Global strong solutions of the full Navier-Stokes and $Q$-tensor system for nematic liquid crystal flows in $2D$: existence and long-time behavior

TL;DR

This paper proves the existence, uniqueness, and long-time behavior of global strong solutions for a coupled Navier-Stokes and Q-tensor system modeling nematic liquid crystal flows in 2D, without smallness restrictions on key parameters.

Contribution

It establishes the first result on global strong solutions for the full system in 2D without smallness assumptions and analyzes their asymptotic convergence.

Findings

Existence and uniqueness of global strong solutions in 2D

Uniform boundedness of solutions over time

Convergence to asymptotic limits with explicit rates

Abstract

We consider a full Navier-Stokes and $Q$-tensor system for incompressible liquid crystal flows of nematic type. In the two dimensional periodic case, we prove the existence and uniqueness of global strong solutions that are uniformly bounded in time. This result is obtained without any smallness assumption on the physical parameter $\xi$ that measures the ratio between tumbling and aligning effects of a shear flow exerting over the liquid crystal directors. Moreover, we show the uniqueness of asymptotic limit for each global strong solution as time goes to infinity and provide an uniform estimate on the convergence rate.