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/root v), 0), 0 < v << 1. We show that if U BL is mono- tonic and concave in Y = y/root v, then 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.