  We address the decay of the norm of weak solutions to the 2D dissipative
quasi-geostrophic equation. When the initial data is in $L^2$ only, we prove
that the $L^2$ norm tends to zero but with no uniform rate, that is, there are
solutions with arbitrarily slow decay. For the initial data in $L^p \cap L^2$,
with $1 \leq p < 2$, we are able to obtain a uniform decay rate in $L^2$. We
also prove that when the $L^{\frac{2}{2 \alpha -1}}$ norm of the initial data
is small enough, the $L^q$ norms, for $q > \frac{2}{2 \alpha -1}$ have uniform
decay rates. This result allows us to prove decay for the $L^q$ norms, for $q
\geq \frac{2}{2 \alpha -1}$, when the initial data is in $L^2 \cap
L^{\frac{2}{2 \alpha -1}}$.
