$(H,H^2)$-smoothing effect of Navier-Stokes equations with additive white noise on two-dimensional torus

This paper is devoted to the regularity of Navier-Stokes (NS) equations with additive white noise on two-dimensional torus $\mathbb T^2$. Under the conditions that the external force $f(x)$ belongs to the phase space $ H$ and the noise intensity function $h(x)$ satisfies $\|\nabla h\|_{L^\infty} \leq \sqrt \pi \nu \lambda_1$, where $ \nu $ is the kinematic viscosity of the fluid and $\lambda_1$ is the first eigenvalue of the Stokes operator, it was proved that the random NS equations possess a tempered $(H,H^2)$-random attractor whose (box-counting) fractal dimension in $H^2$ is finite. This was achieved by establishing, first, an $H^2$ bounded absorbing set and, second, an $(H,H^2)$-smoothing effect of the system which lifts the compactness and finite-dimensionality of the attractor in $H$ to that in $H^2$. Since the force $f$ belongs only to $H$, the $H^2$-regularity of solutions as well as the $H^2$-bounded absorbing set was constructed by an indirect approach of estimating the $H^2$-distance between the solution of the random NS equations and that of the corresponding deterministic equations.