Global strong well-posedness of the CAO-problem introduced by Lions, Temam and Wang

TL;DR

This paper proves the global strong well-posedness and real analyticity of solutions for the coupled CAO-system of two fluids, extending previous weak solution existence results to large data in critical Besov spaces.

Contribution

It establishes the first global strong well-posedness result for the CAO-problem with large data, using advanced boundary and paraproduct analysis in Triebel-Lizorkin spaces.

Findings

Global strong well-posedness for large data

Solutions are real analytic away from the boundary

Boundary terms controlled via paraproduct methods

Abstract

Consider the CAO-problem introduced by Lions, Temam and Wang, which concerns a system of two fluids described by two primitive equations coupled by fully nonlinear interface conditions. They proved in their pioneering work the existence of a weak solution to the CAO-system; its uniqueness remained an open problem. In this article, it is shown that this coupled CAO-system is globally strongly well-posed for large data, even in critical Besov spaces. It is furthermore shown that, away from the boundary, the solution is even real analytic. The approach presented relies on an optimal data result for the boundary terms in the linearized system in terms of time-space Triebel-Lizorkin spaces. Boundary terms are then controlled by paraproduct methods in these spaces.