Non-uniqueness of smooth solutions to the Navier-Stokes equations on torus $\TTT^2$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911461417156608 |
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| author | Miao, Changxing Nie, Yao Ye, Weikui |
| author_facet | Miao, Changxing Nie, Yao Ye, Weikui |
| contents | The local well-posedness theory for the incompressible Navier-Stokes equations in $\BMO^{-1}$ has attracted considerable attention over the past two decades. In a recent breakthrough, Coiculescu and Palasek (Invent. Math., 2025) settled the three-dimensional case by demonstrating the existence of two distinct global solutions, both smooth for $t>0$, evolving from a common initial datum in ${\rm BMO}^{-1}(\mathbb{T}^3)$. However, the two-dimensional case remains open. In this paper, we solve the two-dimensional problem. Unlike its three-dimensional counterpart, the two-dimensional setting presents additional difficulties stemming from the geometric intersections of two-dimensional Mikado flows. To overcome these difficulties, we develop a heat-dominated Fourier mode flow built upon steady two-dimensional Euler flows, and present the proof using a new iterative scheme. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19074 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-uniqueness of smooth solutions to the Navier-Stokes equations on torus $\TTT^2$ Miao, Changxing Nie, Yao Ye, Weikui Analysis of PDEs The local well-posedness theory for the incompressible Navier-Stokes equations in $\BMO^{-1}$ has attracted considerable attention over the past two decades. In a recent breakthrough, Coiculescu and Palasek (Invent. Math., 2025) settled the three-dimensional case by demonstrating the existence of two distinct global solutions, both smooth for $t>0$, evolving from a common initial datum in ${\rm BMO}^{-1}(\mathbb{T}^3)$. However, the two-dimensional case remains open. In this paper, we solve the two-dimensional problem. Unlike its three-dimensional counterpart, the two-dimensional setting presents additional difficulties stemming from the geometric intersections of two-dimensional Mikado flows. To overcome these difficulties, we develop a heat-dominated Fourier mode flow built upon steady two-dimensional Euler flows, and present the proof using a new iterative scheme. |
| title | Non-uniqueness of smooth solutions to the Navier-Stokes equations on torus $\TTT^2$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.19074 |