Rigidity for homogeneous solutions to the two-dimensional Euler equations in sector-type domains
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| Format: | Preprint |
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2025
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| _version_ | 1866917160635334656 |
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| author | Li, Li Yan, Xukai Yang, Zhibo |
| author_facet | Li, Li Yan, Xukai Yang, Zhibo |
| contents | We study the rigidity problem for $(-α)$-homogeneous solutions to the two-dimensional incompressible stationary Euler equations in sector-type domains $Ω_{a, b, θ_0}:= \{(r,θ): a<r<b, \ 0<θ<θ_0\}$, where $α\in\mathbb{R}$, $0\leqslant a < b \leqslant +\infty$ and $0< θ_0 \leqslant 2π$. For each type of domains, depending on whether $a = 0$ or $a > 0$, and $b = +\infty$ or $b < +\infty$, we show that if a solution satisfies some homogeneity assumptions on the boundary of $Ω_{a, b, θ_0}$ and if the radial or angular component of the velocity does not vanish in $\overline{Ω_{a, b, θ_0}}\setminus\{\bm{0}\}$, then it must be homogeneous throughout $\overline{Ω_{a, b, θ_0}}\setminus\{\bm{0}\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_18700 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigidity for homogeneous solutions to the two-dimensional Euler equations in sector-type domains Li, Li Yan, Xukai Yang, Zhibo Analysis of PDEs We study the rigidity problem for $(-α)$-homogeneous solutions to the two-dimensional incompressible stationary Euler equations in sector-type domains $Ω_{a, b, θ_0}:= \{(r,θ): a<r<b, \ 0<θ<θ_0\}$, where $α\in\mathbb{R}$, $0\leqslant a < b \leqslant +\infty$ and $0< θ_0 \leqslant 2π$. For each type of domains, depending on whether $a = 0$ or $a > 0$, and $b = +\infty$ or $b < +\infty$, we show that if a solution satisfies some homogeneity assumptions on the boundary of $Ω_{a, b, θ_0}$ and if the radial or angular component of the velocity does not vanish in $\overline{Ω_{a, b, θ_0}}\setminus\{\bm{0}\}$, then it must be homogeneous throughout $\overline{Ω_{a, b, θ_0}}\setminus\{\bm{0}\}$. |
| title | Rigidity for homogeneous solutions to the two-dimensional Euler equations in sector-type domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.18700 |