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

TL;DR

This paper proves the global existence of weak solutions for a 3D isothermal nematic liquid crystal model that includes stretching effects, addressing mathematical challenges without restrictions on data or parameters.

Contribution

It establishes the well-posedness of a complex PDE system modeling nematic liquid crystals with stretching, overcoming the lack of a maximum principle for the director field.

Findings

Proved existence of global weak solutions.

Handled the coupling between flow and director fields.

Addressed the absence of a maximum principle.

Abstract

The present contribution investigates the well-posedness of a PDE system describing the evolution of a nematic liquid crystal flow under kinematic transports for molecules of different shapes. More in particular, the evolution of the {\em velocity field} $\ub$ is ruled by the Navier-Stokes incompressible system with a stress tensor exhibiting a special coupling between the transport and the induced terms. The dynamic of the {\em director field} $\bd$ is described by a variation of a parabolic Ginzburg-Landau equation with a suitable penalization of the physical constraint $|\bd|=1$. 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 $\bd$ and $\ub$.