Contents Online
Methods and Applications of Analysis
Volume 22 (2015)
Number 2
Global strong solutions to the inhomogeneous incompressible nematic liquid crystal flow
Pages: 201 – 220
DOI: https://dx.doi.org/10.4310/MAA.2015.v22.n2.a4
Author
Abstract
In this paper, we consider the Dirichlet problem to the inhomogeneous incompressible nematic liquid crystal system in bounded smooth domains of two or three dimensions. We prove the global existence and uniqueness of strong solutions to this system, with initial data being of small norm but allowed to have vacuum and large oscillations. More precisely, for the two dimensional case, we only require that the basic energy ${\lVert \sqrt{\rho_0} u_0 \rVert}^{2}_{L^2} + {\lVert \nabla d_0 \rVert}^{2}_{L^2}$ is small, while for the three dimensional case, we ask for the smallness of the production of the basic energy and the quantity ${\lVert \nabla u_0 \rVert}^{2}_{L^2} + {\lVert \Delta d_0 \rVert}^{2}_{L^2}$ We achieve some suitable time independent a priori estimates on the strong solutions, based on which, one can extend the local strong solution to be a global one.
Keywords
existence and uniqueness, global strong solutions, liquid crystal
2010 Mathematics Subject Classification
35D35, 35Q35, 76A15, 76D03
Published 1 June 2015