Existence and decay of solutions of the 2D QG equation in the presence of an obstacle
Leonardo Kosloff Tomas Schonbek
Discrete & Continuous Dynamical Systems - S 2014, 7(5): 1025-1043 doi: 10.3934/dcdss.2014.7.1025
We continue the study initiated in [16] of dissipative differential equations governing fluid motion in the presence of an obstacle, in which the dissipative term is given by the Laplacian, or a fractional power of the Laplacian. Our main tools are the Ikebe-Ramm transform, and the localized version of the fractional Laplacian due to Caffarelli and Silvestre [5] as improved by Stinga and Torrea [21]. We give applications to the problem of existence of weak solutions of the two dimensional dissipative quasi-geostrophic equation and the decay of these solutions in the $L^2$-norm.
keywords: Ikebe-Ramm transform. Navier-Stokes Exterior Laplacian quasi-geostrophic
Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows
Maria Schonbek Tomas Schonbek
Discrete & Continuous Dynamical Systems - A 2005, 13(5): 1277-1304 doi: 10.3934/dcds.2005.13.1277
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.

keywords: Fourier-splitting. decay Quasi-geostrophic

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