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Optimal Prandtl
expansion around a concave boundary layer
David Gérard-Varet, Yasunori Maekawa and Nader Masmoudi |
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Vol. 17 (2024), No. 9, 3125–3187
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Abstract |
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We show an optimal stability result for boundary layer solutions of the Navier–Stokes equation in a half-plane, under a mild concavity condition on the boundary layer profile. The key point is the derivation of sharp Gevrey estimates for the linearized Navier–Stokes equation in vorticity form, on a time interval uniform in . As the nonlocal boundary condition on the vorticity prevents us from deriving direct estimates, we use a novel iteration scheme, similar to a splitting method in numerical analysis. Our result is a big step forward compared to our previous work (Duke Math. J. 167 (2018), 2531–2631), where we proved stability of boundary layer expansions of shear flow type. Indeed, the approach of the present paper is much more robust than the one in that previous work, which was based on the Fourier transform and hence only adapted to expansions independent of the tangential variable. Moreover, we are now able to relax the assumption of strict concavity made in our previous work to obtain the optimal Gevrey stability, which was not satisfied by generic boundary layer expansions. We provide in this way the first justification of unsteady boundary layer theory outside the analytic setting. |
Keywords
boundary layer theory, incompressible Navier–Stokes
equations, Prandtl expansion, inviscid limit
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Mathematical Subject Classification
Primary: 35Q30
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Milestones
Received: 20 April 2022
Revised: 8 May 2023
Accepted: 13 June 2023
Published: 1 November 2024
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Authors |
| David Gérard-Varet | |
| Institut de Mathématiques de
Jussieu-Paris Rive Gauche Université Paris Cité UFR de Mathématiques Paris France |
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| Yasunori Maekawa | |
| Department of Mathematics Graduate School of Science Kyoto University Japan |
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| Nader Masmoudi | |
| Science Division New York University Abu Dhabi Abu Dhabi United Arab Emirates |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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