Decay rates for mild solutions of QGE with critical fractional dissipation in $L^2(\mathbb{R}^2)$

TL;DR

This paper proves that certain decay conditions on initial data are unnecessary for establishing the asymptotic decay rates of solutions to the critical quasi-geostrophic equation in $L^2(R^2)$, using Fourier analysis.

Contribution

It shows that the previously assumed decay condition on initial data is not needed for asymptotic results, broadening the understanding of solution behavior.

Findings

Decay rates for solutions are established without initial decay conditions.

Fourier analysis techniques are effectively applied to the problem.

The results extend previous asymptotic analysis to more general initial data.

Abstract

In \cite{MRSC1} the authors proved some asymptotic results for the global solution of critical Quasi-geostrophic equation with a condition on the decay of $\widehat{\theta_0}$ near at zero. In this paper, we prove that this condition is not necessary. Fourier analysis and standard techniques are used.