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

Jinkai Li (Institute of Mathematical Sciences, Chinese University of Hong Kong; and Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel)

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

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Published 1 June 2015