Polyhomogeneous solutions of wave equations in the radiation regime

Abstract

We study the "hyperboloidal Cauchy problem" for linear and semilinear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behaviour at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a λφ p nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times admitting conformal completions at null infinity, as well to a large class of equations with a similar non-linearity structure. We prove existence of solutions with controlled asymptotic behaviour, and asymptotic expansions for solutions when the initial data have such expansions. In particular we prove that polyhomogeneous initial data (satisfying compatibility conditions) lead to solutions which are polyhomogeneous at the conformal boundary I + of the Minkowski space-time.

Conformal Einstein Evolution

“…Denote by∇ the torsion free connection on U which defines with the connection entering (14), (15) the difference tensor∇ −∇ = S(b) and denote byL the tensor (11) derived from∇. Comparing with (12), we find that equations (14), (15) can be written∇ẋẋ…”

“…The solution is 'rough' at J ′ and it is expected that it admits a polyhomogeneous expansion there ( [60], cf. also [14]). The evolution of data of type (c) has not been studied yet.…”

“…Denote by f a smooth 1-form field. Then, if x(τ ), b(τ ) solve the conformal geodesics equations (14), (15), the pair x(τ ), b(τ ) − f | x(τ ) solves the same equations with ∇ replaced by the connection∇ =∇ + S(f ) and L byL, i.e. x(τ ), in particular its parameter τ , is independent of the Weyl connection in the conformal class which is used to write the equations.…”

“…While it may not be applicable any longer in the desired generality, the underlying idea is viable because the conformal field equations may be used to control also solutions with a 'rough' asymptotic behaviour ( [14], [60], cf. also P. T. Chruściel, this volume).…”

Asymptotically Simple Solutions of the Vacuum Einstein Equations in Even Dimensions

“…Our methods do not apply in odd space-time dimensions, where the situation is rather different in any case, as one generically expects polyhomogeneous expansions with half-integer powers of 1/r, where r is, say, the luminosity distance, compare [11,26,27,30].…”