[Paper Review] Optimal Prandtl expansion around concave boundary layer

David Gérard‐Varet, Yasunori Maekawa+1 more|arXiv (Cornell University)|May 11, 2020

Navier-Stokes equation solutions22 references14 citations

TL;DR

This paper establishes an optimal Gevrey stability result for Prandtl boundary layer expansions in the presence of mild concavity in the boundary layer profile. By analyzing the linearized Navier-Stokes equations around a multiscale asymptotic expansion, the authors prove uniform-in-viscosity estimates in Gevrey regularity classes, extending and improving prior analytic stability results to non-strictly concave settings, thus providing a robust justification of Prandtl's theory under broader conditions.

ABSTRACT

We provide an optimal Gevrey stability result for general boundary layer expansions, under a mild concavity condition on the boundary layer profile. Our result generalizes (and even improves in the non strictly concave case) the one obtained in [Gerard-Varet et al, Duke Math. J. 167 (2018)], restricted to expansions of shear flow type.

Motivation & Objective

• To establish uniform stability estimates for Prandtl boundary layer expansions in the high Reynolds number limit.
• To generalize prior analytic stability results to cases with mild concavity in the boundary layer profile, including non-strictly concave cases.
• To provide a rigorous justification of Prandtl's multiscale asymptotic expansion for Navier-Stokes flows under Gevrey regularity assumptions.
• To overcome limitations of previous analytic frameworks by introducing a refined functional setting that captures the optimal regularity threshold for stability.

Proposed method

• Formulating the Navier-Stokes equations in a high Reynolds number regime with a multiscale asymptotic expansion involving inner (boundary layer) and outer (Euler) profiles.
• Analyzing the perturbation equation derived from the difference between the Navier-Stokes solution and the Prandtl expansion, focusing on linearized dynamics.
• Employing Gevrey regularity classes to characterize initial data and solutions, allowing for a sharp control of high-frequency modes.
• Applying weighted energy estimates and commutator estimates in Sobolev spaces with variable coefficients to control the growth of perturbations.
• Using the Biot-Savart law and streamfunction formulations to relate vorticity to velocity fields under divergence-free and no-slip constraints.
• Introducing a cut-off function and frequency localization to isolate and estimate high-frequency contributions in the perturbation expansion.

Experimental results

Research questions

• RQ1 Can Prandtl boundary layer expansions be justified in Gevrey regularity classes under mild concavity conditions?
• RQ2 What is the optimal regularity threshold for stability of Prandtl expansions beyond analyticity?
• RQ3 How does concavity of the boundary layer profile affect the growth of perturbations in the linearized Navier-Stokes system?
• RQ4 Can the stability framework be extended to non-strictly concave profiles where previous analytic methods fail?

Key findings

• The paper establishes optimal Gevrey stability for Prandtl expansions under a mild concavity condition on the boundary layer profile, generalizing and improving upon earlier analytic results.
• The stability result holds uniformly in the viscosity parameter ν, ensuring that the approximation remains valid over time intervals of order one as ν → 0.
• The method yields uniform estimates in Gevrey norms, with the decay rate of high-frequency modes matching the optimal threshold for stability.
• The analysis extends previous results restricted to shear flows to general boundary layer profiles with concave behavior, including non-strictly concave cases.
• The proof relies on a refined energy estimate in weighted Sobolev spaces and a new application of the Biot-Savart law under boundary constraints.