[Paper Review] On the $L^\infty$ stability of Prandtl expansions in Gevrey class

Qi Chen, Di Wu+1 more|arXiv (Cornell University)|Apr 21, 2020

Navier-Stokes equation solutions35 references4 citations

TL;DR

This paper establishes $L^∞$ stability of Prandtl expansions for monotone, concave shear flows in the Gevrey class $\frac{3}{2}$ by introducing a direct resolvent estimate method for the linearized Orr-Sommerfeld operator, bypassing the Rayleigh-Airy iteration. The key result is a uniform $L^\infty$ bound without the $t^{1/4}$ prefactor that previously caused blowup as $t \to 0$, proving robust stability under small Gevrey-class perturbations.

ABSTRACT

In this paper, we prove the $L^\infty\cap L^2$ stability of Prandtl expansions of shear flow type as $\big(U(y/\sqrtν),0\big)$ for the initial perturbation in the Gevrey class, where $U(y)$ is a monotone and concave function and $ν$ is the viscosity coefficient. To this end, we develop the direct resolvent estimate method for the linearized Orr-Sommerfeld operator instead of the Rayleigh-Airy iteration method introduced by Grenier, Guo and Nguyen.

Motivation & Objective

• To establish $L^\infty$ stability of Prandtl expansions for monotone and concave shear flows in the Gevrey class $\frac{3}{2}$, overcoming limitations of prior methods.
• To remove the $t^{1/4}$ prefactor in $L^\infty$ estimates that causes blowup as $t \to 0$, ensuring uniform stability bounds.
• To develop a direct resolvent estimate method for the linearized Orr-Sommerfeld operator, replacing the Rayleigh-Airy iteration used in earlier works.
• To provide a new framework applicable to other hydrodynamic stability problems involving boundary layers.

Proposed method

• Develops a direct resolvent estimate method for the linearized Orr-Sommerfeld operator, avoiding iterative constructions used in the Rayleigh-Airy method.
• Employs complex analysis and asymptotic estimates for Airy functions to control the behavior of solutions to the resolvent equation.
• Derives decay estimates for the Green's function via bounds on $|A_0(\kappa \eta)|$ and $|A_0'(\kappa \eta)|$, leveraging known properties of Airy functions in the complex plane.
• Uses weighted $L^2$ norms involving powers of $Y$ to control the growth of derivatives and ensure integrability in the resolvent estimates.
• Applies interpolation techniques between $L^2$ and $H^1$ estimates to derive $L^\infty$ bounds from $L^2$ and $H^1$ control.
• Establishes a lower bound on $|\partial_Y \tilde{\Phi}(0)|$ using integral representations and asymptotic decay of Airy functions.

Experimental results

Research questions

• RQ1 Can $L^\infty$ stability of Prandtl expansions be proven without the $t^{1/4}$ prefactor that diverges as $t \to 0$?
• RQ2 Is a direct resolvent estimate method viable for the linearized Orr-Sommerfeld operator, avoiding the Rayleigh-Airy iteration?
• RQ3 Can the stability of monotone and concave shear flows be established in the Gevrey class $\frac{3}{2}$ under small initial perturbations?
• RQ4 What is the sharp decay rate of the Green's function for the linearized operator in the complex frequency plane?

Key findings

• The paper proves $\nu^{1/4}\|v^\nu(t)\|_{L^\infty} \leq C\|a\|_{G_\gamma}$, removing the $t^{1/4}$ factor that previously caused instability at $t \to 0$.
• The direct resolvent estimate method successfully controls the linearized Orr-Sommerfeld operator without relying on the Rayleigh-Airy iteration.
• A lower bound $|\partial_Y \tilde{\Phi}(0)| \geq C^{-1}(1+|\kappa\eta|)^{-1/2}(\kappa+3\alpha)^{-1}$ is established, crucial for the $L^\infty$ estimate.
• Weighted $L^2$ estimates for $Y^k \partial_Y \tilde{\Phi}$ yield decay rates $\|Y^k \partial_Y \tilde{\Phi}\|_{L^2} \leq C\kappa^{-(2k+3)/2}(1+|\kappa\eta|)^{-(2k+3)/4}$, ensuring integrability.
• The method achieves $L^\infty$ stability in the Gevrey class $\frac{3}{2}$, confirming the conjecture that monotone and concave shear flows are stable under such perturbations.
• The framework is generalizable to other hydrodynamic stability problems involving boundary layers and linearized operators.