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

Abstract

We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data θ₀ 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 θ₀ in L ^p ∩ L ², with 1 ≤  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 θ₀ 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 θ₀ is in $L^2 \cap L^{\frac{2}{2\alpha-1}}$ .