Stochastic Forced 3D Navier-Stokes Equations in $\mathbb{H}^{1/2}$-Space
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| Format: | Preprint |
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2025
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| _version_ | 1866912950047997952 |
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| author | Hong, Wei Li, Shihu Liu, Wei |
| author_facet | Hong, Wei Li, Shihu Liu, Wei |
| contents | In the classical work [FK], Fujita and Kato established the local existence of solutions to the 3D Navier-Stokes equations in the critical $\mathbb{H}^{1/2}$-space. In this paper, we are concerned with the global well-posedness of the stochastic forced 3D Navier-Stokes equations in the $\mathbb{H}^{1/2}$-space under general initial conditions, where the stochastic forcing comprises a transport forcing and a nonlocal turbulent forcing. In this setting, the random noise is shown to provide a regularization effect on the energy estimates, which we obtain by constructing suitable Lyapunov functions. However, its nonlocality also brings analytical challenges. We develop a bootstrap type estimate based on the kinematic viscosity together with a delicate stopping time argument to prove the global existence and uniqueness of solutions, as well as continuous dependence on the initial value. Furthermore, we also investigated the long-time behavior of the stochastic forced 3D Navier-Stokes equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_15223 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stochastic Forced 3D Navier-Stokes Equations in $\mathbb{H}^{1/2}$-Space Hong, Wei Li, Shihu Liu, Wei Probability In the classical work [FK], Fujita and Kato established the local existence of solutions to the 3D Navier-Stokes equations in the critical $\mathbb{H}^{1/2}$-space. In this paper, we are concerned with the global well-posedness of the stochastic forced 3D Navier-Stokes equations in the $\mathbb{H}^{1/2}$-space under general initial conditions, where the stochastic forcing comprises a transport forcing and a nonlocal turbulent forcing. In this setting, the random noise is shown to provide a regularization effect on the energy estimates, which we obtain by constructing suitable Lyapunov functions. However, its nonlocality also brings analytical challenges. We develop a bootstrap type estimate based on the kinematic viscosity together with a delicate stopping time argument to prove the global existence and uniqueness of solutions, as well as continuous dependence on the initial value. Furthermore, we also investigated the long-time behavior of the stochastic forced 3D Navier-Stokes equations. |
| title | Stochastic Forced 3D Navier-Stokes Equations in $\mathbb{H}^{1/2}$-Space |
| topic | Probability |
| url | https://arxiv.org/abs/2511.15223 |