Asymptotic study of a global solution of super-critical Quasi-Geostrophic equation

TL;DR

This paper investigates the super-critical Quasi-Geostrophic equation, proving local existence for large data, global existence for small data, and analyzing the decay and blow-up behavior of solutions using Fourier analysis.

Contribution

It provides new results on the existence, blow-up, and decay of solutions to the super-critical QG equation in Gevrey-Sobolev spaces, extending understanding of its long-term behavior.

Findings

Local existence for large initial data

Global existence for small initial data

Solutions decay to zero as time approaches infinity

Abstract

In this paper, we study the super-critical Quasi-Geostrophic equation in Gevrey-Sobolev space. We prove the local existence of $(QG)$ for any large initial data and we give an exponential type of Blow-up to the solution. Moreover, we establish the existence global for a small initial data and we show that $\|\theta\|_{H^s_{a, \alpha^{-1}}}$ decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.