Global Solvability of the Inhomogeneous Ericksen-Leslie System with only Bounded Density

Abstract

Ericksen and Leslie established a theory to model the flow of nematic liquid crystals. This paper is devoted to the Cauchy Problem of a simplified version of their system, which retains most of the properties of the original one. We consider the density-dependent case and we establish the global existence of solutions in the whole space for small initial data. The initial density only has to be bounded and kept far from vacuum, while the initial velocity belongs to some critical Besov Space. Under a little bit more regularity for the initial velocity, we prove also that those solutions are unique.