Global Solution to the Klein--Gordon--Zakharov Equations with Uniform Energy Bounds

Abstract

On the (1+3) dimensional Minkowski spacetime, for small, regular initial data, it is well-known that the Dirac-Klein-Gordon system admits a global solution. In the present paper, we aim to establish the uniform boundedness of the total energy of the solution for this system. The proof relies on Klainerman's vector field and Alinhac's ghost weight methods.The main difficulty originates from the slow decay nature of the Dirac and wave components in three space dimensions. To overcome the difficulty, a sharp understanding of the structure for this system, and a new weighted conformal energy estimate are required. In addition, we also provide a few scattering results for the system.

Nonlinear stability of the totally geodesic wave maps in non-isotropic manifolds

“…However, different from the case in R 1+3 , here the space-time integration controlled on the left-hand side will play essential roles in the following discussion. For the Klein-Gordon equation (c > 0), this is essentially the ghost weight energy introduced in [36,53].…”

“…Then we make some necessary improvements on Soblev decay estimates in Section 8 in order to be adapted to this lower dimensional case. For other type of techniques aimed at wave-Klein-Gordon system with non-compactly supported initial data, the reader is referred to [31,32,33,34,35,36,37,38,39].…”