Convergence to equilibrium of global weak solutions for a Q-tensor problem related to liquid crystals

TL;DR

This paper proves the existence of global weak solutions and their convergence to equilibrium over time for a complex Q-tensor model describing nematic liquid crystals in three dimensions, combining fluid dynamics and elastic forces.

Contribution

It establishes the global existence and long-time convergence of weak solutions for a coupled Navier-Stokes and Allen-Cahn Q-tensor system, advancing mathematical understanding of liquid crystal dynamics.

Findings

Existence of global weak solutions established.

Solutions converge to equilibrium as time approaches infinity.

Utilization of Lojasiewicz-Simon's inequality to prove convergence.

Abstract

We study a Q-tensor problem modeling the dynamic of nematic liquid crystals in 3D domains. The system consists of the Navier-Stokes equations, with an extra stress tensor depending on the elastic forces of the liquid crystal, coupled with an Allen-Cahn system for the Q-tensor variable. This problem has a dissipative in time free-energy which leads, in particular, to prove the existence of global in time weak solutions. We analyze the large-time behavior of the weak solutions. By using a Lojasiewicz-Simon's result, we prove the convergence as time goes to infinity of the whole trajectory to a single equilibrium.