The authors treat here the Cauchy problem for the coupled Dirac-Klein-Gordon (DKG) system on , , . They study local (in time) well-posedness (LWP) for low regularity data.
Theorem. DKG in dimension has LWP for data for all . Here , . Iteration scheme is used in the proof. Let , and .
The proof relies on the null structure of the system, combined with the bilnear space-time estimates: , ,
That is, they obtain for and for satisfying the condition , and prove the estimates of for two norms. means for a constant .