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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.18745 |
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| _version_ | 1866912555060953088 |
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| author | Cui, Hongyong Liu, Hui Xin, Jie |
| author_facet | Cui, Hongyong Liu, Hui Xin, Jie |
| contents | 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 $\|\nabla h\|_{L^\infty} \leq \sqrt πνλ_1$, where $ ν$ is the kinematic viscosity of the fluid and $λ_1$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18745 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $(H,H^2)$-smoothing effect of Navier-Stokes equations with additive white noise on two-dimensional torus Cui, Hongyong Liu, Hui Xin, Jie Analysis of PDEs 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 $\|\nabla h\|_{L^\infty} \leq \sqrt πνλ_1$, where $ ν$ is the kinematic viscosity of the fluid and $λ_1$ 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. |
| title | $(H,H^2)$-smoothing effect of Navier-Stokes equations with additive white noise on two-dimensional torus |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.18745 |