Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation
Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation
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DOI：
10.1007/s00220-004-1254-9
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发表时间：
2004-02
影响因子：
2.4
通讯作者：
T. Gallay;C. E. Wayne
中科院分区：
文献类型：
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作者：
T. Gallay;C. E. Wayne
对流体运动的实验和数值研究都表明，最初的局部涡量区域倾向于演化为孤立的涡旋，然后这些涡旋成为流动的组织中心。在这篇文章中，我们证明了在二维中，涡度的局域区确实向涡旋演化。更确切地说，我们证明了初始涡度分布为可积的二维Navier-Stokes方程的任何解都收敛到一个显式的自相似解，称为“Osee‘s涡”。这意味着对于所有的环流雷诺数，Oseen涡都是动态稳定的，并且我们的方法也表明这些涡是以Dirac质量为初始涡量的二维N-S方程的唯一解。最后，在涡度分布稍强的假设下，我们给出了向涡旋收敛的速率的精确估计。
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 give precise estimates on the rate of convergence toward the vortex.