Trajectory attractors for the Sun-Liu model for nematic liquid crystals in 3D

TL;DR

This paper establishes the existence of a trajectory attractor for a 3D liquid crystal PDE system that models molecular orientation and fluid flow, incorporating stretching effects and boundary conditions.

Contribution

It proves the existence of a trajectory attractor for a complex coupled PDE system modeling nematic liquid crystals with stretching effects in three dimensions.

Findings

Existence of a trajectory attractor for the system.

Development of a suitable trajectory space and metric.

Derivation of a dissipative energy estimate.

Abstract

In this paper we prove the existence of a trajectory attractor (in the sense of V.V. Chepyzhov and M.I. Vishik) for a nonlinear PDE system coming from a 3D liquid crystal model accounting for stretching effects. The system couples a nonlinear evolution equation for the director d (introduced in order to describe the preferred orientation of the molecules) with an incompressible Navier-Stokes equation for the evolution of the velocity field u. The technique is based on the introduction of a suitable trajectory space and of a metric accounting for the double-well type nonlinearity contained in the director equation. Finally, a dissipative estimate is obtained by using a proper integrated energy inequality. Both the cases of (homogeneous) Neumann and (non-homogeneous) Dirichlet boundary conditions for d are considered.