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

Abstract

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.

Keywords

Stationary Navier–Stokes equations / exterior domain / rotation and flux carrier / 35Q35 / 76D05

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