# A Determining Form for the 2D Rayleigh-B\'enard Problem

## TL;DR

This paper develops a new mathematical framework called a determining form for analyzing the long-term behavior of the 2D Rayleigh-Bénard convection system, focusing on velocity trajectories to infer temperature dynamics.

## Contribution

It introduces a novel ODE-based determining form that captures the global attractor of the 2D RB system using velocity trajectories alone.

## Key findings

- Constructed a determining form for the 2D RB system.
- Identified long-time dynamics via zeros of a scalar equation.
- Applicable to systems with no-slip and stress-free boundaries.

## Abstract

We construct a determining form for the 2D Rayleigh-B\'enard (RB) system in a strip with solid horizontal boundaries, in the cases of no-slip and stress-free boundary conditions. The determining form is an ODE in a Banach space of trajectories whose steady states comprise the long-time dynamics of the RB system. In fact, solutions on the global attractor of the RB system can be further identified through the zeros of a scalar equation to which the ODE reduces for each initial trajectory. The twist in this work is that the trajectories are for the velocity field only, which in turn determines the corresponding trajectories of the temperature.