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 \(L^2 \cap L^{\frac{2}{2\alpha-1}}\) .