Decay of Weak Solutions to the 2D Dissipative Quasi-Geostrophic Equation

We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data θ0 is in L2 only, we prove that the L2 norm tends to zero but with no uniform rate, that is, there are solutions with arbitrarily slow decay. For θ0 in Lp ∩ L2, with 1 ≤ p < 2, we are able to obtain a uniform decay rate in L2. We also prove that when the $$L^{\frac{2}{2\alpha-1}}$$ norm of θ0 is small enough, the Lq norms, for $$q > {\frac{2}{2\alpha-1}}$$ , have uniform decay rates. This result allows us to prove decay for the Lq norms, for $$q \geq {\frac{2}{2\alpha-1}}$$ , when θ0 is in $$L^2 \cap L^{\frac{2}{2\alpha-1}}$$ .