On the rigidity of nematic liquid crystal flow on S2S2

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 S2S2, namely the uniform convergence in L2L2 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∞d∞ and any bubble ωiωi (1⩽i⩽l1⩽i⩽l) share a common orientation. As a consequence, the uniformity property holds under various types of small initial data.