Existence de l'application exponentielle riemannienne d'un groupe de difféomorphismes muni d'une métrique de Sobolev

Abstract

We prove the existence of geodesics of the weak Riemannian Lie group ( Diff H ∞ ( R n ) , g H k ) = ( Diff ( R n ) ∩ ( Id + ⋂ k ∈ N H k ( R n ; R n ) ) , g H k ) , where g H k is the weak Sobolev metric of order k. Next, we study the Riemannian exponential mapping induced by this metric. For k = 1 , the result immediately gives the local existence of solution in C ∞ ( ; H ∞ ( R n ; R n ) ) of the n-dimensional analog of the Camassa–Holm's equation on the Euclidean space R n .