Global well-posedness of the Navier-Stokes equations for small initial data in frequency localized Koch-Tataru's space
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866929760059260928 |
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| author | Cheskidov, Alexey Eguchi, Taichi |
| author_facet | Cheskidov, Alexey Eguchi, Taichi |
| contents | We construct global smooth solutions to the incompressible Navier--Stokes equations in $\mathbb{R}^3$ for initial data in $L^2$ satisfying some smallness condition. The high-frequency part is assumed to be small in $BMO^{-1}$, while the low-frequency part is assumed to be small only in $\dot B^{-1}_{\infty,\infty}$. Since $BMO^{-1}$ is strictly embedded in $\dot B^{-1}_{\infty,\infty}$, our assumption is weaker than that of Koch and Tataru (2001), which we also demonstrate with an example of finite energy divergence-free initial data. Also, our solutions attain the initial data in the strong $L^2$ sense, and hence satisfy the energy balance for all time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11642 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global well-posedness of the Navier-Stokes equations for small initial data in frequency localized Koch-Tataru's space Cheskidov, Alexey Eguchi, Taichi Analysis of PDEs 35A01 We construct global smooth solutions to the incompressible Navier--Stokes equations in $\mathbb{R}^3$ for initial data in $L^2$ satisfying some smallness condition. The high-frequency part is assumed to be small in $BMO^{-1}$, while the low-frequency part is assumed to be small only in $\dot B^{-1}_{\infty,\infty}$. Since $BMO^{-1}$ is strictly embedded in $\dot B^{-1}_{\infty,\infty}$, our assumption is weaker than that of Koch and Tataru (2001), which we also demonstrate with an example of finite energy divergence-free initial data. Also, our solutions attain the initial data in the strong $L^2$ sense, and hence satisfy the energy balance for all time. |
| title | Global well-posedness of the Navier-Stokes equations for small initial data in frequency localized Koch-Tataru's space |
| topic | Analysis of PDEs 35A01 |
| url | https://arxiv.org/abs/2503.11642 |