  In this paper, we study the back flow of the two-dimensional unsteady Prandtl
boundary layer under an adverse pressure gradient. The occurrence of back flow
is an important physical event in the evolution of boundary layer, which
eventually leads to separation. For the two-dimensional unsteady Prandtl
boundary layer equations, when the initial tangential velocity is strictly
monotonic with respect to the normal variable, and the pressure gradient of the
outer flow is adverse, we obtain that the first critical point of the
tangential velocity profile with respect to the normal variable, if exists when
the boundary layer evolves in time, must appear on the boundary. Moreover, we
give a condition on the growth rate of the initial tangential velocity such
that there is a back flow point of the Prandtl boundary layer under the adverse
pressure gradient. In the appendix, we introduce two examples showing that back
flow occurs either when the flow distance is long in the streamwise direction
for a given initial monotonic tangential velocity field, or when the initial
tangential velocity grows slowly in a large neighborhood of the boundary for a
fixed flow distance in the streamwise direction.
