Transference of Bilinear Restriction Estimates to Quadratic Variation Norms and the Dirac-Klein-Gordon System

TL;DR

This paper extends bilinear Fourier restriction estimates to quadratic variation norms and applies these results to establish global well-posedness and scattering for the 3D Dirac-Klein-Gordon system, including resonant cases.

Contribution

It introduces a novel extension of bilinear restriction estimates to quadratic variation spaces and applies this to prove new well-posedness results for the Dirac-Klein-Gordon system.

Findings

Extended bilinear restriction estimates to quadratic variation spaces.

Proved global well-posedness for Dirac-Klein-Gordon in 3D with small data.

Established scattering results in resonant and non-resonant regimes.

Abstract

Firstly, bilinear Fourier Restriction estimates --which are well-known for free waves-- are extended to adapted spaces of functions of bounded quadratic variation, under quantitative assumptions on the phase functions. This has applications to nonlinear dispersive equations, in particular in the presence of resonances. Secondly, critical global well-posedness and scattering results for massive Dirac-Klein-Gordon systems in dimension three are obtained, in resonant as well as in non-resonant regimes. The results apply to small initial data in scale-invariant Sobolev spaces exhibiting a small amount of angular regularity.