  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 $L^\infty$ 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 $L^\infty$ norm.
