Global existence of weak solutions of the nematic liquid crystal flow in dimensions three

Abstract

For any bounded smooth domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^3$, we establish the global existence of a weak solution $u:\hat{I}\text{c◯}\times (0,+\infty )\to {\mathbb{R}}^3\times {\mathbb{S}}^2$ of the initial-boundary value (or the Cauchy) problem of the simplified Ericksen-Leslie system (1.1) modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary (or Cauchy) data $(u_0.d_0)\in \mathbf{H}\times H^1(\hat{I}\text{c◯},{\mathbb{S}}^2$), with $d_0(\hat{I}\text{c◯})\subset {\mathbb{S}}_{+}^2$ (the upper hemisphere). Furthermore, ($u,d$) satisfies the global energy inequality (1.4).

24 pages