Existence of solution to a system of PDEs modeling the crystal growth inside lithium batteries

The life-cycle of electric batteries depends on a complex system of interacting electrochemical and growth phenomena that produce dendritic structures during the discharge cycle. We study herein a system of 3 partial differential equations combining an Allen--Cahn phase-field model (simulating the dendrite-electrolyte interface) with the Poisson--Nernst--Planck systems simulating the electrodynamics and leading to the formation of such dendritic structures. We prove novel existence, uniqueness and stability results for this system and use it to produce simulations based on a finite element code.