Long time decay to the solution to the 2D-DQG Equation

In \cite{JAMH1}, we prove the well posedness of the quasi-geostrophic equation $(QG{)}_{\alpha },1\mathrm{/}2<\alpha \le 1$, in the space introduced by Z. Lei and F. Lin in \cite{ZY1}. In this chapter we discuss the long time behaviour. Mainly, we prove that $\mathrm{\parallel }\theta {\mathrm{\parallel }}_{{\mathcal{X}}^{1-2\alpha }}$ decays to zero as time goes to infinity.