The coupled Maxwell-Lorentz system describes feed-back action of electromagnetic fields in classical electrodynamics. When applied to point-charge sources (viewed as limiting cases of charged fluids) the resulting nonlinear weakly hyperbolic system lies beyond the scope of classical distribution theory. Using regularized derivatives in the framework of Colombeau algebras of generalized functions we analyze a two-dimensional analogue of the Maxwell-Lorentz system. After establishing existence and uniqueness of solutions in this setting we derive some results on distributional limits of solutions with 6-like initial values.
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Günther Hörmann, Michael Kunzinger, Regularized Derivatives in a 2-Dimensional Model of Self-Interacting Fields with Singular Data. Z. Anal. Anwend. 19 (2000), no. 1, pp. 147–158