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

Abstract

In this paper we prove the existence of a trajectory attractor (in the sense of Chepyzhov and Vishik) for a nonlinear PDE system obtained 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.