Separation of the two-dimensional unsteady Prandtl boundary layers under adverse pressure gradient

In this paper, we study the separation of the two-dimensional unsteady Prandtl boundary layer under adverse pressure gradient. For monotonic initial datum we prove that the first critical point of the tangential velocity profile with respect to the normal variable, if exists, must appear on the boundary if the pressure gradient of the outer flow is adverse. Moreover, we shall give a condition on the growth rate of the initial tangential velocity such that there is a separation point of the Prandtl boundary layer under the adverse pressure gradient.