Nematic Liquid Crystal Flow with Partially Free Boundary

We study a simplified Ericksen-Leslie system modeling the flow of nematic liquid crystals with partially free boundary conditions. It is coupled between Navier-Stokes equation for fluid velocity transported heat harmonic maps, and both these parabolic equations are critical analysis in two dimensions. The conditions physically natural they correspond to Navier slip condition zero friction field Plateau-Neumann type map. In this paper we construct smooth solutions that blow up finite time at any finitely many given points on or interior domain.