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 $|u_0|_2+|\nabla d_0|_2$ and $|\nabla u_0|_2+|\nabla^2d_0|_2$ is suitably small. This result can be viewed as a supplement to the local existence and blow up criteria discussed in \cite{Hong1}.