Global existence and stability for a hydrodynamic system in thenematic liquid crystal flows

Jihong Zhao; Qiao Liu; Shangbin Cui

doi:10.3934/cpaa.2013.12.341

Article Contents

2013, Volume 12, Issue 1: 341-357. Doi: 10.3934/cpaa.2013.12.341

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Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows

1.
Institute of Applied Mathematics, College of Science, Northwest A\&F University, Yangling, Shaanxi 712100
2.
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081
3.
Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275

Received: June 2011

Revised: September 2011

Published: January 2013

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Abstract

In this paper we consider a coupled hydrodynamical system which involves the Navier-Stokes equations for the velocity field and kinematic transport equations for the molecular orientation field. By applying the Chemin-Lerner's time-space estimates for the heat equation and the Fourier localization technique, we prove that when initial data belongs to the critical Besov spaces with negative-order, there exists a unique local solution, and this solution is global when initial data is small enough. As a corollary, we obtain existence of global self-similar solution. In order to figure out the relation between the solution obtained here and weak solutions of standard sense, we establish a stability result, which yields in a direct way that all global weak solutions associated with the same initial data must coincide with the solution obtained here, namely, weak-strong uniqueness holds.

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Mathematics Subject Classification: Primary: 35A01, 35B35; Secondary: 76A15.

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