Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum

Abstract

We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible nematic liquid crystal flows on the whole space ${\mathbb{R}}^2$ with vacuum as far field density. It is proved that the 2D nonhomogeneous incompressible nematic liquid crystal flows admits a unique global strong solution provided the initial data density and the gradient of orientation decay not too slow at infinity, and the basic energy $\Vert \sqrt{{\rho }_0}{\mathbf{u}}_0{\Vert }_{L^2}^2+\Vert \nabla {\mathbf{d}}_0{\Vert }_{L^2}^2$ is small. In particular, the initial density may contain vacuum states and even have compact support. Moreover, the large time behavior of the solution is also obtained.

23 pages. arXiv admin note: substantial text overlap with arXiv:1507.06471, arXiv:1506.03884, Nonlinearity, 2017

References in corpus (4)

• Global Well-Posedness and Large Time Asymptotic Behavior of Classical Solutions to the Compressible Navier-Stokes Equations with Vacuum
• On Local Strong Solutions to the Cauchy Problem of Two-Dimensional Density-Dependent Magnetohydrodynamic Equations with Vacuum
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• Global Existence and Large Time Asymptotic Behavior of Strong Solutions to the Cauchy Problem of 2D Density-Dependent Magnetohydrodynamic Equations with Vacuum