Channels of energy for the linear radial wave equation

Abstract

Abstract. Exterior channel of energy estimates for the radial wave equation were first considered in three dimensions in [6], and for the 5-dimensional case in [12]. In this paper we find the general form of the channel of energy estimate in all odd dimensions for the radial free wave equation. This will be used in the companion paper [11] to establish the soliton resolution for equivariant wave maps in R 3 exterior to the ball B(0, 1) and in all equivariance classes.

Soliton Resolution for Equivariant Wave Maps on a Wormhole

“…However, this property is very sensitive to dimension and in fact fails in the case R = 0 for general data ( f, g) in even dimensions (see [5]). Proposition 6.5 has been generalized to all odd dimensions d ≥ 3 in the work [15]. We note that the orthogonal projections π R , π ⊥ R are given by…”

“…The method of proof used in the works [18], [14], and [16] to establish the soliton resolution conjecture for (1.9) was the celebrated concentration-compactness/rigidity theorem method pioneered by Kenig and Merle in [12] and [13]. In [14] and [16], the authors used a 'channels of energy' argument based on exterior energy estimates for free waves on R 1+d with d odd to close the argument (see [5] and [15] for these estimates). The proof of our main result, Theorem 1.1, uses a similar methodology which we now briefly overview.…”

Wave maps on a wormhole

“…By mixed analytical and numerical methods it was shown in [1] that in each topological sector there exists a unique stationary solution (harmonic map) which serves as the global attractor in the evolution for any finite-energy smooth initial data. Recently, this conjecture was proved by Kenig, Lawrie, and Schlag [2].…”

“…(3) is truly 1+1 dimensional (in the sense that the radial variable r ranges over the whole real line and there is no singularity at r = 0), yet it inherits strong dispersive decay from the original 3 + 1 dimensional problem. It would be interesting to combine the hyperboloidal approach used by us here with the rigorous methods developed in [2].…”

“…In this paper we introduce a similar model, also involving the equivariant wave maps into the threesphere, but with a geometrically natural domain, namely a wormhole-type spacetime. In this model we give evidence for the analogous soliton resolution conjecture as in [1,2]. In contrast to [1], we employ here the hyperboloidal formulation of the initial value problem as developed by Zenginoglu [3] on the basis of concepts introduced by Friedrich [4] and Frauendiener [5].…”