Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.18480 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916806681165824 |
|---|---|
| author | Liu, Hui Sun, Chengfeng Xin, Jie |
| author_facet | Liu, Hui Sun, Chengfeng Xin, Jie |
| contents | 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 $\|\nabla h\|_{L^\infty} < \sqrt πνλ_1^\frac{5}{4}$, where $ ν$ is the kinematic viscosity of the fluid and $λ_1$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18480 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity of random attractor and fractal dimension of fractional stochastic Navier-Stokes equations on three-dimensional torus Liu, Hui Sun, Chengfeng Xin, Jie Analysis of PDEs Probability 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 $\|\nabla h\|_{L^\infty} < \sqrt πνλ_1^\frac{5}{4}$, where $ ν$ is the kinematic viscosity of the fluid and $λ_1$ 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. |
| title | Regularity of random attractor and fractal dimension of fractional stochastic Navier-Stokes equations on three-dimensional torus |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2506.18480 |