Remarks on the three-dimensional Navier-Stokes equations with Lions' exponent forced by space-time white noise

TL;DR

This paper establishes the global existence and uniqueness of solutions for the three-dimensional Navier-Stokes equations driven by space-time white noise with a fractional Laplacian at the energy-critical exponent, advancing stochastic fluid dynamics theory.

Contribution

It extends the global solution theory for stochastic Navier-Stokes equations to the energy-critical fractional Laplacian case using modern analytical techniques.

Findings

Proved global well-posedness for the stochastic Navier-Stokes with Lions' exponent.

Established a rigorous solution framework for energy-critical stochastic fluid equations.

Extended the approach of Hairer and Rosati to a new critical regime.

Abstract

We study the three-dimensional Navier-Stokes equations forced by space-time white noise and diffused via the fractional Laplacian with Lions' exponent so that it is precisely the energy-critical case. We prove its global solution theory following the approach of Hairer and Rosati (2024, Annals of PDE, \textbf{10}, pp. 1--46).