# A determining form for the subcritical surface quasi-geostrophic equation

## TL;DR

This paper develops a new ODE-based framework called a determining form for the subcritical surface quasi-geostrophic equation, linking its long-term behavior to solutions of this simpler system.

## Contribution

It introduces a Lipschitz continuous determining form that embeds the global attractor of the SQG equation into an ODE, providing elementary proofs of key solution types.

## Key findings

- Global attractor embedded in the determining form
- Existence of time-periodic and steady solutions
- Finitely many determining parameters identified

## Abstract

We construct a determining form for the surface quasi-geostrophic (SQG) equation with subcritical dissipation. In particular, we show that the global attractor for this equation can be embedded in the long-time dynamics of an ordinary differential equation (ODE) called a determining form. Indeed, there is a one-to-one correspondence between the trajectories in the global attractor of the SQG equation and the steady state solutions of the determining form. The determining form is a true ODE in the sense that its vector field is Lipschitz. This is shown by combining De Giorgi techniques and elementary harmonic analysis. Finally, we provide elementary proofs of the existence of time-periodic solutions, steady state solutions, as well as the existence of finitely many determining parameters for the SQG equation.