Asymptotic study of supercritical surface Quasi-Geostrophic equation in critical space

TL;DR

This paper proves that small initial data solutions to the supercritical surface Quasi-Geostrophic equation decay to zero in the critical Sobolev space over time, using Fourier analysis and standard techniques.

Contribution

It establishes the decay of solutions in the critical space for the supercritical surface Quasi-Geostrophic equation, extending understanding of long-term behavior.

Findings

Solutions decay to zero as time approaches infinity.

Decay is proven for small initial data in the critical space.

Fourier analysis techniques are effectively applied.

Abstract

In this paper we prove, if $\theta\in C([0,\infty),H^{2-2\alpha}(\mathbb R^2))$ is a global solution of supercritical surface Quasi-Geostrophic equation with small initial data, then $\|\theta(t)\|_{H^{2-2\alpha}}$ decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.