Sharp non-uniqueness for the Navier-Stokes equations in scaling critical spaces
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911517264314368 |
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| author | Fujii, Mikihiro |
| author_facet | Fujii, Mikihiro |
| contents | It is known that uniqueness of mild solutions to the incompressible Navier-Stokes equations holds in the critical class $C([0,T);L^n(\mathbb{R}^n))$ for $n \geqslant 3$. In this paper, we prove that this result is sharp in the sense that uniqueness fails if $L^n(\mathbb{R}^n)$ is replaced by some scaling critical spaces that are even slightly larger. We achieve this through a complete classification for every pair $(p,q)$ of whether uniqueness of mild solutions in the critical Besov class $C([0,T);\dot{B}_{p,q}^{n/p-1}(\mathbb{R}^n))$ holds or not. Our non-uniqueness mechanism produces infinitely many global solutions emanating even from zero initial state, whose large-time asymptotics are governed by non-trivial stationary flow. To the best of our knowledge, such non-unique solutions provide the first examples of non-dissipative unforced Navier-Stokes flow with critical regularity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19846 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp non-uniqueness for the Navier-Stokes equations in scaling critical spaces Fujii, Mikihiro Analysis of PDEs It is known that uniqueness of mild solutions to the incompressible Navier-Stokes equations holds in the critical class $C([0,T);L^n(\mathbb{R}^n))$ for $n \geqslant 3$. In this paper, we prove that this result is sharp in the sense that uniqueness fails if $L^n(\mathbb{R}^n)$ is replaced by some scaling critical spaces that are even slightly larger. We achieve this through a complete classification for every pair $(p,q)$ of whether uniqueness of mild solutions in the critical Besov class $C([0,T);\dot{B}_{p,q}^{n/p-1}(\mathbb{R}^n))$ holds or not. Our non-uniqueness mechanism produces infinitely many global solutions emanating even from zero initial state, whose large-time asymptotics are governed by non-trivial stationary flow. To the best of our knowledge, such non-unique solutions provide the first examples of non-dissipative unforced Navier-Stokes flow with critical regularity. |
| title | Sharp non-uniqueness for the Navier-Stokes equations in scaling critical spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.19846 |