Geometric aspects of the periodic $\mu$-DP equation
Résumé
We consider the periodic $\mu$-DP equation (a modified version of the Degasperis-Procesi equation) as the geodesic flow of a right-invariant affine connection $\nabla$ on the Fréchet Lie group $Diff^{\infty}(S^1)$ of all smooth and orientation-preserving diffeomorphisms of the circle $S^1=R/Z$.
On the Lie algebra $C^{\infty}(S^1)$ of $Diff^{\infty}(S^1)$, this connection is canonically given by the sum of the Lie bracket and a bilinear operator. For smooth initial data, we show the short time existence of a smooth solution of $\mu$-DP which depends smoothly on time and on the initial data. Furthermore, we prove that the exponential map defined by $\nabla$ is a smooth local diffeomorphism of a neighbourhood of zero in $C^{\infty}(S^1)$
onto a neighbourhood of the unit element in $Diff^{\infty}(S^1)$. Our results follow from a general approach on non-metric Euler equations on Lie groups, a Banach space approximation of the Fréchet space $C^{\infty}(S^1)$, and a sharp spatial regularity result for the geodesic flow.