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

Abstract

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 $\Vert \nabla h{\Vert }_{L^{\infty }}<\sqrt{\pi }\nu {\lambda }_1^{\frac{5}{4}}$, 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 $k\ge \frac{5}{2}$, 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.