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We study a diffuse interface model describing the complex rheology and theinterfacial dynamics during phase separation in a polar liquid-crystallineemulsion. More precisely, the physical systems comprises a two-phase mixtureconsisting in a polar liquid crystal immersed in a Newtonian fluid. Suchcomposite material is a paradigmatic example of complex fluids arising in SoftMatter which exhibits multiscale interplay. Beyond the Ginzburg-Landau andFrank elastic energies for the concentration and the polarization, the freeenergy of the system is characterized by a quadratic anchoring term which tunesthe orientation of the polarization at the interface. This leads to severalquasi-linear nonlinear couplings in the resulting system describing themacroscopic dynamics. In this work, we establish the first mathematical resultsconcerning the global dynamics of two-phase complex fluids with interfacialanchoring mechanism. First, we determine a set of sufficient conditions on theparameters of the system and the initial conditions which guarantee theexistence of global weak solutions in two and three dimensions. Secondly, weshow that weak solutions are unique and globally regular in the two dimensionalcase. Finally, we complement our analysis with some numerical simulations todisplay polarization and interfacial anchoring.

Global Solutions for Two-Phase Complex Fluids with Quadratic Anchoring in Soft Matter Physics