Gevrey stability of Prandtl expansions for 2-dimensional Navier–Stokes flows

Abstract

We investigate the stability of boundary layer solutions of the $2$-dimensional incompressible Navier–Stokes equations. We consider shear flow solutions of Prandtl type: $u^{\nu }(t,x,y)=\left(U^E\right(t,y)+U^{\mathrm{BL}}(t,\frac{y}{\sqrt{\nu }}),0)$, $0<\nu \ll 1.$ We show that if $U^{\mathrm{BL}}$ 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 $U^{\mathrm{BL}}$ 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.