  We study a diffuse interface model describing the complex rheology and the
interfacial dynamics during phase separation in a polar liquid-crystalline
emulsion. More precisely, the physical systems comprises a two-phase mixture
consisting in a polar liquid crystal immersed in a Newtonian fluid. Such
composite material is a paradigmatic example of complex fluids arising in Soft
Matter which exhibits multiscale interplay. Beyond the Ginzburg-Landau and
Frank elastic energies for the concentration and the polarization, the free
energy of the system is characterized by a quadratic anchoring term which tunes
the orientation of the polarization at the interface. This leads to several
quasi-linear nonlinear couplings in the resulting system describing the
macroscopic dynamics. In this work, we establish the first mathematical results
concerning the global dynamics of two-phase complex fluids with interfacial
anchoring mechanism. First, we determine a set of sufficient conditions on the
parameters of the system and the initial conditions which guarantee the
existence of global weak solutions in two and three dimensions. Secondly, we
show that weak solutions are unique and globally regular in the two dimensional
case. Finally, we complement our analysis with some numerical simulations to
display polarization and interfacial anchoring.
