Modeling and analysis of the ericksen-leslie equations for nematic liquid crystal flows

This survey article discusses various aspects of modeling and analysis of the Ericksen-Leslie equations describing nematic liquid crystal flow both in the isothermal and non-isothermal situation. Of special interest is the development of thermodynamically consistent Ericksen-Leslie models in the general situation based on the entropy principle. The full analytical understanding of the dynamics of the underlying system is then based on this principle and gives rise to a rather complete understanding of the dynamics of this system. Furthermore, well-posedness and long-time behavior results in the weak and strong sense are described for the general Ericksen-Leslie system and various simplifications, ranging from the simplified and penalized model, with or without stretching terms, to the case of general Leslie stress both for incompressible and compressible fluids.