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

TL;DR

This work proves global well-posedness for the stochastic forced 3D Navier–Stokes equations in the critical space $\mathbb{H}^{1/2}$ with both transport and nonlocal stochastic forcing, for general initial data. It introduces a Lyapunov-based energy framework and a bootstrap regularity strategy that leverages viscous smoothing together with stopping-time localization to obtain global existence and uniqueness, even in the absence of finite second moments. The results include Feller properties, continuous dependence on initial data, and a long-time behavior analysis: under stronger noise assumptions, the solution decays exponentially to zero and admits a unique invariant measure; the nonlocal forcing is shown to balance energy and enable global solvability while presenting technical nonlocality challenges. The analysis relies on stochastic compactness (Jakubowski–Skorokhod), careful energy estimates, and localization arguments to handle the nonlocal stochastic forcing and to establish ergodicity without resorting to Markov selections.

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.

Theorems & Definitions (67)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Definition 3.1
• Definition 3.2
• Remark 3.1