Stochastic Forced 3D Navier-Stokes Equations in $\mathbb{H}^{1/2}$-Space

TL;DR

This paper proves the global existence, uniqueness, and continuous dependence of solutions for the stochastic forced 3D Navier-Stokes equations in the critical 1/2-space, highlighting the regularization effect of stochastic noise and analyzing long-term dynamics.

Contribution

It establishes the global well-posedness of stochastic 3D Navier-Stokes equations in 1/2-space with general initial data, incorporating nonlocal turbulent forcing and stochastic regularization.

Findings

Stochastic noise provides a regularization effect on energy estimates.

Global existence and uniqueness of solutions are proved.

Long-time behavior of solutions is analyzed.

Abstract

In the classical work [FK], Fujita and Kato established the local existence of solutions to the 3D Navier-Stokes equations in the critical $\mathbb{H}^{1/2}$-space. In this paper, we are concerned with the global well-posedness of the stochastic forced 3D Navier-Stokes equations in the $\mathbb{H}^{1/2}$-space under general initial conditions, where the stochastic forcing comprises a transport forcing and a nonlocal turbulent forcing. In this setting, the random noise is shown to provide a regularization effect on the energy estimates, which we obtain by constructing suitable Lyapunov functions. However, its nonlocality also brings analytical challenges. We develop a bootstrap type estimate based on the kinematic viscosity together with a delicate stopping time argument to prove the global existence and uniqueness of solutions, as well as continuous dependence on the initial value. Furthermore, we also investigated the long-time behavior of the stochastic forced 3D Navier-Stokes equations.