Well-Posedness for Stochastic Fractional Navier–Stokes Equation in the Critical Fourier–Besov Space

Abstract

The well-posedness of stochastic Navier–Stokes equations with various noises is a hot topic in the area of stochastic partial differential equations. Recently, the consideration of stochastic Navier–Stokes equations involving fractional Laplacian has received more and more attention. Due to the scaling-invariant property of the fractional stochastic equations concerned, it is natural and also very important to study the well-posedness of stochastic fractional Navier–Stokes equations in the associated critical Fourier–Besov spaces. In this paper, we are concerned with the three-dimensional stochastic fractional Navier–Stokes equation driven by multiplicative noise. We aim to establish the well-posedness of solutions of the concerned equation. To this end, by utilising the Fourier localisation technique, we first establish the local existence and uniqueness of the solutions in the critical Fourier–Besov space $$\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}$$ B ˙ p , r 4 - 2 α - 3 p . Then, under the condition that the initial date is sufficiently small, we show the global existence of the solutions in the probabilistic sense.