Existence of the planar stationary flow in the presence of interior sources and sinks in an exterior domain
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| Format: | Preprint |
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2024
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| _version_ | 1866915102955929600 |
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| author | Li, Zijin Pan, Xinghong |
| author_facet | Li, Zijin Pan, Xinghong |
| contents | In the paper, we consider the solvability of the two-dimensional Navier-Stokes equations in an exterior unit disk. On the boundary of the disk, the tangential velocity is subject to the perturbation of a rotation, and the normal velocity is subject to the perturbation of an interior sources or sinks. At infinity, the flow stays at rest. We will construct a solution to such problem, whose principal part admits a critical decay $O(|x|^{-1})$. The result is related to an open problem raised by V. I. Yudovich in [{\it Eleven great problems of mathematical hydrodynamics}, Mosc. Math. J. 3 (2003), no. 2, 711--737], where Problem 2b states that: {\em Prove or disprove the global existence of stationary and periodic flows of a viscous incompressible fluid in the presence of interior sources and sinks.} Our result partially gives a positive answer to this open in the exterior disk for the case when the interior source or sink is a perturbation of the constant state. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_18474 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of the planar stationary flow in the presence of interior sources and sinks in an exterior domain Li, Zijin Pan, Xinghong Analysis of PDEs 35Q35, 76D05 In the paper, we consider the solvability of the two-dimensional Navier-Stokes equations in an exterior unit disk. On the boundary of the disk, the tangential velocity is subject to the perturbation of a rotation, and the normal velocity is subject to the perturbation of an interior sources or sinks. At infinity, the flow stays at rest. We will construct a solution to such problem, whose principal part admits a critical decay $O(|x|^{-1})$. The result is related to an open problem raised by V. I. Yudovich in [{\it Eleven great problems of mathematical hydrodynamics}, Mosc. Math. J. 3 (2003), no. 2, 711--737], where Problem 2b states that: {\em Prove or disprove the global existence of stationary and periodic flows of a viscous incompressible fluid in the presence of interior sources and sinks.} Our result partially gives a positive answer to this open in the exterior disk for the case when the interior source or sink is a perturbation of the constant state. |
| title | Existence of the planar stationary flow in the presence of interior sources and sinks in an exterior domain |
| topic | Analysis of PDEs 35Q35, 76D05 |
| url | https://arxiv.org/abs/2412.18474 |