Geometric aspects of the periodic μ-Degasperis-Procesi equation

Abstract

We consider the periodic μDP equation (a modified version of the Degasperis-Procesi equation) as the geodesic flow of a right-invariant affine connection ∇ on the Fréchet Lie group Diff∞(S1) of all smooth and orientation-preserving diffeomorphisms of the circle S1 = ℝ/ℤ. On the Lie algebra C∞(S1) of Diff∞(S1), 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 μDP which depends smoothly on time and on the initial data. Furthermore, we prove that the exponential map defined by ∇ is a smooth local diffeomorphism of a neighbourhood of zero in C∞(S1) onto a neighbourhood of the unit element in Diff∞(S1). Our results follow from a general approach on non-metric Euler equations on Lie groups, a Banach space approximation of the Fréchet space C∞(S1), and a sharp spatial regularity result for the geodesic flow.