  In the lines of a recent paper by Gerard-Varet and Dormy, we establish
various ill-posedness results for the Prandtl equation. By considering
perturbations of stationary shear flows, we show that for some linearizations
of the Prandtl equation and some $C^\infty$ initial data, local in time
$C^\infty$ solutions do not exist. At the nonlinear level, we prove that if a
flow exists in the Sobolev setting, it cannot be Lipschitz continuous. Besides
ill-posedness in time, we also establish some ill-posedness in space, that
casts some light on the results obtained by Oleinik for monotonic data.
