Removable singularity of (-1)-homogeneous solutions of stationary Navier-Stokes equations
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912390776356864 |
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| author | Li, Li Li, YanYan Yan, Xukai |
| author_facet | Li, Li Li, YanYan Yan, Xukai |
| contents | We study the removable singularity problem for $(-1)$-homogeneous solutions of the three-dimensional incompressible stationary Navier-Stokes equations with singular rays. We prove that any local $(-1)$-homogeneous solution $u$ near a potential singular ray from the origin, which passes through a point $P$ on the unit sphere $\mathbb{S}^2$, can be smoothly extended across $P$ on $\mathbb{S}^2$, provided that $u=o(\ln \text{dist} (x, P))$ on $\mathbb{S}^2$. The result is optimal in the sense that for any $α>0$, there exists a local $(-1)$-homogeneous solution near $P$ on $\mathbb{S}^2$, such that $\lim_{x\in \mathbb{S}^2, x\to P}|u(x)|/\ln |x'|=-α$. Furthermore, we discuss the behavior of isolated singularities of $(-1)$-homogeneous solutions and provide examples from the literature that exhibit varying behaviors. We also present an existence result of solutions with any finite number of singular points located anywhere on $\mathbb{S}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11170 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Removable singularity of (-1)-homogeneous solutions of stationary Navier-Stokes equations Li, Li Li, YanYan Yan, Xukai Analysis of PDEs We study the removable singularity problem for $(-1)$-homogeneous solutions of the three-dimensional incompressible stationary Navier-Stokes equations with singular rays. We prove that any local $(-1)$-homogeneous solution $u$ near a potential singular ray from the origin, which passes through a point $P$ on the unit sphere $\mathbb{S}^2$, can be smoothly extended across $P$ on $\mathbb{S}^2$, provided that $u=o(\ln \text{dist} (x, P))$ on $\mathbb{S}^2$. The result is optimal in the sense that for any $α>0$, there exists a local $(-1)$-homogeneous solution near $P$ on $\mathbb{S}^2$, such that $\lim_{x\in \mathbb{S}^2, x\to P}|u(x)|/\ln |x'|=-α$. Furthermore, we discuss the behavior of isolated singularities of $(-1)$-homogeneous solutions and provide examples from the literature that exhibit varying behaviors. We also present an existence result of solutions with any finite number of singular points located anywhere on $\mathbb{S}^2$. |
| title | Removable singularity of (-1)-homogeneous solutions of stationary Navier-Stokes equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.11170 |