  Motivated by the paper by D. Gerard-Varet and E. Dormy [JAMS, 2010] about the
linear ill-posedness for the Prandtl equations around a shear flow with
exponential decay in normal variable, and the recent study of well-posedness on
the Prandtl equations in Sobolev spaces, this paper aims to extend the result
in \cite{GV-D} to the case when the shear flow has general decay. The key
observation is to construct an approximate solution that captures the initial
layer to the linearized problem motivated by the precise formulation of
solutions to the inviscid Prandtl equations.
