Enhanced dissipation for stochastic Navier-Stokes equations with transport noise

TL;DR

This paper demonstrates how transport noise can enhance dissipation in stochastic Navier-Stokes equations, leading to suppression of blow-up in 3D and ensuring global solutions with high probability.

Contribution

It provides new results on dissipation enhancement in 2D and blow-up suppression in 3D stochastic Navier-Stokes equations with quantifiable noise parameter estimates.

Findings

Dissipation is enhanced by transport noise in 2D stochastic Navier-Stokes.

Blow-up suppression is achieved in 3D stochastic Navier-Stokes in vorticity form.

Global solutions exist with high probability under certain noise parameters.

Abstract

The phenomenon of dissipation enhancement by transport noise is shown for stochastic 2D Navier-Stokes equations in velocity form. In the 3D case, suppression of blow-up is proved for stochastic Navier-Stokes equations in vorticity form; in particular, quantitative estimate allows us to choose the parameters of noise, uniformly in initial vorticity bounded in $L^2$-norm, so that global solutions exist with a large probability sufficiently close to 1.