Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation

Bo-Qing Dong; Juan Song

doi:10.3934/dcds.2012.32.57

Article Contents

2012, Volume 32, Issue 1: 57-79. Doi: 10.3934/dcds.2012.32.57

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Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation

Bo-Qing Dong^1, and
Juan Song^1,

1.
School of Mathematical Science, Anhui University, Hefei 230039

Received: June 30, 2010

Revised: December 31, 2010

Published: December 2011

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Abstract

This paper is concerned with the Cauchy problem of three-dimensional modified Navier-Stokes equations with fractional dissipation $ \nu (-\Delta)^{\alpha} u$. The results are three-fold. We first prove the global existence of weak solutions for $0<\alpha\leq1 $ and global smooth solution for $\frac{3}{4}<\alpha\leq1.$ Second, we obtain the optimal decay rates of both weak solutions and the higher-order derivative of the smooth solution. Finally, we investigate the asymptotic stability of the large solution to the system under large initial and external forcing perturbation.

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Mathematics Subject Classification: 35Q30, 76D05.

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