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

TL;DR

This work resolves the global well-posedness of the CAO-system, a coupled two-fluid model with fully nonlinear interface conditions, for large data in critical Besov spaces. The authors develop an optimal data framework for the linearized problem in $L^p$-$L^q$ spaces and vector-valued Triebel-Lizorkin spaces, establish precise boundary-term control via paraproducts, and derive a local well-posedness theory together with a blow-up criterion. These tools enable a global-in-time, unique strong solution in maximal regularity spaces, and extend to initial data in critical Besov spaces, with interior real-analyticity when forcing is absent. The results mark a significant stride in the mathematical understanding of fully nonlinear boundary coupling in geophysical fluid dynamics and provide a robust framework for future analysis of similar multi-physics systems.

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.

Theorems & Definitions (27)

• Theorem 3.1: Global strong well-posedness for the CAO-system
• Remark 1
• Corollary 1: Regularity
• Corollary 2: Analyticity of the solution
• Theorem 4.1
• Remark 2
• Corollary 3
• Lemma 1
• proof
• proof : Proof of \ref{['thm:optdata']}