TY - JOUR

T1 - A Kirchoff-Sobolev parametrix for the wave equation and applications

AU - Klainerman, Sergiu

AU - Rodnianski, Igor

N1 - Funding Information:
The first author is partially supported by NSF grant DMS-0070696. The second author is partially supported by NSF grant DMS-01007791. Part of this work was done while he was visiting Department of Mathematics at MIT.

PY - 2007/9

Y1 - 2007/9

N2 - We construct a first order, physical space, parametrix for solutions to covariant, tensorial, wave equations on a general Lorentzian manifold. The construction is entirely geometric; that is both the parametrix and the error terms generated by it have a purely geometric interpretation. In particular, when the background Lorentzian metric satisfies the Einstein vacuum equations, the error terms, generated at some point p of the space-time, depend, roughly, only on the flux of curvature passing through the boundary of the past causal domain of p. The virtues of our specific geometric construction becomes apparent in applications to realistic problems. Though our main application is to General Relativity, which we discuss in [14], another simpler application shown here is to give a gauge invariant proof of the classical regularity result of Eardley - Moncrief [4, 5] for the Yang - Mills equations in ℝ1+3.

AB - We construct a first order, physical space, parametrix for solutions to covariant, tensorial, wave equations on a general Lorentzian manifold. The construction is entirely geometric; that is both the parametrix and the error terms generated by it have a purely geometric interpretation. In particular, when the background Lorentzian metric satisfies the Einstein vacuum equations, the error terms, generated at some point p of the space-time, depend, roughly, only on the flux of curvature passing through the boundary of the past causal domain of p. The virtues of our specific geometric construction becomes apparent in applications to realistic problems. Though our main application is to General Relativity, which we discuss in [14], another simpler application shown here is to give a gauge invariant proof of the classical regularity result of Eardley - Moncrief [4, 5] for the Yang - Mills equations in ℝ1+3.

KW - Kirchoff

KW - Paramix

KW - Wave equation

KW - Yang-Mills

UR - http://www.scopus.com/inward/record.url?scp=34447277696&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34447277696&partnerID=8YFLogxK

U2 - 10.1142/S0219891607001203

DO - 10.1142/S0219891607001203

M3 - Article

AN - SCOPUS:34447277696

SN - 0219-8916

VL - 4

SP - 401

EP - 433

JO - Journal of Hyperbolic Differential Equations

JF - Journal of Hyperbolic Differential Equations

IS - 3

ER -