Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows

Maria Schonbek; Tomas Schonbek

doi:10.3934/dcds.2005.13.1277

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2005, Volume 13, Issue 5: 1277-1304. Doi: 10.3934/dcds.2005.13.1277

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Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows

Maria Schonbek^1, and
Tomas Schonbek^2,

1.
Department of Mathematics, University of Santa Cruz, Santa Cruz, CA 95064
2.
Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431

Received: July 31, 2004

Revised: March 31, 2005

Published: September 2005

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Abstract

We consider the long time behavior of moments of solutions and of the solutions itself to dissipative Quasi-Geostrophic flow (QG) with sub-critical powers. The flow under consideration is described by the nonlinear scalar equation

$\frac{\partial \theta}{\partial t} + u\cdot \nabla \theta + \kappa (-\Delta)^{\alpha}\theta =f$, $\theta|_{t=0}=\theta_0 $

Rates of decay are obtained for moments of the solutions, and lower bounds of decay rates of the solutions are established.

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Mathematics Subject Classification: 35Q35, 76B03.

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