Smooth attractors for weak solutions of the SQG equation with critical dissipation

TL;DR

This paper studies the long-term behavior of weak solutions to the critically dissipative surface quasi-geostrophic equation, establishing the existence of a global attractor with optimal Sobolev regularity despite potential non-uniqueness.

Contribution

It introduces a new Sobolev estimate involving H"older norms and proves the existence of a global attractor for the set-valued dynamical system associated with the equation.

Findings

Existence of a global attractor with optimal Sobolev regularity.

Development of a new Sobolev estimate involving H"older norms.

Handling of non-uniqueness via a set-valued dynamical system.

Abstract

We consider the evolution of weak vanishing viscosity solutions to the critically dissipative surface quasi-geostrophic equation. Due to the possible non-uniqueness of solutions, we rephrase the problem as a set-valued dynamical system and prove the existence of a global attractor of optimal Sobolev regularity. To achieve this, we derive a new Sobolev estimate involving H\"older norms, which complement the existing estimates based on commutator analysis.