gives a simpler and more natural method to construct the solution compared to the original construction by Le Jan and Sznitman. This new stochastic construction is then used to show the finite-time blowup and non-uniqueness of the initial value problem for the Montgomery-Smith equation. We exploit symmetry properties inherent in our construction to give a simple proof of the global well-posedness results for small initial data in scale-critical Fourier-Besov spaces. We also obtain the pointwise convergence of the Picard's iteration associated with the Fourier-transformed Navier-Stokes equations.