Gevrey Stability of Prandtl Expansions for 2D Navier-Stokes

TL;DR

This paper proves Gevrey stability of Prandtl boundary layer expansions for 2D Navier-Stokes equations under certain monotonicity and concavity conditions, improving classical analytic stability results and establishing optimal Gevrey regularity.

Contribution

It establishes Gevrey stability of Prandtl expansions for 2D Navier-Stokes with sharp resolvent estimates, extending classical analytic results to Gevrey classes.

Findings

Stability holds over a time interval independent of viscosity.

Gevrey regularity is sufficient for stability under specified conditions.

Optimal Gevrey exponent achieved for steady, strictly concave boundary layers.

Abstract

We investigate the stability of boundary layer solutions of the two-dimensional incompressible Navier-Stokes equations. We consider shear flow solutions of Prandtl type : $$ u^\nu(t,x,y) \, = \, \big (U^E(t,y) + U^{BL}(t,\frac{y}{\sqrt{\nu}})\,, \, 0 \big )\, , \quad 0<\nu \ll 1\,. $$ We show that if $U^{BL}$ is monotonic and concave in $Y = y /\sqrt{\nu}$ then $u^\nu$ is stable over some time interval $(0,T)$, $T$ independent of $\nu$, under perturbations with Gevrey regularity in $x$ and Sobolev regularity in $y$. We improve in this way the classical stability results of Sammartino and Caflisch in analytic class (both in $x$ and $y$). Moreover, in the case where $U^{BL}$ is steady and strictly concave, our Gevrey exponent for stability is optimal. The proof relies on new and sharp resolvent estimates for the linearized Orr-Sommerfeld operator.