Freedericksz Transition in Nematic Liquid Crystal Flows in Dimension Two

In this article we are concerned with the dynamical instability of liquid crystals induced by applied magnetic fields, which is the so-called Freedericksz transition. The model under consideration is the simplified Ericksen--Leslie system for a nematic liquid crystal in a magnetic field. We showed that if the magnetic field strength exceeds a threshold, any global solution converges exponentially to a unique nontrivial equilibrium solution. Here we also imposed certain initial and strong anchoring boundary conditions. It is also proved that below the threshold, the asymptotic limit is the trivial state.