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

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^v(t, x,y) = (U ^E(t, y)+U^BL(t, y/√v), 0), 0 < v ≪ 1: We show that if U^BL is monotonic and concave in Y = y/√v then u^v is stable over some time interval (0, T), T independent of v, 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.