New Liouville type theorems for the stationary Navier-Stokes equations
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909450447618048 |
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| author | Tan, Wenke |
| author_facet | Tan, Wenke |
| contents | We mainly research the Liouville type problem for the stationary Navier-Stokes equations (including the fractional case) in $\mathbb{R}^3$. We first establish a new formula for the Dirichlet integral of solutions and show that the globally defined quantity $\int_{\mathbb{R}^3}|\nabla u|^2dx$ is completely determined by the information of the solution $u$ at the origin in frequency space. From this character, we show some new Liouville type theorems for solutions of the stationary Navier-Stokes equations. Then we extend the obtained results for classical stationary Navier-Stokes equations to the stationary fractional Navier-Stokes equations for $\frac{1}{2}\leq s<1$, especially, we solve the Liouville type problem for $s=\frac{5}{6}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03609 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New Liouville type theorems for the stationary Navier-Stokes equations Tan, Wenke Analysis of PDEs We mainly research the Liouville type problem for the stationary Navier-Stokes equations (including the fractional case) in $\mathbb{R}^3$. We first establish a new formula for the Dirichlet integral of solutions and show that the globally defined quantity $\int_{\mathbb{R}^3}|\nabla u|^2dx$ is completely determined by the information of the solution $u$ at the origin in frequency space. From this character, we show some new Liouville type theorems for solutions of the stationary Navier-Stokes equations. Then we extend the obtained results for classical stationary Navier-Stokes equations to the stationary fractional Navier-Stokes equations for $\frac{1}{2}\leq s<1$, especially, we solve the Liouville type problem for $s=\frac{5}{6}$. |
| title | New Liouville type theorems for the stationary Navier-Stokes equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.03609 |