Moving vesicles in elastic tissues: A model with existence and uniqueness of weak solutions

We describe transport of vesicles in an elastic body in terms of a phase field. Besides standard forces implied by macroscopic strain, we account for a balance of microstructural actions induced by vesicle-to-vesicle interactions, vesicle self-actions, membrane bending and related nonlinear pressure- type confinement effects on vesicles. For the resulting balance equations, we prove existence and uniqueness of appropriate weak solutions (the so-called regular weak ones). Also, in the absence of bulk forces and under homogeneous Neumann-type boundary conditions, we prove a weak maximum principle for the vesicle phase-field.