Stability of a coupled wave-Klein-Gordon system with quadratic nonlinearities

Using the hyperboloidal foliation method, we establish stability results for a coupled wave-Klein-Gordon system with quadratic nonlinearities. In particular, we investigate quadratic wave-Klein-Gordon interactions in which there are no derivatives on the massless wave component. By combining hyperboloidal energy estimates with appropriate transformations of our fields, we are able to show global existence of solutions for sufficiently small initial data.Comment: Published version. Both authors are grateful to the anonymous referees, whose comments improve a lot on the presentation and the precision of the articl

Asymptotic stability for the Dirac--Klein-Gordon system in two space dimensions

We study the Dirac--Klein-Gordon system in $1+2$ spacetime dimensions. We show global existence of the solutions, as well as sharp time decay and linear scattering. One key advance is that we provide the first asymptotic stability result for the Dirac--Klein-Gordon system in $1+2$ spacetime dimensions in the case of a massive Klein-Gordon field and a massless Dirac field. The nonlinearities are below-critical in two spatial dimensions, and so our method requires the identification of special structures within the system and novel weighted energy estimates. Another key advance, is that our proof allows us to weaken certain conditions on the nonlinear structures that have been assumed in the literature.Comment: 37 page

Stability of some two dimensional wave maps: a wave--Klein-Gordon model

We are interested in the stability of a class of totally geodesic wave maps, as recently studied by Abbrescia and Chen, and later by Duan and Ma. The relevant equations of motion are a system of coupled semilinear wave and Klein-Gordon equations in $\mathbb{R}^{1+n}$ whose nonlinearities are critical when $n=2$. In this paper we use a pure energy method to show global existence when $n=2$. By carefully examining the structure of the nonlinear terms, we are able to obtain uniform energy bounds at lower orders. This allows us to prove pointwise decay estimates and also to reduce the required regularity.Comment: arXiv admin note: text overlap with arXiv:2011.1199