(This is a joint Analysis -AMC Seminar).
Since the mid-1990’s, a great deal of effort by the research community on partial differential equations has been devoted to the mathematical study of the Ericksen-Leslie equations of liquid crystal flow. Formulated as balance laws of linear and angular momentum for the velocity, pressure and unit director fields, the treatment of flow with defects, that is, locations where the director field vanishes, still present open problems.
We present recent results in modeling and well-posedness of the governing system, focused on the role of the scalar order parameter in the relaxation of singularities. In the conclusion, we discuss a mixture theory approach to model systems of bacteria in liquid crystal flow arising in the design of nano-transport devices.