Long-time dynamics of the nonhomogeneous incompressible flow of nematic liquid crystals

Abstract

AbstractWe study the long-time behavior of global strong solutions to a hydrodynamicsystem for nonhomogeneous incompressible nematic liquid crystal flows driven by twotypes of external forces in a smooth bounded domain in R 2 . For arbitrary large regularinitial data with the initial density being away from vacuum, we prove the decayof the velocity field for both cases. Furthermore, for the case with asymptoticallyautonomous external force, we can prove the convergence of the density function andthe director vector as time goes to infinity. Estimates on convergence rate are alsoprovided.Keywords: Nonhomogeneous nematic liquid crystal flow, long-time behavior,uniqueness of asymptotic limit, convergence rate.AMS Subject Classification: 35B40, 35B41, 35Q35, 76D05. 1 Introduction Liquid crystals are substances that exhibit a phase of matter that has properties betweenthose of a conventional liquid, and those of a solid crystal [5]. The hydrodynamic theory ofliquid crystals due to Ericken and Leslie was developed around 1960’s [6,15,16]. Since then,the mathematical theory is still progressing and the study of the full Ericksen–Leslie modelpresents relevant mathematical difficulties. We consider the following hydrodynamicalmodel for the flow of nematic liquid crystals (cf. [17])ρ