Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation

By Thierry Gallay, C. Wayne. 2004

224 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/f93684d23a2b780d08b9bd0317e1c1770cf5662f

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

End-point maximal regularity and its application to two-dimensional Keller–Segel system (2010, unknown): “where ∇ ⊥ = (∇2, −∇1) denotes the skew adjoint gradient in R 2 (cf, Giga-Miyakawa-Osada [28], Giga-Kambe [26], Gallay-Wayne [ 25 ]).”
Hypocoercive Diffusion Operators (2006, background): “Thierry Gallay and Eugene Wayne [11] have studied this phenomenon rigorously for a two-dimensional incompressible viscous fluid in the whole space: If ω = ωt(x) is the vorticity, the equation is just ∂ω ∂t + BS[ω] · ∇ω = ω, where BS[ω] is the velocity field obtained from the vorticity ω via the Biot”
Existence, Uniqueness and Lipschitz Dependence for Patlak–Keller–Segel and Navier–Stokes in $${\mathbb{R}^2}$$R2 with Measure-Valued Initial Data (2012, methodology): “12 in [36] but the proof deviates in several key places due to the different nature of the linear operator.”
Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data (2012, methodology): “As H1(m + 1) ↪→↪→ L2(m) by the Rellich– Kondrachov embedding theorem, it suffices to prove that ∇wi (τ ) is uniformly bounded in L2(m) for all m > 2, for which we proceed similar to what is done to prove analogous statements in [33,36], with the necessary alterations to deal with the divergence of t”
Non-uniqueness of Leray solutions of the forced Navier-Stokes equations (2021, background): “The infinitesimal structure of these vortex filaments is the two-dimensional Lamb–Oseen vortex, whose stability was demonstrated in [27] (see also [23, 24]).”
Hypocoercivity (2006, methodology): “Oseen vortices are certain self-similar solutions to the two-dimensional incompressible Navier–Stokes equation, in vorticity formulation [24, 25].”