Global Strong Solutions to Incompressible Ericksen-Leslie System in ${\mathbb{R}}^3$

Abstract

In this paper, we consider the Cauchy problem to the Ericksen-Leslie system of liquid crystals in ${\mathbb{R}}^3$. Global well-posedness of strong solutions are obtained under the condition that the product of $\Vert u_0{\Vert }_2+\Vert \nabla d_0{\Vert }_2$ and $\Vert \nabla u_0{\Vert }_2+\Vert {\nabla }^2{d}_0{\Vert }_2$ is suitably small. This result can be viewed as a supplement to the local existence and blow up criteria discussed in \cite{Hong1}.