Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation
Thierry Gallay,C. Eugene Wayne +1 more
TLDR
In this article, it was shown 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.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 give precise estimates on the rate of convergence toward the vortex.read more
Citations
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Optimal Transport: Old and New
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
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Emergence of order from turbulence in an isolated planar superfluid.
TL;DR: This work demonstrates that after a period of vortex annihilation the remaining vortices self-organize into two macroscopic coherent "Onsager vortex" clusters that are stable indefinitely--despite the absence of driving or external dissipation in the dynamics.
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Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier–Stokes equations
Margaret Beck,C. Eugene Wayne +1 more
TL;DR: In this article, a dynamical system explanation of the metastability of an explicit family of solutions, referred to as bar states, of the two-dimensional incompressible Navier-Stokes equation on the torus was proposed.
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Space-time decay of Navier-Stokes flows invariant under rotations
TL;DR: In this article, it was shown that the solutions to the nonstationary Navier-Stokes equations in (d = 2,3) which are left invariant under the action of discrete subgroups of the orthogonal group O(d) decay much faster as than than in generic case.
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Space-time decay of Navier-Stokes flows invariant under rotations
TL;DR: In this paper, it was shown that the solutions to the nonstationary Navier-Stokes equations in R^d, $d=2,3$ which are left invariant under the action of discrete subgroups of the orthogonal group $O(d)$ decay much faster as $|x|\to\infty$ or $ t \to ∞$ than in the generic case and for each subgroup, the precise decay rates in space-time of the velocity field.
References
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Theory of Ordinary Differential Equations
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