New Challenges in the Kinetic Theory of Complex Fluids

Overview

Complex fluids such as liquid crystals exhibit rich behavior that calls for models capable of capturing both microscopic structure and macroscopic dynamics. In this talk, I will present recent progress on a kinetic approach to complex fluids developed in collaboration with Giovanni Russo, José Carrillo, Patrick Farrell, and Andrea Medaglia. Beginning with Capriz’s notion of order parameter manifolds, I will describe how an extended BBGKY hierarchy leads to a Boltzmann-type equation (the CFM model), and discuss new challenges that arise from the interplay of slowly and rapidly varying coordinates. I will highlight the role of embedding approaches, weak order interactions, and alternative metrics such as the Bregman distance.

I will then turn to questions of relaxation and equilibrium. In particular, I will present results on the uniqueness of collision invariants when the order parameter manifold is transitive, and outline the open problem of what happens in the non-transitive case. Finally, I will discuss hydrodynamic limits and a new energy functional for calamitic fluids, addressing difficulties related to the lack of lower semicontinuity and coercivity, as well as connections to Nash equilibria and Lavrentiev phenomena. If time permits, I will conclude with a brief overview of a diffuse-interface model for ordered fluids and phase transitions.