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In \cite{JAMH1}, we prove the well posedness of the quasi-geostrophicequation $(QG)_{\alpha}\;,1/2<\alpha\leq 1$, in the space introduced by Z. Leiand F. Lin in \cite{ZY1}. In this chapter we discuss the long time behaviour.Mainly, we prove that $\|\theta\|_{{\mathcal X}^{1-2\alpha}}$ decays to zero astime goes to infinity.

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