Two-dimensional Navier-Stokes flow with measures as initial vorticity

Giga, Yoshikazu; Miyakawa, Tetsuro; Osada, Hirofumi

doi:10.1007/bf00281355

Cited by 151 publications

(191 citation statements)

References 26 publications

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“…Similarly, (18) implies that the quantities above are O(t −(2− 1 p ) ) and O(t −( 3 2 − 1 q ) ), respectively. The rest of this paper is organized as follows.…”

Section: Introductionmentioning

confidence: 95%

See 1 more Smart Citation

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

Gallay

Wayne

2005

Commun. Math. Phys.

156

210

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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.

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“…Similarly, (18) implies that the quantities above are O(t −(2− 1 p ) ) and O(t −( 3 2 − 1 q ) ), respectively. The rest of this paper is organized as follows.…”

Section: Introductionmentioning

confidence: 95%

“…Then w ∈ C 1 ([0, ∞), S(R 2 )) is a classical solution of (12) in S(R 2 ). Moreover, w(ξ, τ ) satisfies a Gaussian lower bound for any τ > 0, see [24] or ( [18], Theorem 3.1). More precisely, there exist positive constants γ and C γ (depending only on |w 0 | 1 ) such that, for all ξ ∈ R 2 and all τ > 0,…”

Section: A Pair Of Lyapunov Functionsmentioning

confidence: 99%

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

Gallay

Wayne

2005

Commun. Math. Phys.

156

210

View full text Add to dashboard Cite

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“…Combining the results 1-3, we can then proceed along similar lines as [GMO88]. The uniqueness of the solutions from the above theorem seems to be a difficult open problem.…”

Section: Introductionmentioning

confidence: 74%

“…The existence problem becomes more difficult and was solved only in the 1980s in [C86,GMO88], see also [BA94,K94]. Uniqueness is again a subtle issue and is known only in the class (1.10) of 2d solutions, see [GG05].…”

Section: Introductionmentioning

confidence: 99%

On the Cauchy Problem for Axi-Symmetric Vortex Rings

Feng

Šverák

2014

Arch Rational Mech Anal

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We consider the classical Cauchy problem for the 3d Navier-Stokes equation with the initial vorticity ω 0 concentrated on a circle, or more generally, a linear combination of such data for circles with common axis of symmetry. We show that natural approximations of the problem obtained by smoothing the initial data satisfy good a-priori estimates which enable us to conclude that the original problem with the singular initial distribution of vorticity has a solution. We impose no restriction on the size of the initial data.

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“…On the other hand, the two-dimensional Navier-Stokes equations are known to be globally wellposed when the initial vorticity field is given by the point vortices of the form (2.7) below. In particular, the uniqueness of solutions is also available, which is proved in [57,76] under smallness condition on the total variation N i= |α i |, and in [42] without any smallness condition on the size of α i . The inviscid limit problem for point vortices is rigorously analyzed in [105,106,43] …”

Section: )mentioning

confidence: 99%

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

Maekawa¹,

Mazzucato²

2016

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

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The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.

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Two-dimensional Navier-Stokes flow with measures as initial vorticity

Cited by 151 publications

References 26 publications

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

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

On the Cauchy Problem for Axi-Symmetric Vortex Rings

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

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