Long time well-posedness of Prandtl equations in Sobolev space

In this paper, we study the long time well-posedness for the nonlinear Prandtl boundary layer equation on the half plane. While the initial data are small perturbations of some monotone shear profile, we prove the existence, uniqueness and stability of solutions in weighted Sobolev space by energy methods. The key point is that the life span of the solution could be any large T as long as its initial datum is a perturbation around the monotonic shear profile of smell size e(-T). The nonlinear cancellation properties of Prandtl equations under the monotonic assumption are the main ingredients to establish a new energy estimate. (C) 2017 Elsevier Inc. All rights reserved.