Stochastic 3D Leray-$\alpha$ Model with Fractional Dissipation

In this paper, we establish the global well-posedness of stochastic 3D Leray-$\alpha$ model with general fractional dissipation driven by multiplicative noise. This model is the stochastic 3D Navier-Stokes equation regularized through a smoothing kernel of order $\theta_1$ in the nonlinear term and a $\theta_2$-fractional Laplacian. In the case of $\theta_1 \ge 0, \theta_2 > 0$ and $\theta_1+\theta_2 \geq\frac{5}{4}$, we prove the global existence and uniqueness of strong solutions. The main results cover many existing works in the deterministic cases, and also generalize some known results of stochastic models as our special cases such as stochastic hyperviscous Navier-Stokes equation and classical stochastic 3D Leray-$\alpha$ model.