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<p>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 source or sink. At infinity, the flow stays at rest. We will construct a solution to such a problem, whose principal part admits a critical decay <i>O</i>(∣<i>x</i>∣<sup>−1</sup>). The result is related to an open problem raised by V. I. Yudovich in [Mosc. Math. J., 2003, 3(2): 711–737], where Problem 2b states that: <i>Prove or disprove the global existence of stationary and periodic flows of a viscous incompressible fluid in the presence of interior sources and sinks</i>. Our result partially gives a positive answer to this open problem in the exterior disk for the case when the interior source or sink is a perturbation of the constant state.</p>

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Existence of the Planar Stationary Flow in the Presence of Interior Sources and Sinks in an Exterior Domain

  • 期刊论文

  • 发表日期：

    2026年7月5日

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  • Zijin Li,
  • Xinghong Pan

摘要

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 source or sink. At infinity, the flow stays at rest. We will construct a solution to such a 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 [Mosc. Math. J., 2003, 3(2): 711–737], where Problem 2b states that: 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 problem in the exterior disk for the case when the interior source or sink is a perturbation of the constant state.