Long time behavior and critical limit of subcritical SQG equations in scale-invariant Sobolev spaces

TL;DR

This paper establishes the existence of a global attractor with optimal regularity for subcritical SQG equations in scale-invariant Sobolev spaces, introducing new energy estimates and analyzing the critical limit behavior.

Contribution

It introduces a novel energy estimate in Sobolev spaces for subcritical SQG equations, enabling the proof of global attractors and their stability in the critical limit.

Findings

Existence of a global attractor with optimal regularity

New energy estimates based on nonlinear lower bounds

Stability and upper-semicontinuity of attractors in the critical limit

Abstract

We consider the subcritical SQG equation in its natural scale invariant Sobolev space and prove the existence of a global attractor of optimal regularity. The proof is based on a new energy estimate in Sobolev spaces to bootstrap the regularity to the optimal level, derived by means of nonlinear lower bounds on the fractional laplacian. This estimate appears to be new in the literature, and allows a sharp use of the subcritical nature of the $L^\infty$ bounds for this problem. As a byproduct, we obtain attractors for weak solutions as well. Moreover, we study the critical limit of the attractors and prove their stability and upper-semicontinuity with respect to the strength of the diffusion.