Well-posedness of Maxwell–Born–Infeld dynamics with prescribed moving point charges

Prove that the Maxwell–Born–Infeld field equations with prescribed reasonably moving point charge sources form a mathematically well-posed initial value problem (existence, uniqueness, and continuous dependence on initial data).

Background

Beyond finiteness of energy, a central mathematical requirement for a physical field theory is well-posedness of the initial value problem. The authors note that for MBI with given point-charge motions, this status has not yet been established. Demonstrating well-posedness would provide a rigorous foundation for analyzing dynamical MBI phenomena with point sources.

The contrast with BLTP is highlighted later in the paper, where the authors discuss progress toward well-posedness and practical computations; the question for MBI remains open.

References

In this vein, even the Maxwell--Born--Infeld field theory with prescribed reasonably moving point charge sources is not yet established as being mathematically well-posed, though conceivably and hopefully it will be in the not too distant future.

— Radiation-Reaction on the Straight-Line Motion of a Point Charge accelerated by a constant applied Electric Field in an Electromagnetic Bopp-Landé-Thomas-Podolsky vacuum (2506.08799 - McGuigan et al., 10 Jun 2025) in Section 1 (Introduction)