# American Institute of Mathematical Sciences

September  2017, 37(9): 4907-4922. doi: 10.3934/dcds.2017211

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

 College of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

* Corresponding author: Xiaoli Li

Received  April 2016 Revised  April 2017 Published  June 2017

Fund Project: The author is supported in part by the National Natural Science Foundation of China under grants 11401036,11271052 and 11471050.

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 $\mathbb{R}^2$. 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.

Citation: Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
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