On asymptotic isotropy for a hydrodynamic model of liquid crystals

TL;DR

This paper analyzes a PDE model for liquid crystals, demonstrating that solutions tend to an isotropic state over time at a specific decay rate using Fourier splitting methods.

Contribution

It establishes the asymptotic isotropy of solutions in a coupled Q-tensor and Navier-Stokes model, providing decay rates for the approach to equilibrium.

Findings

Solutions tend to isotropic state as t → ∞

Decay rate of (1 + t)^{-3/2} for solutions

Method of Fourier splitting applied to PDE system

Abstract

We study a PDE system describing the motion of liquid crystals by means of the $Q-$tensor description for the crystals coupled with the incompressible Navier-Stokes system. Using the method of Fourier splitting, we show that solutions of the system tend to the isotropic state at the rate $(1 + t)^{-3/2}$ as $t \to \infty$.