Asymptotic study of supercritical generalized SQG equations in critical Sobolev spaces

TL;DR

This paper analyzes the long-term behavior of solutions to supercritical generalized SQG equations in critical Sobolev spaces, proving decay of solutions under small initial data.

Contribution

It establishes the global existence and decay of solutions for supercritical gSQG equations with small initial data in critical Sobolev spaces.

Findings

Existence of unique global solutions under small initial data.

Critical Sobolev norm of solutions decays to zero over time.

Solutions satisfy an energy inequality ensuring stability.

Abstract

We study the long time behavior of regular solutions of the supercritical gSQG equations in the fully nonlinear regime. More precisely, under the assumption of small initial data in the critical Sobolev norm, we prove the existence of the unique global solution that satisfies the energy inequality (1.3) and for which the critical norm $||{\theta(t)}||_{H^{1+\beta-2\alpha}}$ decays to 0 as time goes to infinity.