2025/06/23 by Liu, Hui, Sun, Chengfeng, Xin, Jie
#Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2506.18480
In this paper we will study the asymptotic dynamics of fractional Navier-Stokes (NS) equations with additive white noise on three-dimensional torus \mathbb T3. Under the conditions that the external forces f(x) belong to the phase space H and the noise intensity function h(x) satisfies ‖∇ h‖L^∞ < √ πνλ1^(5)/(4), where ν is the kinematic viscosity of the fluid and λ1 is the first eigenvalue of the Stokes operator, we shown that the random fractional three-dimensional NS equations possess a tempered (H,H^(5)/(2))-random attractor whose fractal dimension in H^(5)/(2) is finite. This was proved by establishing, first, an H^(5)/(2) bounded absorbing set and, second, a local (H,H^(5)/(2))-Lipschitz continuity in initial values from which the (H,H^(5)/(2))-asymptotic compactness of the system follows. Since the forces f belong only to H, the H^(5)/(2) bounded absorbing set was constructed by an indirect approach of estimating the H^(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 Hk-(5)/(4) and the noise intensity function h(x) belong to Hk+(5)/(4) for k≥(5)/(2), we shown that the random fractional three-dimensional NS equations possess a tempered (H,Hk)-random attractor whose fractal dimension in Hk is finite. This was proved by using iterative methods and establishing, first, an Hk bounded absorbing set and, second, a local (H,Hk)-Lipschitz continuity in initial values from which the (H,Hk)-asymptotic compactness of the system follows.