On the ill-posedness of the Prandtl equation

By Davdi Gerard-Varet, Emmanuel Dormy. 2009

310 citations recorded by the source corpus.

The concern of this paper is the Cauchy problem for the Prandtl equation. This problem is known to be well-posed for analytic data or for data with monotonicity properties. We prove here that it is linearly ill-posed in Sobolev type spaces. The key of the analysis is the construction, at high tangential frequencies, of unstable quasimodes for the linearization around solutions with non-degenerate critical points. Interestingly, the strong instability is due to vicosity, which is coherent with well-posedness results obtained for the inviscid version of the equation. A numerical study of this instability is also provided.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5efdb4645bbc9ae3de7f4e0ddf30ac486974e88f

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Citations in the corpus

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”
Almost Global Existence for the Prandtl Boundary Layer Equations (2015, methodology): “Note that this Gevreyexponent is not in contradictionwith the ill-posedness in Sobloev spaces established by Gerard-Varet and Dormy in [16] for the linearized Prandtl equation around a non-monotonic shear flow.”
A note on Prandtl boundary layers (2010, methodology): “Our work is based on a recent result of Gérard-Varet and Dormy [2] where they established illposedness for the Cauchy problem of the linearized Prandtl equation around non-monotonic shear flows.”
Global existence and decay of solutions to Prandtl system with small analytic data. (2019, unknown): “Gérard-Varet and Dormy [10] proved the ill-posedness in Sobloev spaces for the linearized Prandtl system around non-monotonic shear flows.”
Well‐Posedness in Gevrey Function Space for 3D Prandtl Equations without Structural Assumption (2020, background): “Here we only mention the works [10] about the construction of blowup solutions (see also [5,6]), [17] on the unstable Euler shear flow that yields instability of Prandtl equation, [12, 16, 20] about the instability around a shear flow with a nondegenerate critical point, [18] on the instabilility ev”
Well-posedness of the Prandtl equation without any structural assumption (2018, methodology): “Specifically, paper [9] centers on the linearization of (1)-(2) around shear flows, given by (U, V ) = (Us(y), 0).”
On the local existence of analytic solutions to the Prandtl boundary layer equations (2013, unknown): “Additionally, in [12, 15] it is proven that if one linearizes the system ( 1.”
Well-posedness of the hydrostatic Navier–Stokes equations (2018, background): “arity, and if furthermore the initial data are monotonic in y, (1.5) has local in time Sobolev solutions [34, 33]. On the other hand, without monotonicity, system (1.5) is ill-posed in Sobolev spaces [12, 15]. Local in time wellposedness can be achieved when the initial datum is real analytic [39, 2”
Well-posedness for the Prandtl system without analyticity or monotonicity (2013, methodology): “This non-locality explains the ill-posedness of the Prandtl equation in the Sobolev setting, as can be seen from the mechanism described by the first author and Dormy [6].”
Gevrey stability of Prandtl expansions for 2-dimensional Navier–Stokes flows (2016, background): “ author and Dormy established the ill-posedness of (1.0.3) in the Sobolev setting. More recently, local Gevrey wellposedness for data with non-degenerate critical points was shown in [6]. We refer to [5, 7, 13, 15] for more on the study of the Prandtl equation itself. The present paper is devoted to”
Uniform Regularity and Vanishing Viscosity Limit for the Free Surface Navier–Stokes Equations (2012, background): “Nevertheless, the equation that governs the boundary layers that appear in the formal expansion, the Prandtl equation, can be ill-posed in Sobolev spaces [17], thus it is not always possible to even constuct an approximate solution.”
Well-posedness of the Prandtl equation in Sobolev spaces (2012, background): “PRANDTL EQUATION 3 As mentioned in [2, 6, 7], the local in time well-posedness of the Prandtl equation for initial data in Sobolev space remains an open problem.”
Local‐in‐Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods (2012, background): “More recently, Gérad-Varet and Dormy [7] proved ill-posedness for the linearized Prandtl equation around a nonmonotonic shear flow (see also [10, 8]).”
On the Inviscid Limit for the Compressible Navier–Stokes System in an Impermeable Bounded Domain (2012, unknown): “Yet this approach seems to fail to justify the inviscid limit in general, see [17, 20] and the references therein.”
The Inviscid Limit of Navier–Stokes Equations for Analytic Data on the Half-Space (2017, background): “31, 32, 1, 29]) or data with Gevrey or analytic regularity ([35, 11, 22], among others). In the case of non-monotonic data with Sobolev regularity, the Prandtl boundary layer equations are ill-posed ([8, 12, 20]). The validity of Prandtl’s Ansatz (1.3), and hence the inviscid limit, were established”
On the Inviscid Limit Problem of the Vorticity Equations for Viscous Incompressible Flows in the Half‐Plane (2012, methodology): “Indeed, for the linearized Prandtl equations the ill-posedness in the Sobolev framework is shown in [15], and this result is applied to prove the invalidity of the Lipschitz dependence of solutions on the initial data for the nonlinear Prandtl equations [16, 19] .”