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

Zijin Li, Xinghong Pan

TL;DR

The paper addresses global existence of stationary viscous flows for the 2D Navier–Stokes equations in the exterior of a unit disk under perturbations that rotate the boundary tangential velocity and introduce interior sources or sinks. By transforming to polar coordinates and performing a Fourier mode decomposition, the authors reduce the problem to linear per-mode analyses of zero and nonzero modes, coupling them with a nonlinear fixed-point framework in subcritical decay spaces. Under a small-data regime and a spectral constraint relating ν and μ, they construct a unique solution whose principal part decays like $O(r^{-1})$, i.e., $\boldsymbol{u}=\frac{ν}{r}\mathbf{e}_r+\frac{μ}{r}\mathbf{e}_θ+\boldsymbol{v}$ with $\boldsymbol{v}$ in a decay space. This provides a partial positive answer to Yudovich’s Problem 2b in exterior domains for perturbations of the constant state, and connects to the broader theory of exterior-domain flow and D-solutions in two dimensions.

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.

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

TL;DR

The paper addresses global existence of stationary viscous flows for the 2D Navier–Stokes equations in the exterior of a unit disk under perturbations that rotate the boundary tangential velocity and introduce interior sources or sinks. By transforming to polar coordinates and performing a Fourier mode decomposition, the authors reduce the problem to linear per-mode analyses of zero and nonzero modes, coupling them with a nonlinear fixed-point framework in subcritical decay spaces. Under a small-data regime and a spectral constraint relating ν and μ, they construct a unique solution whose principal part decays like , i.e., with in a decay space. This provides a partial positive answer to Yudovich’s Problem 2b in exterior domains for perturbations of the constant state, and connects to the broader theory of exterior-domain flow and D-solutions in two dimensions.

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 . 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.

Paper Structure

This paper contains 4 sections, 3 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Formulation of the linear system in Polar coordinates and Fourier mode
  3. Solvability of the Fourier mode in subcritically decayed spaces
  4. The nonlinear solution

Key Result

Theorem 1.2

Assume that or alternatively There exists $\epsilon>0$, being sufficiently small and depending on $\mu$ and $\nu$, such that if then problem PoNS has a unique solution $\boldsymbol{u}$ such that where $\boldsymbol{v}\in\widetilde{\mathcal{B}}_\lambda\,$ that satisfies for some constants $\lambda=3^+$ and $C>0$, depending on $\mu$ and $\nu$. Here $3^+$ denotes a constant which is larger but cl

Theorems & Definitions (8)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 3.1
  • Lemma 3.2