Global Existence and Exponential Decay of Strong Solutions to the 2D Density-dependent Nematic Liquid Crystal Flows with Vacuum

Abstract

This paper deals with the 2D incompressible nematic liquid crystal flows with density-dependent viscosity in bounded domain. The global well-posedness of strong solutions are established in the vacuum cases, provided the assumption that $\overline{\rho} + \|\nabla d_0\|_{L^2}$ is suitably small with large velocity, which extends the recent work [Discrete Contin. Dyn. Syst. 37 (2017), 4907--4922] and [Methods Appl. Anal. 22 (2015), 201--220] to the case of variable viscosity. Furthermore, the exponential decay of the solution is also obtained.