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In this paper, the authors first establish the global well-posedness ofstrong solutions of the simplified Ericksen-Leslie model for nonhomogeneousincompressible nematic liquid crystal flows in two dimensions if the initialdata satisfies some smallness condition. It is worth pointing out that theinitial density is allowed to contain vacuum states and the initial velocitycan be arbitrarily large. We also present a Serrin's type criterion, dependingonly on $\nabla d$, for the breakdown of local strong solutions. As abyproduct, the global strong solutions with large initial data are obtained,provided the macroscopic molecular orientation of the liquid crystal materialssatisfies a natural geometric angle condition (cf. [19])

Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density