On nonlinear instability of Prandtl's boundary layers: The case of Rayleigh's stable shear flows

In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of Navier Stokes equations near a boundary as the viscosity goes to 0. His Ansatz has later been justified for analytic data by R.E. Caflisch and M. Sammartino. In this paper, we prove that his expansion is false, up to O(nu(1/4)) order terms in L-infinity norm, in the case of solutions with Sobolev regularity, even in cases where the Prandlt's equation is well posed in Sobolev spaces. In addition, we also prove that monotonic boundary layer profiles, which are stable when nu = 0, are nonlinearly unstable when nu > 0, provided nu is small enough, up to O(nu(1/4)) terms in L-infinity norm. (c) 2024 Elsevier Masson SAS. All rights reserved.