Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit

Constantin, Peter; Vicol, Vlad

doi:10.1007/s00332-017-9424-z

Cited by 45 publications

(48 citation statements)

References 19 publications

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“…The total velocity increment w q+1 is defined by 39) and is by construction mean zero and divergence-free. The new velocity field v q+1 is defined as…”

Section: The Velocity Increment and Verification Of The Inductive Estmentioning

confidence: 99%

Convex integration and phenomenologies in turbulence

Buckmaster

Vicol

2020

EMS Surv. Math. Sci.

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114

145

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In this review article we discuss a number of recent results concerning wild weak solutions of the incompressible Euler and Navier-Stokes equations. These results build on the groundbreaking works of De Lellis and Székelyhidi Jr., who extended Nash's fundamental ideas on C 1 flexible isometric embeddings, into the realm of fluid dynamics. These techniques, which go under the umbrella name convex integration, have fundamental analogies with the phenomenological theories of hydrodynamic turbulence [47,188,50,51]. Mathematical problems arising in turbulence (such as the Onsager conjecture) have not only sparked new interest in convex integration, but certain experimentally observed features of turbulent flows (such as intermittency) have also informed new convex integration constructions.First, we give an elementary construction of nonconservative C 0+ x,t weak solutions of the Euler equations, first proven by De Lellis-Székelyhidi Jr. [49,48]. Second, we present Isett's [102] recent resolution of the flexible side of the Onsager conjecture. Here, we in fact follow the joint work [18] of De Lellis-Székelyhidi Jr. and the authors of this paper, in which weak solutions of the Euler equations in the regularity class C 1 /3− x,t are constructed, attaining any energy profile. Third, we give a concise proof of the authors' recent result [20], which proves the existence of infinitely many weak solutions of the Navier-Stokes in the regularity classx . We conclude the article by mentioning a number of open problems at the intersection of convex integration and hydrodynamic turbulence.

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“…The total velocity increment w q+1 is defined by 39) and is by construction mean zero and divergence-free. The new velocity field v q+1 is defined as…”

Section: The Velocity Increment and Verification Of The Inductive Estmentioning

confidence: 99%

Convex integration and phenomenologies in turbulence

Buckmaster

Vicol

2020

EMS Surv. Math. Sci.

Self Cite

114

145

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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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“…The conclusion of the theorem then follows from the above, in view of formulas (3) and (4). The theorem is proved.…”

Section: Proof Of Theorem 12mentioning

confidence: 61%

“…There have been a large amount of research work on vanishing viscosity limit for incompressible Navier-Stokes equations. See for instance [1], [2], [4], [5], [6], [7], [10], [11], [12], [20], [21], [26], [27], [33] and [36].…”

Section: Introductionmentioning

confidence: 99%

Vanishing viscosity limit for homogeneous axisymmetric no-swirl solutions of stationary Navier-Stokes equations

Yan

2019

Journal of Functional Analysis

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1)-homogeneous axisymmetric no-swirl solutions of three dimensional incompressible stationary Navier-Stokes equations which are smooth on the unit sphere minus the north and south poles have been classified. In this paper we study the vanishing viscosity limit of sequences of these solutions. As the viscosity tends to zero, some sequences of solutions C m loc converge to solutions of Euler equations on the sphere minus the poles, while for other sequences of solutions, transition layer behaviors occur. For every latitude circle, there are sequences which C m loc converge respectively to different solutions of the Euler equations on the spherical caps above and below the latitude circle. We give detailed analysis of these convergence and transition layer behaviors.

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Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit

Cited by 45 publications

References 19 publications

Convex integration and phenomenologies in turbulence

Convex integration and phenomenologies in turbulence

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

Vanishing viscosity limit for homogeneous axisymmetric no-swirl solutions of stationary Navier-Stokes equations

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