Right-invariant Sobolev metrics ${H}^{s}$ on the diffeomorphisms group of the circle

Joachim Escher (1) , Boris Kolev (2)

1 IFAM - Institute for Applied Mathematics
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm $H^{s}$ for $s \ge 1/2$), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the Euler-Weil-Petersson equation.

Mots clés: Euler equation; diffeomorphisms group; fractional Sobolev metrics

Domaines: Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]

Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics ${H}^{s}$ on the diffeomorphisms group of the circle. 2012. ⟨hal-00673137v3⟩