Local and global solutions on arcs for the Ericksen–Leslie problem in RN$\mathbb {R}^N

The work deals with the Ericksen–Leslie system for nematic liquid crystals on the space RN$\mathbb {R}^N$ with N≥3$N\ge 3$. In our work, we suppose the initial condition v0$v_0$ stays on an arc connecting two fixed orthogonal vectors on the unit sphere. Thanks to this geometric assumption, we prove through energy a priori estimates the local existence and the global existence for small initial data of a solution u∈L∞((0,T);Hs(RN)),∇u∈L2((0,T);Hs(RN)),$$\begin{equation*} \hspace*{60pt}u\in L^\infty ((0,T);H^s(\mathbb {R}^N)),\quad \nabla u\in L^2((0,T);H^s(\mathbb {R}^N)), \end{equation*}$$∇v∈L∞((0,T);Hs(RN)),∇2v∈L2((0,T);Hs(RN))$$\begin{equation*} \hspace*{55pt}\nabla v\in L^\infty ((0,T);H^s(\mathbb {R}^N)),\quad \nabla ^2v\in L^2((0,T);H^s(\mathbb {R}^N)) \end{equation*}$$for s>N2−1$s>\frac{N}{2}-1$, asking low regularity assumptions on u0$u_0$ and v0$v_0$.