A 3D isothermal model for nematic liquid crystals with delay terms

In this paper we consider a model describing the evolution of a nematic liquid crystal flow with delay external forces. We analyze the evolution of the velocity field u which is ruled by the 3D incompressible Navier-Stokes system containing a delay term and with a stress tensor expressing the coupling between the transport and the induced terms. The dynamics of the director field d is described by a modified Allen-Cahn equation with a suitable penalization of the physical constraint vertical bar d vertical bar = 1. We prove the existence of global in time weak solutions under appropriate assumptions, which in some cases requires the delay term to be small with respect to the viscosity parameter.