Long time well-posedness of Prandtl system with small and analytic initial data

In this paper, we investigate the long time existence and uniqueness of small solution to d, for d=2,3, dimensional Prandtl system with small initial data which is analytic in the horizontal variables. In particular, we prove that d dimensional Prandtl system has a unique solution with the life-span of which is greater than ε−43 if the initial data is of size ε and the value on the boundary of the tangential velocity of the outflow are of size ε53. We mention that the tool developed in [4,5] to make the analytical type estimates and the special structure of the nonlinear terms to this system play an essential role in the proof of this result.