Corrigendum: On the Soliton Resolution for Equivariant Wave Maps to the Sphere

Abstract

Consider a sequence E n D . 0;n ; 1;n / bounded in H L 2 : it admits a linear profile decomposition . E V j;L I . n ; t n // as stated in Theorem 2.12 of [2], from which we use the notation and references. The following Pythagorean expansions are also stated in Theorem 2.12:These formulas were taken from the paper by Duyckaerts, Kenig, and Merle [5, eqs. (2.6) and (2.7), p. 645], but they are in fact false, as the same authors noted in the corrected version [6]. 1 The failure of the above Pythagorean expansions (2.1)-(2.2) created a gap in the proofs of [5], which the corrected version [6] filled. That failure also has an impact in the context of equivariant wave maps [2-4], which [6] also corrects, mutatis mutandis. For the convenience of the reader, we provide here an independent, shorter correction that applies to [2].1 They provide the following counterexample. Consider E L .t/ a solution to the linear flow (LW`/ bounded in H L 2 and a sequence of times t n ! C1. Then the sequence E n WD .2 L .t n /; 0/ admits the profile decompositionLet us, however, recall that the sum of (2.1) and (2.2) does hold; that is,