Decay of weak solutions to the 2D dissipative quasi-geostrophic equation

Details Decay of weak solutions to the 2D dissipative quasi-geostrophic equation Decay of weak solutions to the 2D dissipative quasi-geostrophic equation

2006-05-22

arxiv.org/abs/math/0605576v2

Mathematics

Analysis of PDEs

A paragraph describing work by Carrillo and Ferreira proving results directly related to the ones in this paper is added in th

Scientific paper

10.1007/s00220-007-0327-y

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

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