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

Abstract

In this paper, we first establish the global well-posedness of strong solutions of the simplified Ericksen-Leslie model for nonhomogeneous incompressible nematic liquid crystal flows in dimensions two, if the initial data satisfies some smallness condition. It is worth pointing out that the initial density is allowed to contain vacuum states and the initial velocity can be arbitrarily large. Next, we present a Serrin's type criterion, depending only on $\nabla d$, for the breakdown of local strong solutions. As a byproduct, the global strong solutions with large initial data are obtained, provided the macroscopic molecular orientation of the liquid crystal materials satisfies a natural geometric angle condition (cf. [19]).