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

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 $\R^3$ with an arbitrary parameter $\xi\in\R$, 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 enables us to deal with the additional nonlinear difficulties arising from the parameter $\xi$. Compared with the known results, our finding reveals that the criterion of weak-strong uniqueness for $\xi\ne 0$ is a sub-regime of the one for the corotational case. The associated regularity assumption rises with the nonlinearity of the model. Moreover, we establish the global well-posedness of this model for small initial data in $H^s$-framework.