Decay estimates for spherically symmetric Yang-Mills fields in Minkowski space-time

Abstract

We consider Yang–Mills fields in Minkowski space-time ${\mathrm{R}}^{3+1}$. We prove global existence and establish decay estimates for any initial data which is sufficiently small in a specified energy norm and which is spherically symmetric in the sense of principal bundles. We consider initial data which lead to bounded solutions near the central line. The proof of the global result is achieved using wighted Sobolev norms of the Klainerman type. They not only provide global existence, but also sharp asymptotic behaviour of the solutions in time. The initial data considered here does not include Coulomb charges but includes dipole radiation, a situation which cannot be acomodated in the framework of the conformai method. A new class of solutions, which have not been considered before, is covered then by the theorem.

Résumé

On démontre un théorème global d’existence d’une solution du problème de Cauchy pour les équations de Yang–Mills sur l’espace de Minkowski, pour des données petites et avec symétrie radiale, dans le sens d’une fibration principale.