Well-posedness of the Prandtl equation in Sobolev space without monotonicity

Chao-Jiang Xu, Xu Zhang

Published 2015-11-16Version 1

We study the well-posedness theory for the Prandtl boundary layer equation on the half plane with initial data in Sobolev spaces. We consider a class of initial data which admit the non-degenerate critical points, so it is not monotonic. For this kind of initial data, we prove the local-in-time existence, uniqueness and stability of solutions for the nonlinear Prandtl equation in weighted Sobolev space. We use the energy method to prove the existence of solution by a parabolic regularizing approximation. The nonlinear cancellation properties of the Prandtl equations and non-degeneracy of the critical points are the main ingredients to establish a new energy estimate. Our result improves the classical local well-posedness results for the initial data that are monotone, analytic or Gevery class, and it will also help us to understand the ill-posedness and instability results for the Prandtl equation.