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STABILITY FOR NEMATIC LIQUID CRYSTALS WITH STRETCHING TERMS

We study a nematic crystal model that appeared in [Liu et al., 2007], modeling stretching effects depending on the different shapes of the microscopic molecules of the material, under periodic boundary conditions. The aim of the present article is two-fold: to extend the results given in [Sun & Liu, 2009], to a model with more complete stretching terms and to obtain some stability and asymptotic stability properties for this model.

Time-periodic solutions for a generalized Boussinesq model with Neumann boundary conditions for temperature

The aim of this work is to prove the existence of regular time-periodic solutions for a generalized Boussinesq model (with nonlinear diffusion for the equations of velocity and temperature). The main idea is to obtain higher regularity (of H3 type) for temperature than for velocity (of H2 type), using specifically the Neumann boundary condition for temperature. In fact, the case of Dirichlet condition for temperature remains as an open problem.

A review on mathematical analysis for nematic and smectic-A liquid crystal models

We review the mathematical analysis of some uniaxial, liquid crystal phases. First, we state the models for the two different studied phases: nematic and smectic-A liquid crystals. The spatial and temporal profiles of the liquid crystal configurations will be described by means of strongly nonlinear parabolic partial differential systems, which are presented at the same time. Then, we will state some results about existence, regularity, time-periodicity and stability of solutions at infinite time for both models. It is our aim is to show that, although nematic and smectic-A phases have different physical properties and are modeled by different nonlinear parabolic problems, there exists a common mathematical machinery to rewrite the models and to obtain the analytical results.

Long-Time Behavior of a Cahn-Hilliard-Navier-Stokes Vesicle-Fluid Interaction Model

A model about the dynamic of vesicle membranes in incompressible viscous fluids is introduced. The system consists of the Navier-Stokes equations with an extra stress depending on the membrane, coupled with a Cahn-Hilliard phase-field equation in 3D domains. This problem has a time dissipative energy which leads, in particular, to the existence of global in time weak solutions. By using some extra regular estimates, we prove that every weak solution is strong and unique for sufficiently large times. Moreover, the asymptotic behavior of these solutions is analyzed. We prove that the w-limit set is a subset of the set of equilibrium points. By using a Lojasiewic-Simon type inequality and a continuity result with respect to the initial values, we demonstrate the convergence of the whole trajectory to a single equilibrium.