Well-posedness of The Prandtl Equation in Sobolev Spaces

TL;DR

This paper establishes the well-posedness of the Prandtl equation in Sobolev spaces by employing a direct energy method and Nash-Moser iteration, avoiding traditional transformations and handling degeneracy with a monotonicity condition.

Contribution

It introduces a new approach using energy methods and Nash-Moser iteration to prove well-posedness of the Prandtl equation in Sobolev spaces under monotonicity assumptions.

Findings

Proved well-posedness for the nonlinear Prandtl equation with small perturbations.

Developed a framework for handling degeneracy via Nash-Moser iteration.

Analyzed the linearized equation in weighted Sobolev spaces.

Abstract

We develop a new approach to study the well-posedness theory of the Prandtl equation in Sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the Crocco transformation. Precisely, we firstly investigate the linearized Prandtl equation in some weighted Sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable. Then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation, we apply the Nash-Moser-Hormander iteration to obtain a well-posedness theory of classical solutions to the nonlinear Prandtl equation when the initial data is a small perturbation of a monotonic shear flow.