Existence and uniqueness for the DFN model in two and three dimensions

Establish existence and uniqueness of weak solutions to the Doyle–Fuller–Newman (DFN) lithium‑ion battery model—a fully coupled nonlinear elliptic–parabolic system for electrolyte potential φ1(x,t), electrode potential φ2(x,t), electrolyte concentration c1(x,t) on the electrolyte domain Ω1, and solid-phase lithium concentration c2(x,r,t) on the electrode domain Ω2×(0,R_s(x)) with the specified boundary and interface conditions—in two and three spatial dimensions (N=2,3), thereby extending the known well‑posedness results from one spatial dimension.

Background

The paper defines a weak solution for the DFN model, a multiscale, multi-domain nonlinear elliptic–parabolic PDE system coupling electrolyte and solid-phase dynamics via a Butler–Volmer-type reaction term and interface/boundary conditions. Prior work has established local existence and uniqueness for one-dimensional domains, with global uniqueness under additional conditions.

The authors note that extending these well‑posedness results beyond one dimension faces major challenges, including strong nonlinear source terms with singular behavior, discontinuous and nonlinear coefficients across subdomains, non-smooth boundaries, and the pseudo-(N+1)-dimensional radial equation posed on Ω2×(0,R_s(x)).