On a 3D isothermal model for nematic liquid crystals accounting for stretching terms

Abstract

In the present contribution we study a PDE system describing the evolution of a nematic liquid crystals flow under kinematic transports for molecules of different shapes. More in particular, the evolution of the velocity field u is ruled by the Navier-Stokes incompressible system with a stress tensor exhibiting a special coupling between the transport and the induced terms. The dynamics of the director field d is described by a variation of a parabolic Ginzburg-Landau equation with a suitable penalization of the physical constraint |d| = 1. Such equation accounts for both the kinematic transport by the flow field and the internal relaxation due to the elastic energy. The main aim of this contribution is to overcome the lack of a maximum principle for the director equation and prove (without any restriction on the data and on the physical constants of the problem) the existence of global in time weak solutions under physically meaningful boundary conditions on d and u.

Weak solutions for an initial–boundary Q-tensor problem related to liquid crystals

“…A similar argument is used in [7] for a nematic model in (u, d ). In [5] a different Galerkin approximation is considered, where the Q-system is maintained at infinity dimension, adding a regularized viscous tensor of p-laplacian type.…”

Global Strong Solutions of the Full Navier--Stokes and $Q$-Tensor System for Nematic Liquid Crystal Flows in Two Dimensions

“…We refer to [6,7,20,26,38,42] and the references therein for mathematical analysis of the general EricksenLeslie system either under the unit length constraint of the molecule director or with Ginzburg-Landau approximation of the free energy.…”

A New Approach to Non-Isothermal Models for Nematic Liquid Crystals

“…In order to prevent this failure, Sun and Liu [24] introduced a variant of the model proposed by Lin and Liu, where the stretching term is included in the system and a new component is added to the stress tensor in order to save the total energy balance. A related isothermal model accounting for the stretching contribution has been recently analyzed in [3] and in [20]. In [3], the existence of a global solution is proved in the case of homogeneous Dirichlet boundary conditions for u and Neumann or non-homogeneous Dirichlet boundary conditions for d, while in [20] the long time behaviour of solutions is investigated in the three dimensional case without assuming any restriction on the size of the viscosity coefficient μ and possibly considering a non-analytic nonlinearity f .…”

“…A related isothermal model accounting for the stretching contribution has been recently analyzed in [3] and in [20]. In [3], the existence of a global solution is proved in the case of homogeneous Dirichlet boundary conditions for u and Neumann or non-homogeneous Dirichlet boundary conditions for d, while in [20] the long time behaviour of solutions is investigated in the three dimensional case without assuming any restriction on the size of the viscosity coefficient μ and possibly considering a non-analytic nonlinearity f . A result similar to the one obtained in [3] is also proved in [4], where, however, only formal estimates are proved and no rigorous existence statement is provided.…”

“…In [3], the existence of a global solution is proved in the case of homogeneous Dirichlet boundary conditions for u and Neumann or non-homogeneous Dirichlet boundary conditions for d, while in [20] the long time behaviour of solutions is investigated in the three dimensional case without assuming any restriction on the size of the viscosity coefficient μ and possibly considering a non-analytic nonlinearity f . A result similar to the one obtained in [3] is also proved in [4], where, however, only formal estimates are proved and no rigorous existence statement is provided. The results in [3,4,20] generalize the ones previously obtained in [26], where the two-dimensional case and the three-dimensional case with large viscosity μ are dealt with.…”