Dynamic boundary conditions with noise for energy balance models coupled to geophysical flows

TL;DR

This work analyzes a coupled 2D energy balance model with a 3D ocean governed by the primitive equations, incorporating dynamic boundary conditions for surface temperature and, optionally, boundary noise. The authors develop an abstract $L^p$-framework, proving global, strong well-posedness for deterministic data in $W^{2(1-\frac{1}{p}),p}$ ($p\in[2,\infty)$, $p\neq3$) and stochastic data in $H^1$ via maximal regularity, $\mathcal{H}^\infty$-calculus, and the Da Prato–Debussche approach. Key technical ingredients include a bounded $\mathcal{H}^\infty$-calculus for the temperature-EBM operator, a maximum principle to control the surface temperature for energy estimates, and stochastic maximal regularity to handle boundary noise. The results lay a rigorous analytical foundation for strong, global solutions of climate-fluid systems with dynamic boundary conditions and stochastic forcing, providing a framework for future extensions to more realistic geometries and non-periodic domains.

Abstract

This article investigates an energy balance model coupled to the primitive equations by a dynamic boundary condition with and without noise on the boundary. It is shown that this system is globally strongly well-posed both in the deterministic setting for arbitrary large data in $W^{2(1-\frac{1}{p}),p}$ for $p \in [2,\infty)$ and in the stochastic setting for arbitrary large data in $H^1$.

Theorems & Definitions (24)

• Theorem 4.1: Local and Global Well-Posedness of \ref{['eq: primitive + EBM simplified']}
• Corollary 1: Regularity
• Theorem 4.2: Global well-posedness of \ref{['eq: primitive + EBM simplified1']}
• Remark 1
• Remark 2
• Lemma 1
• Lemma 2
• proof
• Remark 3
• Corollary 2