Pathological solutions of Navier-Stokes equations on $\mathbb{T}^2$ with gradients in Hardy spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911145664708608 |
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| author | Burczak, Jan Hidalgo-Torné, Antonio |
| author_facet | Burczak, Jan Hidalgo-Torné, Antonio |
| contents | For an arbitrary smooth initial datum, we construct multiple nonzero solutions to the $2$d Navier-Stokes equations, with their gradients in the Hardy space $\mathcal{H}^p$ with any $p \in (0,1)$. Thus, in terms of the path space $C(\mathcal{H}^p)$ for vorticity, $p=1$ is the threshold value distinguishing between non-uniqueness and uniqueness regimes. In order to obtain our result, we develop the needed theory of Hardy spaces on periodic domains. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_08168 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pathological solutions of Navier-Stokes equations on $\mathbb{T}^2$ with gradients in Hardy spaces Burczak, Jan Hidalgo-Torné, Antonio Analysis of PDEs Functional Analysis For an arbitrary smooth initial datum, we construct multiple nonzero solutions to the $2$d Navier-Stokes equations, with their gradients in the Hardy space $\mathcal{H}^p$ with any $p \in (0,1)$. Thus, in terms of the path space $C(\mathcal{H}^p)$ for vorticity, $p=1$ is the threshold value distinguishing between non-uniqueness and uniqueness regimes. In order to obtain our result, we develop the needed theory of Hardy spaces on periodic domains. |
| title | Pathological solutions of Navier-Stokes equations on $\mathbb{T}^2$ with gradients in Hardy spaces |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2509.08168 |