Long time dynamics of forced critical SQG

TL;DR

This paper establishes the existence of a finite-dimensional global attractor for the forced critical SQG equation by providing a new proof of global regularity and analyzing long-term dynamics, including decay rates and attractor properties.

Contribution

It introduces a novel proof of global regularity for critical SQG and demonstrates the existence and finite dimensionality of the global attractor for its dynamics.

Findings

Existence of a compact global attractor for forced critical SQG.

Finite fractal (box-counting) dimension of the attractor.

Exponential decay of volume elements in the attractor.

Abstract

We prove the existence of a compact global attractor for the dynamics of the forced critical surface quasi-geostrophic equation (SQG) and prove that it has finite fractal (box-counting) dimension. In order to do so we give a new proof of global regularity for critical SQG. The main ingredient is the nonlinear maximum principle in the form of a nonlinear lower bound on the fractional Laplacian, which is used to bootstrap the regularity directly from $L^\infty$ to $C^\alpha$, without the use of De Giorgi techniques. We prove that for large time, the norm of the solution measured in a sufficiently strong topology becomes bounded with bounds that depend solely on norms of the force, which is assumed to belong merely to $L^\infty \cap H^1$. Using the fact that the solution is bounded independently of the initial data after a transient time, in spaces conferring enough regularity, we prove the existence of a compact absorbing set for the dynamics in $H^1$, obtain the compactness of the linearization and the continuous differentiability of the solution map. We then prove exponential decay of high yet finite dimensional volume elements in $H^1$ along solution trajectories, and use this property to bound the dimension of the global attractor.