Conformal-type energy estimates on hyperboloids and the wave-Klein-Gordon model of self-gravitating massive fields

Senhao Duan; Yue MA; Weidong Zhang; Senhao Duan; Yue MA; Weidong Zhang

doi:10.3934/cam.2023007

Communications in Analysis and Mechanics

2023, Volume 15, Issue 2: 111-131. doi: 10.3934/cam.2023007

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Conformal-type energy estimates on hyperboloids and the wave-Klein-Gordon model of self-gravitating massive fields

1.
Département ingénierie Mathématique et informatique, École des Ponts ParisTech, Paris, 77420, France
2.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, China

Received: 31 December 2022 Revised: 20 March 2023 Accepted: 04 April 2023 Published: 22 April 2023
35L72, 35L05, 83C05

In this article we revisit the global existence result of the wave-Klein-Gordon model of the system of the self-gravitating massive field. Our new observation is that, by applying the conformal energy estimates on hyperboloids, we obtain mildly increasing energy estimate up to the top order for the Klein-Gordon component, which clarify the question on the hierarchy of the energy bounds of the Klein-Gordon component in our previous work. Furthermore, a uniform-in-time energy estimate is established for the wave component up to the top order, as well as a scattering result. These improvements indicate that the partial conformal symmetry of the Einstein-massive scalar system will play an important role in the global analysis.
- conformal energy estimate,
- partial conformal symmetry,
- Einstein-massive field system,
- wave-Klein-Gordon system,
- nonlinear stability
Citation: Senhao Duan, Yue MA, Weidong Zhang. Conformal-type energy estimates on hyperboloids and the wave-Klein-Gordon model of self-gravitating massive fields[J]. Communications in Analysis and Mechanics, 2023, 15(2): 111-131. doi: 10.3934/cam.2023007

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Abstract

In this article we revisit the global existence result of the wave-Klein-Gordon model of the system of the self-gravitating massive field. Our new observation is that, by applying the conformal energy estimates on hyperboloids, we obtain mildly increasing energy estimate up to the top order for the Klein-Gordon component, which clarify the question on the hierarchy of the energy bounds of the Klein-Gordon component in our previous work. Furthermore, a uniform-in-time energy estimate is established for the wave component up to the top order, as well as a scattering result. These improvements indicate that the partial conformal symmetry of the Einstein-massive scalar system will play an important role in the global analysis.

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