Energy Dissipation and Regularity for a Coupled Navier-Stokes and Q-Tensor System

TL;DR

This paper investigates a coupled Navier-Stokes and Q-tensor system modeling nematic liquid crystals, establishing global weak solutions, higher regularity in 2D, and weak-strong uniqueness.

Contribution

It proves the existence of global weak solutions for the coupled system and demonstrates higher regularity and weak-strong uniqueness in two dimensions.

Findings

Existence of global weak solutions in 2D and 3D.

Higher regularity results in 2D.

Weak-strong uniqueness in 2D.

Abstract

We study a complex non-newtonian fluid that models the flow of nematic liquid crystals. The fluid is described by a system that couples a forced Navier-Stokes system with a parabolic-type system. We prove the existence of global weak solutions in dimensions two and three. We show the existence of a Lyapunov functional for the smooth solutions of the coupled system and use the cancellations that allow its existence to prove higher global regularity, in dimension two. We also show the weak-strong uniqueness in dimension two.