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

2024/12/24 by Li, Zijin, Pan, Xinghong
#35Q35 #76D05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2412.18474

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.

Related

• Existence of planar non-symmetric stationary flows with large flux in an exterior disk
• Existence and non-uniqueness of classical solutions to the axially symmetric stationary Navier-Stokes equations in an exterior cylinder
• Stability of planar exterior stationary flows with suction
• On the existence of solutions to the planar exterior Navier Stokes system
• Vanishing viscosity limit of the 2D stationary Navier-Stokes equations outside a rotating disc and its application