Weak-strong uniqueness of the full coupled Navier-Stokes and Q-tensor system in dimension three

TL;DR

This paper proves weak-strong uniqueness for the 3D full coupled Navier-Stokes and Q-tensor system modeling nematic liquid crystals, introducing a new criterion and establishing global well-posedness for small initial data.

Contribution

It introduces a novel uniqueness criterion involving $( riangle Q, abla u)$ and extends weak-strong uniqueness results to a new parameter regime, including the simplified case $\xi=0$.

Findings

Established weak-strong uniqueness in 3D for the coupled system.

Proposed a new criterion based on $( riangle Q, abla u)$ with specific regularity.

Proved global well-posedness for small initial data.

Abstract

In this paper, we study the weak-strong uniqueness for the Leray-Hopf type weak solutions to the Beris-Edwards model of nematic liquid crystals in $\mathbb{R}^3$ with an arbitrary parameter $\xi\in\mathbb{R}$, which measures the ratio of tumbling and alignment effects caused by the flow. This result is obtained by proposing a new uniqueness criterion in terms of $(\Delta Q,\nabla u)$ with regularity $L_t^qL_x^p$ for $\frac{2}{q}+\frac{3}{p}=\frac{3}{2}$ and $2\leq p\leq 6$, which enable us to deal with the additional nonlinear difficulties arising from the parameter $\xi$. Comparing with the results of related literature, our finding also reveals a new regime of weak-strong uniqueness for the simplified case of $\xi=0$. Moreover, we establish the global well-posedness of this model for small initial data in $H^s$-framework.