Communications in Mathematical Sciences

Volume 13 (2015)

Number 6

Weighted decay for the surface quasi-geostrophic equation

Pages: 1599 – 1614

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

Authors

Igor Kukavica (Department of Mathematics, University of Southern California, Los Angeles, Calif., U.S.A.)

Fei Wang (Department of Mathematics, University of Southern California, Los Angeles, Calif., U.S.A.)

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

Published 13 May 2015