Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system

Abstract

We prove almost optimal local well-posedness for the coupled Dirac-Klein-Gordon (DKG) system of equations in 1 + 3 dimensions. The proof relies on the null structure of the system, combined with bilinear spacetime estimates of Klainerman-Machedon type. It has been known for some time that the Klein-Gordon part of the system has a null structure; here we uncover an additional null structure in the Dirac equation, which cannot be seen directly, but appears after a duality argument.

Global Well-posedness Below the Charge Norm for the Dirac-Klein-Gordon System in One Space Dimension

“…The intrinsic null structure of DKG manifests itself through the different signs in the right hand side of the last equation; the same structure is in fact encoded in the first two equations, as becomes apparent via a duality argument; see [5,11].…”

“…; see [5] for more details about these spaces. We shall need the following basic estimates (see [5] for further references):…”

“…To prove the nonlinear smoothing we rely on the null structure of the DKG system, which was recently completed by P. D'Ancona, D. Foschi and the present author in [5], and on bilinear estimates in Bourgain-Klainerman-Machedon spaces.…”

“…In order to derive sufficiently strong spacetime estimates for the solution, we need to exploit the null structure inherent in DKG; to reveal this structure we decompose the spinor ψ using the eigenspace projections of the Dirac operator αD x , following [5,11]. The symbol αξ of αD x has eigenvalues ±ξ and corresponding eigenspace projections…”

Global Well-Posedness of the 1d Dirac–klein–gordon System in Sobolev Spaces of Negative Index

“…The estimates we present in the following two lemmas are a priori estimates for the solutions of the massive Dirac and Klein-Gordon Cauchy problems, and they 1 If we set m = 0 in (8), then the constant C in the energy estimate (11) will depend on t.…”