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Global stability of vortex solutions of the two-dimensional Navier-Stokes equation

Thierry Gallay, C. Eugene Wayne

Abstract

Both experimental and numerical studies of fluid motion indicate that initially localized regions of vorticity tend to evolve into isolated vortices and that these vortices then serve as organizing centers for the flow. In this paper we prove that in two dimensions localized regions of vorticity do evolve toward a vortex. More precisely we prove that any solution of the two-dimensional Navier-Stokes equation whose initial vorticity distribution is integrable converges to an explicit self-similar solution called ``Oseen's vortex''. This implies that the Oseen vortices are dynamically stable for all values of the circulation Reynolds number, and our approach also shows that these vortices are the only solutions of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. Finally, under slightly stronger assumptions on the vorticity distribution, we also give precise estimates on the rate of convergence toward the vortex.

Global stability of vortex solutions of the two-dimensional Navier-Stokes equation

Abstract

Both experimental and numerical studies of fluid motion indicate that initially localized regions of vorticity tend to evolve into isolated vortices and that these vortices then serve as organizing centers for the flow. In this paper we prove that in two dimensions localized regions of vorticity do evolve toward a vortex. More precisely we prove that any solution of the two-dimensional Navier-Stokes equation whose initial vorticity distribution is integrable converges to an explicit self-similar solution called ``Oseen's vortex''. This implies that the Oseen vortices are dynamically stable for all values of the circulation Reynolds number, and our approach also shows that these vortices are the only solutions of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. Finally, under slightly stronger assumptions on the vorticity distribution, we also give precise estimates on the rate of convergence toward the vortex.

Paper Structure

This paper contains 17 sections, 29 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Smoothing and compactness properties
  3. Compactness in $L^2(m)$
  4. Compactness in $L^1({\bf R}^2)$
  5. Preservation of positivity
  6. Global convergence results
  7. A pair of Lyapunov functions
  8. Convergence in $L^2(m)$
  9. Convergence in $L^1({\bf R}^2)$
  10. Convergence rate for positive solutions
  11. Local convergence rates
  12. Eigenvalue estimates
  13. Polar coordinates
  14. Gaussian decay of eigenfunctions
  15. Localization of eigenvalues
  16. ...and 2 more sections

Key Result

Theorem 1.1

For all initial data $\omega_0 \in L^1({\bf R}^2)$, equation (V) has a unique global solution $\omega \in C^0([0,\infty),L^1({\bf R}^2)) \cap C^0((0,\infty),L^\infty({\bf R}^2))$ such that $\omega(0) = \omega_0$. Moreover, for all $p \in [1,+\infty]$, there exists $C_p > 0$ such that

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4: gallay:2001a, Theorem 3.2
  • Proposition 1.5
  • Proposition 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.4
  • ...and 28 more