Analysis of a Cahn-Hilliard-Canham-Helfrich system for the evolution of a two-phase membrane

The coupling of the evolution of a surface with evolution equations defined on that surface is of relevance in many applications and has been in the focus of interest in the analysis of parabolic PDEs in recent years. In applications the evolution of two-phase vesicles and biomembranes is governed by flows decreasing an energy which involves Canham-Helfrich-type curvature energies coupled to a Ginzburg-Landau energy. We derive a new Cahn-Hilliard-Canham-Helfrich system for the evolution of two-phase membranes. The resulting system is highly non-linear and we use the theory of quasi-linear parabolic evolution equations in weighted $L_p$-spaces to show the existence of a strong local-in-time solution and hence demonstrate that the derived system is well-posed.