November  2010, 9(6): 1473-1493. doi: 10.3934/cpaa.2010.9.1473

Global in time solution and time-periodicity for a smectic-A liquid crystal model

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain

Received  March 2009 Revised  April 2010 Published  August 2010

In this paper some results are obtained for a smectic-A liquid crystal model with time-dependent boundary Dirichlet data for the so-called layer variable $\varphi$ (the level sets of $\varphi$ describe the layer structure of the smectic-A liquid crystal). First, the initial-boundary problem for arbitrary initial data is considered, obtaining the existence of weak solutions which are bounded up to infinity time. Second, the existence of time-periodic weak solutions is proved. Afterwards, the problem of the global in time regularity is attacked, obtaining the existence and uniqueness of regular solutions (up to infinity time) for both problems, i.e. the initial-valued problem and the time-periodic one, but assuming a dominant viscosity coefficient in the linear part of the diffusion tensor.
Citation: Blanca Climent-Ezquerra, Francisco Guillén-González. Global in time solution and time-periodicity for a smectic-A liquid crystal model. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1473-1493. doi: 10.3934/cpaa.2010.9.1473
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