Local-in-Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods

Abstract

We prove local existence and uniqueness for the two-dimensional Prandtl system in weighted Sobolev spaces under the Oleinik's monotonicity assumption. In particular we do not use the Crocco transform. Our proof is based on a new nonlinear energy estimate for the Prandtl system. This new energy estimate is based on a cancellation property which is valid under the monotonicity assumption. To construct the solution, we use a regularization of the system that preserves this nonlinear structure. This new nonlinear structure may give some insight on the convergence properties from Navier-Stokes system to Euler system when the viscosity goes to zero.