On the rigidity of nematic liquid crystal flow on $S^2$

In this paper we establish the uniformity property of a simplified Ericksen-Leslie system modelling the hydrodynamics of nematic liquid crystals on the two dimensional unit sphere $S^2$, namely the uniform convergence in $L^2$ to a steady state exponentially as t tends to infinity. The main assumption, similar to Topping [15], concerns the equation of liquid crystal director $d$ and states that at infinity time, a weak limit $d_\infty$ and any bubble $\omega_i$($1\le i\le l$) share a common orientation. As consequences, the uniformity property holds under various types of small initial data.