Pullback $V$-Attractors of the Stochastic Calmed 3$D$ Navier-Stokes Equations

In this paper, we investigate a calmed version of the 3$D$ rotational Navier-Stokes equations driven by additive noise. First, we use the Ornstein-Uhlenbeck process to transform the equation into a random one. By using the Galerkin approximation, we establish the global well-posedness of solutions for the calmed system. Then, we demonstrate the existence of a closed, measurable $\mathcal{D}_V$-pullback absorbing set. Finally, by proving the pullback flattening property, we obtain the existence of a $\mathcal{D}_V$-pullback attractor in $V$.