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 epsilon(-4/3). if the initial data is of size epsilon and the value on the boundary of the tangential velocity of the outflow are of size epsilon(5/3). 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. (C) 2016 Elsevier Inc. All rights reserved.NSF of China [11371347, 11371039, 11421101]; Chinese Academy of Sciences; National Center for Mathematics and Interdisciplinary Sciences; MCMSCI(E)ARTICLEzp@amss.ac.cn; zfzhang@math.pku.edu.cn72591-261527