On one multidimensional compressible nonlocal model of the dissipative QG equations

Shu Wang; Zhonglin Wu; Linrui Li; Shengtao Chen

doi:10.3934/dcdss.2014.7.1111

Article Contents

2014, Volume 7, Issue 5: 1111-1132. Doi: 10.3934/dcdss.2014.7.1111

This issue Previous Article Lower and upper bounds to the change of vorticity by transition from slip- to no-slip fluid flow Next Article

On one multidimensional compressible nonlocal model of the dissipative QG equations

1.
College of Applied Sciences, Beijing University of Technology, PingLeYuan100, Chaoyang District, Beijing 100124
2.
Basic Courses Department, Institute of Disaster Prevention, Yanjiao, Sanhe City, Hebei Province, 065201

Received: December 31, 2012

Revised: May 31, 2013

Published: September 2014

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Abstract

In this paper we study the Cauchy problem for one multidimensional compressible nonlocal model of the dissipative quasi-geostrophic equations and discuss the effect of the sign of initial data on the wellposedness of this model. First, we prove the existence and uniqueness of local smooth solutions for the Cauchy problem for the model with the nonnegative initial data, which seems to imply that whether the well-posedness of this model holds or not depends heavily upon the sign of the initial data even for the subcritical case. Secondly, for the sub-critical case $1<\alpha\leq 2$, we obtain the global existence and uniqueness results of the nonnegative smooth solution. Next, we prove the global existence of the weak solution for $0<\alpha\le 2$ and $\nu>0$. Finally, for the sub-critical case $1<\alpha\leq 2$, we establish $H^\beta(\beta\geq 0)$ and $L^p(p\geq 2)$ decay rates of the smooth solution as $t\to\infty$. A inequality for the Riesz transformation is also established.

Keywords:

Multidimensional compressible nonlocal model,
dissipative quasi-geostrophic equations,
sub-critical case,
Riesz transformation,
Cauchy problem.

Mathematics Subject Classification: 35A01, 35L45, 35L60, 35Q35.

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