Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two

Abstract

This paper is devoted to the study of the initial-boundary value problem for density-dependent incompressible nematic liquid crystal flows with vacuum in a bounded smooth domain of \begin{document}$\mathbb{R}^2$\end{document} . The system consists of the Navier-Stokes equations, describing the evolution of an incompressible viscous fluid, coupled with various kinematic transport equations for the molecular orientations. Assuming the initial data are sufficiently regular and satisfy a natural compatibility condition, the existence and uniqueness are established for the global strong solution if the initial data are small. We make use of a critical Sobolev inequality of logarithmic type to improve the regularity of the solution. Our result relaxes the assumption in all previous work that the initial density needs to be bounded away from zero.