Global Well-posedness and Regularity of Weak Solutions to the Prandtl's System

TL;DR

This paper establishes the global existence, uniqueness, and regularity of weak solutions to the 2D Prandtl's system under favorable pressure conditions, advancing understanding of boundary layer equations.

Contribution

It provides a new direct BV estimate method for proving global weak solution existence and demonstrates the smoothness and continuous dependence of solutions.

Findings

Existence of global weak solutions proved

Uniqueness and continuous dependence established

Weak solutions shown to be smooth and globally existing

Abstract

We continue our study on the global solution to the two-dimensional Prandtl's system for unsteady boundary layers in the class considered by Oleinik provided that the pressure is favorable. First, by using a different method from [13], we gave a direct proof of existence of a global weak solution by a direct BV estimate. Then we prove the uniqueness and continuous dependence on data of such a weak solution to the initial-boundary value problem. Finally, we show the smoothness of the weak solutions and then the global existence of smooth solutions.