On optimal boundary control of Ericksen–Leslie system in dimension two

• DOI: 10.1007/s00526-019-1676-z
• Publication Date: 2020-01-25T19:03:01Z
• Authors: Qiao Liu; Changyou Wang; Xiaotao Zhang; Jianfeng Zhou
• Keywords: Mathematics - Analysis of PDEs; 35B65, 35Q35; FOS: Mathematics; 0101 mathematics; 01 natural sciences; Analysis of PDEs (math.AP)

In this paper, we consider the boundary value problem of a simplified Ericksen-Leslie system in dimension two with non-slip boundary condition for the velocity field $u$ and time-dependent boundary condition for the director field $d$ of unit length. For such a system, we first establish the existence of a global weak solution that is smooth away from finitely many singular times, then establish the existence of a unique global strong solution that is smooth for $t>0$ under the assumption that the image of boundary data is contained in the hemisphere $\mathbb S^2_+$. Finally, we apply these theorems to the problem of optimal boundary control of the simplified Ericksen-Leslie system and show both the existence and a necessary condition of an optimal boundary control.<br/>70 pages<br/>