Existence of the planar stationary flow in the presence of interior sources and sinks in an exterior domain

• Published in: arXiv.org
• Main Authors: Li, Zijin, Pan, Xinghong
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 24.12.2024
• Subjects: Fluid flow; Incompressible flow; Incompressible fluids; Perturbation; Rotating disks; Two dimensional flow
• ISSN: 2331-8422

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.