Regularization by noise of an averaged version of the Navier-Stokes equations

TL;DR

This paper investigates whether adding stochastic transport noise can regularize an averaged version of the 3D Navier-Stokes equations that typically exhibits finite-time blow-up, building on Tao's 2016 construction.

Contribution

It analyzes the potential of stochastic transport noise to prevent or delay blow-up in an averaged 3D Navier-Stokes model, extending previous deterministic and stochastic PDE studies.

Findings

Noise shows potential to regularize the averaged 3D NSE

Stochastic perturbations may delay finite-time blow-up

Provides insights into noise-induced regularization mechanisms

Abstract

In Tao 2016, the author constructs an averaged version of the deterministic three-dimensional Navier-Stokes equations (3D NSE) which experiences blow-up in finite time. In the last decades, various works have studied suitable perturbations of ill-behaved deterministic PDEs in order to prevent or delay such behavior. A promising example is given by a particular choice of stochastic transport noise closely studied in Flandoli et al. 2021. We analyze the model in Tao 2016 in view of these results and discuss the regularization skills of this noise in the context of the averaged 3D NSE.