Growth of Sobolev norms for the Maxwell–Dirac and Dirac–Klein–Gordon systems in R 1 + 1

Abstract

We obtain the growth of Sobolev norms of the solution to the Maxwell–Dirac equations in R 1 + 1 by applying elementary techniques. In particular, we estimate L ∞ bound of the solution by making use of a local energy conservation. The similar idea can be applied to the Dirac–Klein–Gordon equations.