Decay for Skyrme wave maps

TL;DR

This paper proves that solutions to the Skyrme and Adkins-Nappi equations exhibit energy decay outside the light cone and for small solutions with bounded weighted energies, using new virial estimates.

Contribution

It introduces three new virial type estimates to establish decay properties of solutions to the Skyrme and Adkins-Nappi equations.

Findings

Energy decays outside the light cone for bounded solutions

Small solutions with bounded weighted energies decay polynomially

New virial identities are developed for decay analysis

Abstract

We consider the decay problem for global solutions of the Skyrme and Adkins-Nappi equations. We prove that the energy associated to any bounded energy solution of the Skyrme (or Adkins-Nappi) equation decays to zero outside the light cone (in the radial coordinate). Furthermore, we prove that suitable polynomial weighted energies of any small solution decays to zero when these energies are bounded. The proof consists of finding three new virial type estimates, one for the exterior of the light cone, based on the energy of the solution, and a more subtle virial identity for the weighted energies, based on a modification of momentum type quantities.