# American Institute of Mathematical Sciences

January  2013, 12(1): 341-357. doi: 10.3934/cpaa.2013.12.341

## 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, China 2 Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China 3 Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275

Received  June 2011 Revised  September 2011 Published  September 2012

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.
Citation: Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341
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