Fractional Navier-Stokes Equations with Caputo Derivative Driven by Hermite Noise
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910122542891008 |
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| author | Lechiheb, Atef |
| author_facet | Lechiheb, Atef |
| contents | We study time-fractional stochastic Navier-Stokes equations on a bounded domain of $\R^2$ (the restriction to dimension two is essential for the bilinear estimates via Sobolev embeddings) driven by a Hermite process $Z_H^k$ of order $k\ge1$ and Hurst parameter $H\in(1/2,1)$. This class of noises generalizes fractional Brownian motion ($k=1$) and the Rosenblatt process ($k=2$). We construct the Wiener integral with respect to $Z_H^k$ and establish sharp $L^p$ estimates via hypercontractivity, explicitly capturing the dependence on $k$. Using a refined Hilbert-Schmidt estimate for the Mittag-Leffler operator, we prove that the stochastic convolution belongs to $\dot{H}^ν$ under the condition $\al(1-ν)+2H>2$. A fixed-point argument in a weighted space yields the existence, uniqueness, and Hölder regularity of mild solutions. We also prove a non-central limit theorem linking the solution to discrete approximations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10602 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fractional Navier-Stokes Equations with Caputo Derivative Driven by Hermite Noise Lechiheb, Atef Probability 60H15 (Primary), 35R60, 76D06, 26A33 (Secondary) We study time-fractional stochastic Navier-Stokes equations on a bounded domain of $\R^2$ (the restriction to dimension two is essential for the bilinear estimates via Sobolev embeddings) driven by a Hermite process $Z_H^k$ of order $k\ge1$ and Hurst parameter $H\in(1/2,1)$. This class of noises generalizes fractional Brownian motion ($k=1$) and the Rosenblatt process ($k=2$). We construct the Wiener integral with respect to $Z_H^k$ and establish sharp $L^p$ estimates via hypercontractivity, explicitly capturing the dependence on $k$. Using a refined Hilbert-Schmidt estimate for the Mittag-Leffler operator, we prove that the stochastic convolution belongs to $\dot{H}^ν$ under the condition $\al(1-ν)+2H>2$. A fixed-point argument in a weighted space yields the existence, uniqueness, and Hölder regularity of mild solutions. We also prove a non-central limit theorem linking the solution to discrete approximations. |
| title | Fractional Navier-Stokes Equations with Caputo Derivative Driven by Hermite Noise |
| topic | Probability 60H15 (Primary), 35R60, 76D06, 26A33 (Secondary) |
| url | https://arxiv.org/abs/2604.10602 |