Vorticity Measures and the Inviscid Limit

Constantin, Peter; Filho, Milton Lopes; Lopes, Helena J. Nussenzveig; Vicol, Vlad

doi:10.1007/s00205-019-01398-1

Cited by 14 publications

(11 citation statements)

References 29 publications

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“…We emphasize that to date, even the question of whether the weak L 2 t L 2 x inviscid limit holds (against test functions compactly supported in the interior of the domain), remains open. Conditional results have been established recently in terms of interior structure functions [9,11], or in terms of interior vorticity concentration measures [8].…”

Section: Introductionmentioning

confidence: 99%

The Inviscid Limit for the Navier–Stokes Equations with Data Analytic Only Near the Boundary

Kukavica

Vicol

Wang

2020

Arch Rational Mech Anal

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We address the inviscid limit for the Navier-Stokes equations in a half space, with initial datum that is analytic only close to the boundary of the domain, and has finite Sobolev regularity in the complement. We prove that for such data the solution of the Navier-Stokes equations converges in the vanishing viscosity limit to the solution of the Euler equation, on a constant time interval.April 12, 2019 1 for the solution u of (1.1)-(1.4). Hereū denotes the real-analytic solution of the Euler equations, and u P is the real-analytic solution of the Prandtl boundary layer equations. We refer the reader to [1,10,17,28,35,36,37,39,47,51,52] for the well-posedness theory for the Prandtl equations, to [14,26,18,38] for the identification of ill-posed regimes, and to [19,20,21,22,23] for recent works which show the invalidity of the Prandtl expansion at the level of Sobolev regularity. In [52, 53] Sammartino-Caflisch carefully analyze the error terms in the expansion (1.6), and show that they remain O( √ ν) for an O(1) time interval, by appealing to real-analyticity and an abstract Cauchy-Kowalevski theorem. This strategy has been proven successful for treating the case of a channel [40,34] and the exterior of a disk [5]. Subsequently, in a remarkable work [44], Maekawa proved that the inviscid limit also holds for initial datum whose associated vorticity is Sobolev smooth and is supported at an O(1) distance away from the boundary of the domain. The main new device in [44] is the use of the vorticity boundary condition in the case of the half space [2, 43], using which one may actually establish the validity of the expansion (1.6). Using conormal Sobolev spaces, the authors of [55] have obtained an energy based proof for the Caflisch-Sammartino result, while in [12, 13] it is shown that Maekawa's result can also be proven solely using energy methods, in 2D and 3D respectively. More recently, Nguyen-Nguyen have found in [50] a very elegant proof of the Sammartino-Caflisch result, which for the first time completely avoids the usage of Prandtl boundary layer correctors. Instead, Nguyen-Nguyen appeal to the boundary vorticity formulation, precise bounds for the associated Green's function, and an analysis in boundary-layer weighted spaces. In this paper we use a number of estimates from [50], chief among which are the ones for the Green's function for the Stokes system (see Lemma 3.4 below). Lastly, we mention that in a recent remarkable result [16], Gerard-Varet-Maekawa-Masmoudi establish the stability in a Gevrey topology in x and a Sobolev topology in y, of Euler+Prandtl shear flows (cf. (1.6)), when the Prandtl shear flow is both monotonic and concave. It is worth noting that in all the above cases the Prandtl expansion (1.6) is valid, and thus the Kato-criterion (1.5) holds. However, in general there is a large discrepancy between the question of the vanishing viscosity limit in the energy norm, and the problem of the validity of the Prandtl expansion. It is not clear to which degree these two problems are related.

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Section: Introductionmentioning

confidence: 99%

The Inviscid Limit for the Navier–Stokes Equations with Data Analytic Only Near the Boundary

Kukavica

Vicol

Wang

2020

Arch Rational Mech Anal

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“…x solutions u E of the Euler equations can be obtained as weak L 2 t,x limits of u ν . In [12], the authors further extend the result of [13] to allow certain interior vorticity concentration.…”

Section: Introductionmentioning

confidence: 86%

A Kato-type criterion for vanishing viscosity near the Onsager's critical regularity

Chen¹,

Liang²,

Wang³

2020

Preprint

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We consider a vanishing viscosity sequence of weak solutions of the threedimensional Navier-Stokes equations on a bounded domain. In a seminal paper [25] Kato showed that for sufficiently regular solutions, the vanishing viscosity limit is equivalent to having vanishing viscous dissipation in a boundary layer of width proportional to the viscosity. We prove that Kato's criterion holds for Hölder continuous solutions with the regularity index arbitrarily close to the Onsager's critical exponent through a new boundary layer foliation and a global mollification.

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“…, and the second is to apply the multiplier 𝑣 𝑦𝑦𝑦𝑦 𝑤 2 0 . The multiplier 𝑞 0 leads to a delicate interaction between the 𝜕 4 𝑦 operator and the Rayleigh term −𝜕 𝑦 {𝑢 2 𝑠 𝜕 𝑦 𝑞 0 }. The key estimate we prove in this direction is the positivity…”

Section: Overview Of Proofmentioning

confidence: 99%

“…The related question of 𝐿 2 (in space) convergence of Navier-Stokes flows to an Euler flow has been studied by several authors. We refer the reader to [4][5][6][7]36], and [53] for some works in this direction.…”

Section: Other Workmentioning

confidence: 99%

Validity of steady Prandtl layer expansions

Guo

Iyer

2023

Comm Pure Appl Math

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Let the viscosity for the 2D steady Navier‐Stokes equations in the region and with no slip boundary conditions at . For , we justify the validity of the steady Prandtl layer expansion for scaled Prandtl layers, including the celebrated Blasius boundary layer. Our uniform estimates in ε are achieved through a fixed‐point scheme: for solving the Navier‐Stokes equations, where are the tangential and normal velocities at , DNS stands for of the vorticity equation for the normal velocity v, and the compatibility ODE for at .

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Vorticity Measures and the Inviscid Limit

Cited by 14 publications

References 29 publications

The Inviscid Limit for the Navier–Stokes Equations with Data Analytic Only Near the Boundary

The Inviscid Limit for the Navier–Stokes Equations with Data Analytic Only Near the Boundary

A Kato-type criterion for vanishing viscosity near the Onsager's critical regularity

Validity of steady Prandtl layer expansions

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