The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

Maekawa, Yasunori; Mazzucato, Anna L.

doi:10.1007/978-3-319-13344-7_15

Cited by 31 publications

(23 citation statements)

References 142 publications

(233 reference statements)

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“…For unsteady flows, the validity of an asymptotic expansion of the form (1.5) has been established in the analyticity framework, [As91], [SC98], [SC98], in the Gevrey setting in [GVMM16], for initial vorticity bounded away from the origin in [Mae14], and for special flows in [MT08]. Giving a more exhaustive survey of results in the unsteady setting would lead us astray, and so we refer the reader to the review articles of [E00], [GJT16], and [MM17] and the references therein.…”

Section: Main Theoremmentioning

confidence: 99%

Steady Prandtl Layers Over a Moving Boundary: Nonshear Euler Flows

Iyer¹

2019

SIAM J. Math. Anal.

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Section: Main Theoremmentioning

confidence: 99%

Steady Prandtl Layers Over a Moving Boundary: Nonshear Euler Flows

Iyer¹

2019

SIAM J. Math. Anal.

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“…is equivalent to the validity of the strong inviscid limit in the energy norm. Refinements and extensions based on Kato's original argument of introducing a boundary layer corrector were obtained for instance in [3,6,7,31,33,46,54,56]; see also the recent review [45] and references therein. These results are important because they yield explicit properties that the sequence of Navier-Stokes solutions must obey as ν → 0 in order for them to have a strong L ∞ t L 2…”

Section: Introductionmentioning

confidence: 99%

“…Finally, we mention that the vanishing viscosity limit is also known to hold in the presence of certain symmetry assumptions on the initial data, which is maintained by the flow; see e.g. [4,27,32,41,42,45,48,49] and references therein. This symmetry implies that the influence of the Prandtl layer to the bulk flow is weak, and thus in these situations the vanishing viscosity limit may be established by verifying Kato's criterion (1.5).…”

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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“…A first step towards establishing the validity or the invalidity of the Prandtl expansion (1.1) is a detailed understanding of the Prandtl system (1.2) itself. The well-and ill-posedness of the Prandtl equations has a long history of which we only provide a very brief summary (see the reviews [2,52] for further references). Under the monotonicity assumption ∂ y u P | t=0 > 0, Oleinik [59,60] obtained global in time, regular solutions on the domain [0, L] × R + for small L, and local in time regular solutions for arbitrary finite L. The aforementioned results rely on the Crocco transform, which is available from the monotonicity hypothesis.…”

Section: B)mentioning

confidence: 99%

Real Analytic Local Well‐Posedness for the Triple Deck

Iyer

Vicol

2020

Comm Pure Appl Math

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The Triple Deck model is a classical high order boundary layer model that has been proposed to describe flow regimes where the Prandtl theory is expected to fail. At first sight the model appears to lose two derivatives through the pressure-displacement relation which links pressure to the tangential slip. In order to overcome this, we split the Triple Deck system into two coupled equations: a Prandtl type system on H and a Benjamin-Ono type equation on R. This splitting enables us to extract a crucial leading order cancellation at the top of the lower deck. We develop a functional framework to subsequently extend this cancellation into the interior of the lower deck, which enables us to prove the local well-posedness of the model in tangentially real analytic spaces.

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The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

Cited by 31 publications

References 142 publications

Steady Prandtl Layers Over a Moving Boundary: Nonshear Euler Flows

Steady Prandtl Layers Over a Moving Boundary: Nonshear Euler Flows

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

Real Analytic Local Well‐Posedness for the Triple Deck

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