Intermittent singular solutions of the stationary 2D Navier-Stokes equations in sharp Sobolev spaces
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909631873286144 |
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| author | Ashkarian, Estepan Bhargava, Ataleshvara Gismondi, Nicholas Novack, Matthew |
| author_facet | Ashkarian, Estepan Bhargava, Ataleshvara Gismondi, Nicholas Novack, Matthew |
| contents | In this paper we construct non-trivial solutions to the stationary Navier-Stokes equations on the two dimensional torus which lie in $\bigcap_{ε\in (0,1)} L^{2-ε}(\mathbb{T}^2) \cap \dot H^{-ε}(\mathbb{T}^2)$. Due to the fact that our solutions are not square integrable, we must redefine the notion of solution. Our result gives a sharp extension of recent work of Lemarié-Rieusset, who proved a similar result in the space $\dot{H}^{-1} \cap {BMO}^{-1}$. The main new ingredient is the incorporation of intermittency into the construction of the solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_00841 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Intermittent singular solutions of the stationary 2D Navier-Stokes equations in sharp Sobolev spaces Ashkarian, Estepan Bhargava, Ataleshvara Gismondi, Nicholas Novack, Matthew Analysis of PDEs In this paper we construct non-trivial solutions to the stationary Navier-Stokes equations on the two dimensional torus which lie in $\bigcap_{ε\in (0,1)} L^{2-ε}(\mathbb{T}^2) \cap \dot H^{-ε}(\mathbb{T}^2)$. Due to the fact that our solutions are not square integrable, we must redefine the notion of solution. Our result gives a sharp extension of recent work of Lemarié-Rieusset, who proved a similar result in the space $\dot{H}^{-1} \cap {BMO}^{-1}$. The main new ingredient is the incorporation of intermittency into the construction of the solutions. |
| title | Intermittent singular solutions of the stationary 2D Navier-Stokes equations in sharp Sobolev spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.00841 |