Polyhomogeneous solutions of wave equations in the radiation regime

Abstract

While the physical properties of the gravitational field in the radiation regime are reasonably well understood, several mathematical questions remain unanswered. The question here is that of existence and properties of gravitational fields with asymptotic behavior compatible with existence of gravitational radiation. A framework to study those questions has been proposed by R. Penrose (R. Penrose, “Zero rest-mass fields including gravitation”, Proc. Roy. Soc. London A284 (1965), 159-203), and developed by H. Friedrich (H. Friedrich, “Cauchy problem for the conformal vacuum field equations in general relativity”, Commun. Math. Phys. 91 (1983), 445-472.), (H. Friedrich, “On the existence of n-geodesically complete or future complete solutions of Einstein's field equations with smooth asymptotic structure”, Commun. Math. Phys. 107 (1986), 587-609.), (-,“On the global existence and the asymptotic behavior of solutions to the Einstein-Maxwell-Yang-Mills equations”, Jour. Diff. Geom. 34 (1991), 275-345) using conformal completions techniques. In this conformal approach one has to 1) construct initial data, which satisfy the general relativistic constraint equations, with appropriate behavior near the conformal boundary, and 2) show a local (and perhaps also a global) existence theorem for the associated evolution problem. In this context solutions of the constraint equations can be found by solving a nonlinear elliptic system of equations, one of which resembles the Yamabe equation (and coincides with this equation in some cases), with the system degenerating near the conformal boundary. In the first part of the talk I (PTC) will describe the existence and boundary regularity results about this system obtained some years ago in collaboration with Helmut Friedrich and Lars Andersson. Some new applications of those techniques are also presented. In the second part of the talk I will describe some new results, obtained in collaboration with Olivier Lengard, concerning the evolution problem.