Global existence of Maxwell–Klein–Gordon fields in (2+1)-dimensional space-time

We study the global existence problem for the Maxwell–Klein–Gordon equations in (2+1)-dimensional, Minkowski spacetime. We first establish local existence, in a suitable Sobolev space, by specializing to the Lorentz gauge and applying standard techniques. We then prove global existence by showing that an appropriate norm of the solutions cannot blow up in a finite time. An essential step in the proof involves showing that a certain second order ’’energy’’ does not blow up.