Mathematisches Kolloquium

Zusammenfassung: I will report on a series of results on the solution theory of the coupled Maxwell and Lorentz equations describing the motion of N classical charges. I will start the discussion with a system of N rigid charges whose solutions can be obtained in terms of an initial value problem and explain the major obstacles in passing to the physically desirable point charge limit. In particular, I will demonstrate that already the Maxwell-Lorentz equations for point charges without self-interaction do not admit an initial value problem in the common sense as generic initial values lead to irregularities in the electromagnetic fields that travel along the light-cones of the initial charge positions and prevent global existence. Global smooth solutions can only be obtained by imposing an infinite system of constraints which effectively turn the Maxwell-Lorentz system into a system of delay differential equations. This system of delay equations is neutral, non-linear and involves state-dependent delays which renders its mathematical study very cumbersome. I will close with a discussion of recent results concerning existence and uniqueness of its solutions.

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