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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2605.29934 |
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| _version_ | 1866911728396140544 |
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| author | Chen, Zipeng Liu, Song Yin, Zhaoyang |
| author_facet | Chen, Zipeng Liu, Song Yin, Zhaoyang |
| contents | In this paper, we consider the generalized Navier-Stokes equations with fritional dissipation $(-Δ)^β$ with $β>\frac{1}{2}$. When $β\in(1,2)$, We prove that smooth solutions of the generalized Navier-Stokes equations are non-unique with arbitrarily small initial data in $\dot{B}^{-β-α}_{\infty,1}(\mathbb{T}^d)$ for any $α>0$. It is worth pointing out that the space $\dot{B}^{-β-α}_{\infty,1}(\mathbb{T}^d)$ is subcritical for $0<α<β-1$. To the best of our knowledge, this is the first non-uniqueness result of Navier-Stokes equations with initial data at the critical regularity. To show the sharpness of the above results, for $β>\frac{1}{2}$, we establish the local well-poseness of the generalized Navier-Stokes equations with small initial data in $\dot{B}^{-β-α}_{\infty,\infty}(\mathbb{T}^d)$ with $α<0$ and $α\leqβ-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29934 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-uniqueness of generalized Navier-Stokes equations in subcritical spaces Chen, Zipeng Liu, Song Yin, Zhaoyang Analysis of PDEs In this paper, we consider the generalized Navier-Stokes equations with fritional dissipation $(-Δ)^β$ with $β>\frac{1}{2}$. When $β\in(1,2)$, We prove that smooth solutions of the generalized Navier-Stokes equations are non-unique with arbitrarily small initial data in $\dot{B}^{-β-α}_{\infty,1}(\mathbb{T}^d)$ for any $α>0$. It is worth pointing out that the space $\dot{B}^{-β-α}_{\infty,1}(\mathbb{T}^d)$ is subcritical for $0<α<β-1$. To the best of our knowledge, this is the first non-uniqueness result of Navier-Stokes equations with initial data at the critical regularity. To show the sharpness of the above results, for $β>\frac{1}{2}$, we establish the local well-poseness of the generalized Navier-Stokes equations with small initial data in $\dot{B}^{-β-α}_{\infty,\infty}(\mathbb{T}^d)$ with $α<0$ and $α\leqβ-1$. |
| title | Non-uniqueness of generalized Navier-Stokes equations in subcritical spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.29934 |