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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ {\boldsymbol u} $\end{document}</tex-math></inline-formula> which is ruled by the 3D incompressible Navier-Stokes system containing a delay term and with a stress tensor expressing the coupling between the transport and the induced terms. The dynamics of the director field <inline-formula><tex-math id="M2">\begin{document}$ \boldsymbol{d} $\end{document}</tex-math></inline-formula> is described by a modified Allen-Cahn equation with a suitable penalization of the physical constraint <inline-formula><tex-math id="M3">\begin{document}$ | \boldsymbol{d}| = 1 $\end{document}</tex-math></inline-formula>. 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.</p>