Regularity of random attractor and fractal dimension of fractional stochastic Navier-Stokes equations on three-dimensional torus

In this paper we will study the asymptotic dynamics of fractional Navier-Stokes (NS) equations with additive white noise on three-dimensional torus $\mathbb T^3$. Under the conditions that the external forces $f(x)$ belong to the phase space $ H$ and the noise intensity function $h(x)$ satisfies $\|\nabla h\|_{L^\infty} < \sqrt \pi \nu \lambda_1^\frac{5}{4}$, where $ \nu $ is the kinematic viscosity of the fluid and $\lambda_1$ is the first eigenvalue of the Stokes operator, we shown that the random fractional three-dimensional NS equations possess a tempered $(H,H^\frac{5}{2})$-random attractor whose fractal dimension in $H^\frac{5}{2}$ is finite. This was proved by establishing, first, an $H^\frac{5}{2}$ bounded absorbing set and, second, a local $(H,H^\frac{5}{2})$-Lipschitz continuity in initial values from which the $(H,H^\frac{5}{2})$-asymptotic compactness of the system follows. Since the forces $f$ belong only to $H$, the $H^\frac{5}{2}$ bounded absorbing set was constructed by an indirect approach of estimating the $H^\frac{5}{2}$-distance between the solutions of the random fractional three-dimensional NS equations and that of the corresponding deterministic equations. Furthermore, under the conditions that the external forces $f(x)$ belong to the $ H^{k-\frac{5}{4}}$ and the noise intensity function $h(x)$ belong to $H^{k+\frac{5}{4}}$ for $k\geq\frac{5}{2}$, we shown that the random fractional three-dimensional NS equations possess a tempered $(H,H^k)$-random attractor whose fractal dimension in $H^k$ is finite. This was proved by using iterative methods and establishing, first, an $H^k$ bounded absorbing set and, second, a local $(H,H^k)$-Lipschitz continuity in initial values from which the $(H,H^k)$-asymptotic compactness of the system follows.