Near Resonant Approximation of the Rotating Stratified Boussinesq system on a 3-Torus

TL;DR

This paper introduces a new near resonant approximation for the rotating stratified Boussinesq system on a 3-torus, capturing coupling effects between modes and ensuring global existence for large initial data.

Contribution

It presents a novel near resonance approach that balances mode interactions and regularity, extending the analysis beyond exact resonance methods.

Findings

Global existence for large initial data.

Accurate coupling of slow and fast modes.

Enhanced regularity from interaction coefficient analysis.

Abstract

Based on a novel treatment of near resonances, we introduce a new approximation for the rotating stratified Boussinesq system on three-dimensional tori with arbitrary aspect ratios. The rotation and stratification parameters are arbitrary and not equal. We obtain global existence for the proposed nonlinear system for arbitrarily large initial data. This system is sufficiently accurate, with an important feature of coupling effects between slow and fast modes. The key to global existence is a sharp counting of the relevant number of nonlinear interactions. An additional regularity advantage arises from a careful examination of some mixed type interaction coefficients. In a wider context, the significance of our near resonant approach is a delicate balance between the inclusion of more interacting modes and the improvement of regularity properties, compared to the well-studied singular limit approach based on exact resonance.