Parametrix for wave equations on a rough background IV: control of the error term

2012/04/08 by Szeftel, Jeremie
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc)

paper · doi:10.48550/arxiv.1204.1771

Abstract

This is the last of a sequence of four papers \citeparam1, \citeparam2, \citeparam3, \citeparam4 dedicated to the construction and the control of a parametrix to the homogeneous wave equation \square_{\bf g} ϕ=0, where \bf g is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes L² bounds on the curvature tensor \bf R of \bf g is a major step of the proof of the bounded L² curvature conjecture proposed in \citeKl:2000, and solved by S. Klainerman, I. Rodnianski and the author in \citeboundedl2. On a more general level, this sequence of papers deals with the control of the eikonal equation on a rough background, and with the derivation of L² bounds for Fourier integral operators on manifolds with rough phases and symbols, and as such is also of independent interest.

Related

• Parametrix for wave equations on a rough background II: construction and control at initial time
• Parametrix for wave equations on a rough background I: regularity of the phase at initial time
• Parametrix for wave equations on a rough background III: space-time regularity of the phase
• Sharp Strichartz estimates for the wave equation on a rough background
• The Bounded L2 Curvature Conjecture