Existence of energy minimizers and well-posedness of the coupled gradient flow

Establish the existence of energy minimizers and the well-posedness of the coupled $L^2$ gradient-flow dynamics associated with the thermodynamically consistent diffuse-interface model for multicomponent vesicles.

Background

The paper formulates a coupled free-energy functional incorporating protein-dependent bending elasticity, surface tension, a volume penalty, and a membrane-associated Ohta–Kawasaki energy. Applying the Onsager variational principle yields a coupled nonconservative $L^2$ gradient-flow system for the membrane phase field and the membrane-associated protein density.

The authors note that the variational structure is suitable for rigorous analysis analogous to existing work, but the paper does not establish either the existence of minimizers for the coupled free energy or the well-posedness of the resulting coupled dynamics. These analytical questions are explicitly deferred to future work.