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

TL;DR

This paper proves the global existence and uniqueness of solutions for a simplified inhomogeneous Ericksen-Leslie system modeling nematic liquid crystal flow, with bounded initial density and small initial velocity.

Contribution

It establishes the first global solvability results for the inhomogeneous Ericksen-Leslie system with only bounded density and small initial data.

Findings

Global solutions exist for small initial data with bounded density

Solutions are unique under additional regularity assumptions

The initial density must be bounded and away from vacuum

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.