On the steady Navier–Stokes equations in 2D exterior domains

Korobkov, Mikhail V.; Pileckas, Konstantin; Russo, Roberto

doi:10.1016/j.jde.2020.01.012

Cited by 25 publications

(17 citation statements)

References 22 publications

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“…The assertion of the above theorem is also true when q = 2. This was obtained by Theorem 6] and Korobkov-Plieckas-Russo [12], while they proved estimates for ω and ∇v:…”

Section: Introductionmentioning

confidence: 86%

“…Later on Amick [1] and Korobkov-Plieckas-Russo [12] proved that every solution v of (1.1) with the finite Dirichlet integral (1.2) is bounded as in (1.4), and so necessarily satisfies (1.5) and (1.6). Recently, Korobkov-Plieckas-Russo [13] succeeded to obtain a remarkable result which states that every solution v of (1.1) with (1.2) converges uniformly at infinity, i.e., (1.8) lim…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic properties of steady and nonsteady solutions to the 2D Navier-Stokes equations with finite generalized Dirichlet integral

Kozono¹,

Terasawa²,

Wakasugi³

2019

Preprint

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We consider the stationary and non-stationary Navier-Stokes equations in the whole plane R 2 and in the exterior domain outside of the large circle. The solution v is handled in the class with ∇v ∈ L q for q ≥ 2. Since we deal with the case q ≥ 2, our class is larger in the sense of spacial decay at infinity than that of the finite Dirichlet integral, i.e., for q = 2 where a number of results such as asymptotic behavior of solutions have been observed.For the stationary problem we shall show that ω(x) = o(|x|2 ) as |x| → ∞, where ω ≡ rot v. For the non-stationary problem, a generalized L q -energy identity is clarified. As an application, we prove the Liouville type theorems under the assumption that ω ∈ L q (R 2 ) and ω ∈ L q (0, T ; L q (R 2 )) for the stationary and the non-stationary equations, respectively.

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“…The assertion of the above theorem is also true when q = 2. This was obtained by Theorem 6] and Korobkov-Plieckas-Russo [12], while they proved estimates for ω and ∇v:…”

Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Asymptotic properties of steady and nonsteady solutions to the 2D Navier-Stokes equations with finite generalized Dirichlet integral

Kozono¹,

Terasawa²,

Wakasugi³

2019

Preprint

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show abstract

“…The convergence of w k to w L and the basic properties of w L are summarized in the following lemma. Note that the local uniform convergence is a standard fact, and the properties of w L has been established through the papers [11,2,18,17].…”

Section: Lemma 3 ([11]mentioning

confidence: 99%

Leray's plane stationary solutions at small Reynolds numbers

Korobkov¹,

Ren²

2021

Preprint

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In the celebrated 1933 paper [22] by J. Leray, the invading domains method was proposed to construct D-solutions for the stationary Navier-Stokes flow around obstacle problem. In two dimensions, whether Leray's D-solution achieves the prescribed limiting velocity at spatial infinity became a major open problem since then. In this paper, we solve this problem at small Reynolds numbers. The proof builds on a novel blow-down argument which rescales the invading domains to the unit disc, as well as the ideas developed in the recent works [19,20].

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“…Recently M.Korobkov, K.Pileckas and R.Russo [7] simplified the issue and proved that the first claim (i) holds in the general case of D-solutions without (1.10) assumption: Theorem 1.1 ( [7]). Let u be a D-solution to the Navier-Stokes system (1.2) in the exterior domain Ω ⊂ R 2 .…”

Section: Introductionmentioning

confidence: 99%

On Convergence of Arbitrary D-Solution of Steady Navier–Stokes System in 2D Exterior Domains

Korobkov

Pileckas

Russo

2019

Arch Rational Mech Anal

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We study solutions to stationary Navier-Stokes system in two dimensional exterior domain. We prove that any such solution with finite Dirichlet integral converges to a constant vector at infinity uniformly. No additional condition (on symmetry or smallness, etc.) are assumed. The proofs based on arguments of the classical Amick's article (Acta Math. 1988) and on results of a recent paper by authors (arXiv 1711:02400) where the uniform boundedness of these solutions was established.

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On the steady Navier–Stokes equations in 2D exterior domains

Abstract: We study the boundary value problem for the stationary Navier-Stokes system in two dimensional exterior domain. We prove that any solution of this problem with finite Dirichlet integral is uniformly bounded. Also we prove the existence theorem under zero total flux assumption.

Cited by 25 publications

References 22 publications

Asymptotic properties of steady and nonsteady solutions to the 2D Navier-Stokes equations with finite generalized Dirichlet integral

Asymptotic properties of steady and nonsteady solutions to the 2D Navier-Stokes equations with finite generalized Dirichlet integral

Leray's plane stationary solutions at small Reynolds numbers

On Convergence of Arbitrary D-Solution of Steady Navier–Stokes System in 2D Exterior Domains

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