Well-posedness for the Prandtl system without analyticity or monotonicity

By Davdi Gerard-Varet, Nader Masmoudi. 2013

203 citations recorded by the source corpus.

It has been thought for a while that the Prandtl system is only well-posed under the Oleinik monotonicity assumption or under an analyticity assumption. We show that the Prandtl system is actually locally well-posed for data that belong to the Gevrey class 7/4 in the horizontal variable x. Our result improves the classical local well-posedness result for data that are analytic in x (that is Gevrey class 1). The proof uses new estimates, based on non-quadratic energy functionals.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/11a5324ea7e03194803ac122d7620a8805f09a77

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In Applied Mathematics

In Computational Mechanics

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Citations in the corpus

Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations (2013, background): “It is worth noting that unlike, for example, [33] where the Gevrey regularity is required due to the linear growth of high frequencies, here (and [68]) the Gevrey regularity is required because of a potential nonlinear frequency cascade.”
Well‐Posedness in Gevrey Function Space for 3D Prandtl Equations without Structural Assumption (2020, methodology): “ce dimensional [9]. The well-posedness theories in function spaces of smooth functions was proved in [31] with justiﬁcation of the Prandtl ansatz when the data is analytic; and then it was studied in [15] for two space dimension with Gevrey index = 7 4 that was improved in [22] to the Gevrey index i”
Gevrey stability of Prandtl expansions for 2-dimensional Navier–Stokes flows (2016, background): “More recently, local Gevrey wellposedness for data with non-degenerate critical points was shown in [6].”
Well-posedness of the Prandtl equation without any structural assumption (2018, background): “This is for instance the reason why article [12] is limited to the special case UE = 0 and decaying initial data.”
Local‐in‐Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods (2012, methodology): “We also point out that the method developed in this paper was applied to prove the well-posedness for Prandtl equations in other classes of data in [9, 20] very recently.”
Almost Global Existence for the Prandtl Boundary Layer Equations (2015, background): “11) Inspired by [31] (see also [17,23]), we tackle this issue by considering the linearlygood unknown g(t, x, y) = ω(t, x, y) − u(t, x, y)a(t, y) (2.”
Global existence and decay of solutions to Prandtl system with small analytic data. (2019, unknown): “Lately, Gérvard-Varet and Masmoudi [11] proved the well-posedness of (1.”
Landau Damping: Paraproducts and Gevrey Regularity (2013, background): “Norms such as Gλ(t),σ ;s are common when dealing with analytic or Gevrey regularity, for example, see the works [10,21,22,25,27,45,50,67].”
The Inviscid Limit for the Navier–Stokes Equations with Data Analytic Only Near the Boundary (2019, methodology): “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”
Well-posedness of the hydrostatic Navier–Stokes equations (2018, background): “.5) is ill-posed in Sobolev spaces [12, 15]. Local in time wellposedness can be achieved when the initial datum is real analytic [39, 25], and even under the milder condition of Gevrey regularity in x[14]. We refer to [10, 42, 13, 20, 26, 9] and references therein for more results on the Prandtl sys”
The Inviscid Limit of Navier–Stokes Equations for Analytic Data on the Half-Space (2017, unknown): “Notably, solutions to the Prandtl equations have been constructed for monotonic data ([31, 32, 1, 29]) or data with Gevrey or analytic regularity ([35, 11, 22], among others).”
On the inviscid limit of the Navier-Stokes equations (2014, background): “(d) The initial vorticity has a single curve of non-degenerate critical points, and it lies in the Gevreyclass 7/4 with respect to the tangential variable [GVM13].”