Boundedness of the conformal hyperboloidal energy for a wave-Klein-Gordon model

TL;DR

This paper proves that a weighted energy for a coupled wave-Klein-Gordon system remains nearly bounded over time by employing a hierarchy of fractional Morawetz estimates and the Hyperboloidal Foliation Method.

Contribution

It introduces a novel hierarchy of fractional Morawetz energy estimates derived from conformal transformations to establish boundedness of solutions.

Findings

Weighted energy remains almost bounded for all times.

Hierarchy of fractional Morawetz estimates is effective.

Optimal estimates achieved using the scaling vector field.

Abstract

We consider the global evolution problem for a model which couples together a nonlinear wave equation and a nonlinear Klein-Gordon equation, and was independently introduced by LeFloch and Y. Ma and by Q. Wang. By revisiting the Hyperboloidal Foliation Method, we establish that a weighted energy of the solutions remains (almost) bounded for all times. The new ingredient in the proof is a hierarchy of fractional Morawetz energy estimates (for the wave component of the system) which is defined from two conformal transformations. The optimal case for these energy estimates corresponds to using the scaling vector field as a multiplier for the wave component.