A note on Prandtl boundary layers

By Yan Guo, Toan T. Nguyen. 2010

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This note concerns nonlinear ill‐posedness of the Prandtl equation and an invalidity of asymptotic boundary layer expansions of incompressible fluid flows near a solid boundary. Our analysis is built upon recent remarkable linear illposedness results established by Gérard‐Varet and Dormy and an analysis by Guo and Tice. We show that the asymptotic boundary layer expansion is not valid for nonmonotonic shear layer flows in Sobolev spaces. We also introduce a notion of weak well‐posedness and prove that the nonlinear Prandtl equation is not well‐posed in this sense near nonstationary and nonmonotonic shear flows. On the other hand, we are able to verify that Oleinik's monotonic solutions are well‐posed. © 2011 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ea0457e3a2bde9b977c4662e80e1380f9758c9ca

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Citations in the corpus

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]).”
Recent progresses in boundary layer theory (2015, background): “See also various perspectives in singular perturbations and boundary layers in [1, 12, 13, 14, 25, 26, 31, 53, 47, 48, 90, 99, 101, 120, 137, 142, 143].”
Global existence and decay of solutions to Prandtl system with small analytic data. (2019, unknown): “The nonlinear ill-posedness was also established in [12, 13] in the sense of non-Lipschtiz continuity of the flow.”
Well-posedness of the Prandtl equation in Sobolev spaces (2012, background): “Along this direction, Guo and Nguyen in [9] proved that the nonlinear Prandtl equation is ill-posed near non-stationary and non-monotonic shear flows, and showed that the asymptotic boundary-layer expansion is not valid for nonmonotonic shear layer flows in Sobolev spaces.”
Well-posedness of the Prandtl equation without any structural assumption (2018, background): “Surprisingly, it was shown in [9] that the Prandtl system is ill-posed in the Sobolev setting ( cf. [13, 18, 24] for improvements).”
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”
Almost Global Existence for the Prandtl Boundary Layer Equations (2015, background): “7) can be justified rigorously [12,30,38], or to show that it fails [8–11,13,16,18].”
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”
Uniform Regularity and Vanishing Viscosity Limit for the Free Surface Navier–Stokes Equations (2012, background): “ot always possible to even constuct an approximate solution. Moreover, even when this is possible, some instabilities make impossible the control of the remainder on a suitable interval of time [22], [27]. Consequently, in a domain with ﬁxed boundary with homogeneous Dirichlet boundary condition, th”
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] .”