Stochastic Forced 3D Navier-Stokes Equations in ℍ^1/2-Space

2025/11/19 by Hong, Wei, Li, Shihu, Liu, Wei
Mathematics · Engineering · Economics, Econometrics and Finance · #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #Stochastic processes and financial applications

paper · doi:10.48550/arxiv.2511.15223

Abstract

In the classical work [FK], Fujita and Kato established the local existence of solutions to the 3D Navier-Stokes equations in the critical 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 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.

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