Global well-posedness and decay rates for the three dimensional incompressible active liquid crystals
This paper investigates the global well-posedness and large-time behavior of 3D incompressible active liquid crystals under constant activity, modeled by a coupled system of forced incompressible Navier-Stokes equations for the velocity and a parabolic system for the $Q$-tensor order parameter. By employing refined commutator estimates, the existence and uniqueness of global strong solutions are proved for small initial data $(Q_0,u_0)\in H^{s+1}\times H^s$ $(s\geq 2)$ with activity $c>c_\star$, which improves a previous result in \cite{active-limit}. In addition, if the initial data further belong to $L^1$ and $s\geq 4$, we obtain a mixing decay estimate on $\|\partial^kQ(t)\|_{L^2}$ that combines both an extra exponential decay factor at a rate proportional to $(c-c_\star)\Gamma$ and the optimal algebraic decay rate that coincides with that of the heat kernel, where $k\leq s-1$. This result reveals that, in the high activity regime, active nematics become isotropic with an activity-dependent exponential convergence rate, and the estimate is stable in the infinite rotational viscosity limit, as $\Gamma\rightarrow 0$. Meanwhile, the sharp decay estimate on $\|\partial^ku(t)\|_{L^2}$ is also derived for $k\leq s-2$ with an additional initial assumption. The proof is established via a combination of the Green's function method and the time-weighted energy method. To the best of our knowledge, these results are the first reported for active/passive nematic liquid crystals within the Beris-Edwards framework, and the enhanced decay effect of the orientational field is essentially derived from the free energy. Furthermore, in the passive setting, our result implies the phase transition of thermotropic liquid crystals at high temperatures.