Global solvability of the inhomogeneous Ericksen–Leslie system with only bounded density

Abstract

The most established theory for modeling the dynamics of nematic liquid crystals is the celebrated Ericksen–Leslie system, which presents some major analytical challenges. We study a simplified version of the system, which still exhibits the major difficulties. We consider the density-dependent case and study the Cauchy problem in the whole space. We establish the global existence of solutions for small initial data by assuming only that the initial density is bounded and kept away far from vacuum, while the initial velocity and the gradient of the initial director field belong to certain critical Besov spaces. Under slightly more assumptions on the initial velocity and the director field, we also prove that the solutions are unique.