Gevrey Stability of Prandtl Expansions for 2D Navier-Stokes

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 : $$u^{\nu }(t,x,y)=(U^E(t,y)+U^{BL}(t,\frac{y}{\sqrt{\nu }}),0),0<\nu \ll 1.$$ 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.

References in corpus (2)

• On the ill-posedness of the Prandtl equations in three space dimensions
• Vanishing Viscosity Limits and Boundary Layers for Circularly Symmetric 2D Flows

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