  In this paper, we study the full regularity and well-posedness of classical
solutions to the nonlinear unsteady Prandtl equations with Robin or Dirichlet
boundary condition in half space. Under Oleinik's monotonicity assumption, we
prove the large time existence of classical solutions to the nonlinear Prandtl
equations with Robin or Dirichlet boundary condition, when both initial
vorticity and the general Euler flow are sufficiently small. For the general
Euler flow, the vertical velocity of the Prandtl flow is unbounded. We prove
that the Prandtl solutions preserve the full regularities in our solution
spaces. The uniqueness and stability are also proved in the weighted Sobolev
spaces.
