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

TL;DR

This paper proves the existence of a planar stationary flow in an exterior domain with interior sources and sinks, addressing an open problem in mathematical hydrodynamics by constructing solutions with specific decay properties.

Contribution

It demonstrates the existence of stationary solutions to the 2D Navier-Stokes equations in an exterior disk with interior sources/sinks, partially resolving an open problem posed by Yudovich.

Findings

Constructed solutions with decay rate O(|x|^{-1})

Addressed an open problem on stationary flows with sources and sinks

Provided a positive partial answer for perturbations of constant states

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 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.