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# Communications in Mathematical Sciences

## Volume 13 (2015)

### Number 6

### Weighted decay for the surface quasi-geostrophic equation

Pages: 1599 – 1614

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n6.a11

#### Authors

#### Abstract

We address the weighted decay for solutions of the surface quasi-geostrophic (SQG) equation which is given by\[\theta_t + u \cdot \nabla \theta + \Lambda^{2\alpha} \theta = 0 \, \textrm{,}\]where $\Lambda = {(- \Delta)}^{1/2}$. The first moment decay ${\lVert \lvert x \rvert \theta \rVert}_{L^2}$ was obtained by M. and T. Schonbek in [M. Schonbek and T. Schonbek, Discrete Contin. Dyn. Syst., 13(5), 1277–1304, 2005]. Here we obtain the decay rates of ${\lVert \lvert x \rvert {}^b \theta \rVert}_{L^2}$ for any $b \in (0,1)$ and the rate of increase of this quantity for $b \in [1, 1+ \alpha)$ under natural assumptions on the initial data.

#### Keywords

surface quasi-geostrophic equations, weighted norm, long time behavior, decay

#### 2010 Mathematics Subject Classification

35Q30, 35R35, 76D05