Nematic liquid crystal flow with partially free boundary

Abstract

We study a simplified Ericksen-Leslie system modeling the flow of nematic liquid crystals with partially free boundary conditions. It is a coupled system between the Navier-Stokes equation for the fluid velocity with a transported heat flow of harmonic maps, and both of these parabolic equations are critical for analysis in two dimensions. The boundary conditions are physically natural and they correspond to the Navier slip boundary condition with zero friction for the velocity field and a Plateau-Neumann type boundary condition for the map. In this paper we construct smooth solutions of this coupled system that blow up in finite time at any finitely many given points on the boundary or in the interior of the domain. Contents Introduction Main resultsRoadmap to the construction 2. Reflection and symmetry of the full system 3. Notations and preliminaries 4. Approximation and improvement 5. Gluing system and derivation of the dynamics for parameters 6. Linear theories for linearized harmonic map heat flow and Stokes system 7. Solving the partially free boundary system 7.1. Couplings in the full system 7.2. Inner-outer gluing system and reduced problems 7.3. Weighted topologies and fixed point argument Appendix A. Derivation of the partially free boundary system Appendix B. Multiple bubbles: analysis of the interactions Appendix C. Analysis of the Stokes operator