A multi-phase Mullins-Sekerka system: matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem

We propose a generalisation of the Mullins-Sekerka problem to model phase separation in multi-component systems. The model includes equilibrium equations in bulk, the Gibbs-Thomson relation on the interfaces, Young's law at triple junctions, together with a dynamic law of Stefan type. Using formal asymptotic expansions, we establish the relationship to a transition layer model known as the Cahn-Hilliard system. We introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds, together with measure theoretical tools. Through an implicit time discretisation, we construct approximate solutions by stepwise minimisation. Under the assumption that there is no loss of area as the time step tends to zero, we show the existence of a weak solution.