Joachim Escher
email: escher@ifam.uni-hannover.de

Boris Kolev
email: boris.kolev@math.cnrs.fr

RIGHT-INVARIANT SOBOLEV METRICS OF FRACTIONAL ORDER ON THE DIFFEOMORPHISM GROUP OF THE CIRCLE

Keywords: Mathematics Subject Classification. 58D05, 35Q53 Euler equation, diffeomorphism group of the circle, Sobolev metrics of fractional order

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 ≥ 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.

Introduction

The interest for geodesic flows on diffeomorphism groups goes back to Arnold [START_REF] Arnold | Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits[END_REF]. He recast the Euler equations of hydrodynamics of an ideal fluid as the geodesic flow for the L 2 right invariant Riemannian metric on the volume preserving diffeomorphism group. Arnold's paper was, somehow, rather formal from the analytical point of view. The well-posedness of the geodesic flow was established, subsequently, by Ebin and Marsden in [START_REF] Ebin | Groups of diffeomorphisms and the motion of an incompressible fluid[END_REF]. To do so, they introduced Hilbert manifolds of diffeomorphisms of class H q , and used them to approximate the Fréchet manifold of smooth diffeomorphisms. This framework was extended thereafter to other equations of physical relevance [START_REF] Constantin | Geodesic flow on the diffeomorphism group of the circle[END_REF][START_REF] Holm | The Euler-Poincaré equations and semidirect products with applications to continuum theories[END_REF][START_REF] Khesin | Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms[END_REF][START_REF] Kolev | Lie groups and mechanics: an introduction[END_REF][START_REF] Misio | A shallow water equation as a geodesic flow on the Bott-Virasoro group[END_REF][START_REF] Shkoller | Geometry and curvature of diffeomorphism groups with H 1 metric and mean hydrodynamics[END_REF][START_REF] Escher | The geometry of a vorticity model equation[END_REF]. Among these studies, right invariant metrics induced by H k Sobolev norms (k ∈ N) on the diffeomorphism group of the circle, Diff ∞ (S 1 ), have been extensively investigated [START_REF] Constantin | Geodesic flow on the diffeomorphism group of the circle[END_REF][START_REF] Khesin | Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms[END_REF][START_REF] Kolev | Lie groups and mechanics: an introduction[END_REF][START_REF] Shkoller | Geometry and curvature of diffeomorphism groups with H 1 metric and mean hydrodynamics[END_REF]. In [START_REF] Escher | The geometry of a vorticity model equation[END_REF], well-posedness of the geodesic flow for the homogeneous H 1/2 right-invariant metric on the homogeneous space Diff ∞ (S 1 )/Rot(S 1 ) was established. The homogeneous H 3/2 right-invariant metric on Diff ∞ (S 1 )/PSL(2, R) was considered in [START_REF] Gay-Balmaz | Infinite dimensional geodesic flows and the universal Teichmüller space[END_REF].

It is the aim of the present paper to study well-posedness of geodesic flows and their corresponding Euler equation for H s Sobolev norms on Diff ∞ (S 1 ), where s ∈ R + . One of the main difficulty which arise immediately, in this context, is that the inertia operator is non-local. More precisely, such a right-invariant metric on Diff ∞ (S 1 ) is induced by an inner product

u, v = S 1
(Au)v dx, on Vect(S 1 ) = C ∞ (S 1 ), where A : C ∞ (S 1 ) → C ∞ (S 1 ) is a (non-local) Fourier multiplier. To be able to make use of the framework proposed by Ebin and Marsden, one needs to extend the metric smoothly on D q (S 1 ), the Hilbert manifold of diffeomorphisms of Sobolev class H q . When A is of finite order r ≥ 0, it extends to a bounded linear operator from H q (S 1 ) to H q-r (S 1 ) for q large enough, and the smoothness of the metric is reduced to the following question, where

R ϕ : v → v • ϕ, ϕ ∈ D q (S 1 ), v ∈ H q (S 1 ).
Problem. Given a Fourier multiplier A of order r ≥ 0, under which conditions is the mapping

(1) ϕ → A ϕ := R ϕ • A • R ϕ -1 ,
D q (S 1 ) → L(H q (S 1 ), H q-r (S 1 ))

smooth?

Note that the problem is not trivial in general, because the mapping (ϕ, v) → R ϕ (v), D q (S 1 ) × H q (S 1 ) → H q (S 1 )

is not differentiable (see [START_REF] Ebin | Groups of diffeomorphisms and the motion of an incompressible fluid[END_REF] for instance), and the mapping ϕ → R ϕ is not even continuous (see remark B.4). It is however a well-known fact that when A is a differential operator of order r, A ϕ is a linear differential operator whose coefficients are polynomial expressions of 1/ϕ x and the derivatives of ϕ up to order r (e.g. D ϕ = (1/ϕ x )D), see [START_REF] Ebin | Groups of diffeomorphisms and the motion of an incompressible fluid[END_REF][START_REF] Escher | The Degasperis-Procesi equation as a non-metric Euler equation[END_REF] for instance. In that case, ϕ → A ϕ is smooth (in fact real analytic) for q ≥ r.

However, for a general Fourier multiplier, we are not aware of any results in this direction. In theorem 3.7, however, we give a sufficient condition on the symbol of A which ensures that the mapping [START_REF] Arnold | Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits[END_REF] is smooth. This answers a question raised in [9, Appendix A], at least in the case of the diffeomorphism group of the circle. Up to the authors knowledge, these results are new.

Remark 1.1. Of course, there are Fourier multiplication operators A, of order less than 1, for which the mapping ϕ → A ϕ , D q (S 1 ) → L(H q (S 1 ), H q-r (S 1 )) is smooth. However, the present proof of theorem 3.7 works only for r ≥ 1. So far, the authors have not been able to exhibit a counter-example which would show that the conclusion of theorem 3.7 is false for 0 ≤ r < 1. They are not aware either of an example of a Fourier multiplier for which the conclusion of theorem 3.7 fails for all q ≥ 0. Theorem 3.7 applies, in particular, to the inertia operator Λ 2s of the Sobolev metric H s on Diff ∞ (S 1 ) for s ∈ R and s ≥ 1/2 (corollary 3.9). This allows us to prove that the corresponding weak Riemannian metric and its geodesic spray can be smoothly extended to the Hilbert manifold approximation D q (S 1 ) for sufficiently large q ∈ R. As a corollary, we are able to prove local existence and uniqueness of geodesics on D q (S 1 ) and Diff ∞ (S 1 ) (theorem 4.3), as well as the well-posedness of the corresponding Euler equation (corollary 4.4).

It is a well known result that the group exponential map on Diff ∞ (S 1 ) is not locally surjective [START_REF] Milnor | Remarks on infinite-dimensional Lie groups[END_REF]. In [START_REF] Constantin | Geodesic flow on the diffeomorphism group of the circle[END_REF], it was shown, moreover, that the Riemannian exponential map for the L 2 metric on Diff ∞ (S 1 ) was not a local diffeomorphism. However, due to the fact that the spray of the H s metric is smooth for s ≥ 1/2 (theorem 3.10), we are able to prove that the exponential map on Diff ∞ (S 1 ) is a local diffeomorphism, in that case (theorem 5.1). From this fact, we can deduce that given two nearby diffeomorphisms, there is a unique geodesic which joins them and that this geodesic is a local minimum of both the arc-length and the energy functionals. For s = 1/2, this local minimum is, however, not a global minimum [START_REF] Bauer | Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group[END_REF]. This exhibits a surprising difference with finite dimensional Riemannian geometry.

We close our study by extending our results to Euler equations on some homogeneous spaces of Diff ∞ (S 1 ), namely Diff ∞ (S 1 )/Rot(S 1 ), where Rot(S 1 ) is the subgroup of all rigid rotations of the circle S 1 and Diff ∞ (S 1 )/PSL(2, R), where PSL(2, R) is the subgroup of all rigid Möbius transformations which preserve the circle S 1 . The first case includes the Constantin-Lax-Majda equation [START_REF] Escher | The geometry of a vorticity model equation[END_REF][START_REF] Escher | Restrictions on the geometry of the periodic vorticity equation[END_REF]. The second case includes the Euler-Weil-Petersson equation, which is related to the Weil-Petersson metric on the universal Teichmüller space T (1) [START_REF] Nag | Teichmüller theory and the universal period mapping via quantum calculus and the H 1/2 space on the circle[END_REF][START_REF] Takhtajan | Weil-Petersson metric on the universal Teichmüller space[END_REF].

The plan of the paper is as follows. In Section 2, we recall basic materials on right-invariant metrics on the diffeomorphism group. Section 3 is devoted to the study of the smoothness of the extended metric an its spray on the Hilbert manifolds D q (S 1 ). In Section 4, we prove local existence and uniqueness of the initial value problem for the geodesics of the rightinvariant H s metric on Diff ∞ (S 1 ) and well-posedness of the corresponding Euler equation. In Section 5 we deal with the Riemannian exponential map and discuss the problem of minimization of the arc-length and the energy. In Section 6 we extend our study to the homogeneous spaces Diff ∞ (S 1 )/Rot(S 1 ) and Diff ∞ (S 1 )/PSL(2, R). We prove well-posedness for the corresponding Euler equations. In Appendix A, we prove some lemmas on Fourier multipliers, while in Appendix B, we provide estimates and local boundedness properties for the right translation operator R ϕ .

Geometric Framework

Let Diff ∞ (S 1 ) be the group of all smooth and orientation preserving diffeomorphisms on the circle. This group is equipped with a Fréchet manifold structure, modelled on the Fréchet vector space C ∞ (S 1 ) (see Guieu and Roger [18]). Since, moreover, composition and inversion are smooth for this structure, we say that Diff ∞ (S 1 ) is a Fréchet-Lie group, cf. [START_REF] Hamilton | The inverse function theorem of Nash and Moser[END_REF]. Its Lie algebra, Vect(S 1 ), is the space of smooth vector fields on the circle. It is isomorphic to C ∞ (S 1 ) with the Lie bracket given by

[u, w] = u x w -uw x . Let A : C ∞ (S 1 ) → C ∞ (S 1
) be a L 2 -symmetric, positive definite, continuous linear operator on C ∞ (S 1 ), we define the following inner product on the Lie algebra Vect(S

1 ) = C ∞ (S 1 ) u, w := S 1 (Au)w dx = S 1 u(Aw) dx.
Translating this inner product on each tangent space, we get one on each tangent space T ϕ Diff ∞ (S 1 ), given by

(2) v 1 , v 2 ϕ = v 1 • ϕ -1 , v 2 • ϕ -1 id = S 1 v 1 (A ϕ v 2 )ϕ x dx, where v 1 , v 2 ∈ T ϕ Diff ∞ (S 1 ), A ϕ = R ϕ • A • R ϕ -1 , and R ϕ (v) := v • ϕ.
One generates this way a smooth, weak Riemannian metric on Diff ∞ (S 1 ). For historical reasons going back to Euler [START_REF] Euler | Du mouvement de rotation des corps solides autour d'un axe variable[END_REF], A is called the inertia operator of the right-invariant metric.

A covariant derivative on a Fréchet manifold is a way of differentiating vector fields along paths. In general, a torsion-free, covariant derivative, compatible with a weak Riemannian metric does not exists but if it does, it is unique. According to Arnold, a necessary and sufficient for the existence of a covariant derivative compatible with a right-invariant metric on Diff ∞ (S 1 ) is the existence of the Arnold bilinear operator

B(u, v) = 1 2 ad ⊤ u v + ad ⊤ v u ,
where u, v ∈ C ∞ (S 1 ) and ad ⊤ u is the adjoint of the operator ad u , with respect to A (see [START_REF] Arnold | Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits[END_REF] for instance). If ϕ(t) is any path in Diff ∞ (S 1 ) and ξ(t) is a field of tangent vectors along the path, we define the covariant derivative along the path to be

D t ξ = R ϕ w t + 1 2 [u, w] + B(u, w) ,
where u(t) := ϕ t • ϕ -1 and w(t) := ξ(t) • ϕ -1 . One can check that this covariant derivative is torsion-free and metric-compatible.

Lemma 2.1. If A : C ∞ (S 1 ) → C ∞ (S 1
) is invertible and commutes with d/dx, then, the map ad ⊤ u is well defined and given by ad

⊤ u w = A -1 [2(Aw)u x + (Aw) x u] , for u, w ∈ C ∞ (S 1 ).
Proof. We have

< ad u v, w >= S 1 (Aw)(u x v -uv x ) dx = S 1 [2(Aw)u x + (Aw) x u)] v dx where u, v, w ∈ C ∞ (S 1 ). But since A : C ∞ (S 1 ) → C ∞ (S 1 ) is invertible, we get, finally ad ⊤ u w = A -1 [2(Aw)u x + (Aw) x u] .
A geodesic is a path ϕ(t) in Diff ∞ (S 1 ), which is an extremal curve of the energy functional

E := 1 2 1 0 u(t), u(t) dt,
where u(t) = ϕ t • ϕ -1 . The corresponding Euler-Lagrange equation

D t ϕ t = 0
is equivalent to the following first order equation ( 3)

u t = -B(u, u) = -A -1 {(Au) x u + 2(Au)u x } ,
called the Euler equation.

Example. For the L 2 -metric (A = I), the corresponding Euler equation ( 3) is the inviscid Burgers equation

u t + 3uu x = 0.
Example. For the H 1 -metric (A = I -d 2 /dx 2 ), the corresponding Euler equation ( 3) is the Camassa-Holm equation

u t -u txx + 3uu x -2u x u xx -uu xxx = 0,
a model in the theory of shallow water waves [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF][START_REF] Fokas | Symplectic structures, their Bäcklund transformations and hereditary symmetries[END_REF].

Let ϕ be the flow of the time dependent vector field u i.e., ϕ t = u • ϕ and let v = ϕ t . Then u solves the Euler equation ( 3), if and only if, (ϕ, v) is a solution of (4)

ϕ t = v, v t = S ϕ (v),
where

S ϕ (v) := R ϕ • S • R ϕ -1 (v), and
S(u) := A -1 {[A, u]u x -2(Au)u x } .
The second order vector field on Diff ∞ (S 1 ), defined by ( 5)

F : (ϕ, v) → (ϕ, v, v, S ϕ (v))
is called the geodesic spray, following Lang [START_REF] Lang | Fundamentals of Differential Geometry[END_REF]. Suppose now that A is a differential operator of order r. Then, the right hand side of the Euler equation is of order 1. It is however quite surprising that, in Lagrangian coordinates, the propagator of evolution equation of the geodesic flow has better mapping properties, provided that the order r of A is not less than 1. Indeed, in that case, the quadratic operator

S(u) := A -1 {[A, u]u x -2(Au)u x }
is of order 0 because the commutator [A, u] is of order less than ≤ r -1. One might expect, that for a larger class of operators A, the quadratic operator S to be of order 0 and the second order system (4) to be the local expression of an ODE on some suitable Banach manifold.

This observation is at the root of a strategy proposed in the 70' by Ebin and Marsden [START_REF] Ebin | Groups of diffeomorphisms and the motion of an incompressible fluid[END_REF] to study well-posedness of the Euler equation. Following their approach, if we can prove local existence and uniqueness of geodesics (ODE) on diffeomorphism groups then the PDE (Euler equation) is wellposed. To do so, it is necessary to introduce an approximation of the Fréchet-Lie group Diff ∞ (S 1 ) by Hilbert manifolds. Let H q (S 1 ) be the completion of C ∞ (S 1 ) for the norm

u H q := k∈Z (1 + k 2 ) q |û k | 2 1/2
, where q ∈ R, q ≥ 0. We recall that H q (S 1 ) is a multiplicative algebra for q > 1/2 (cf. [START_REF] Triebel | Theory of Function Spaces[END_REF]Theorem 2.8.3]). This means that uv H q (S 1 )

u H q (S 1 ) v H q (S 1 ) , u, v ∈ H q (S 1 ).

Definition 2.2. We say that a C 1 diffeomorphism ϕ of S 1 is of class H q if for any of its lifts to R, φ, we have φ -id ∈ H q (S 1 ).

For q > 3/2, the set D q (S 1 ) of C 1 -diffeomorphisms of the circle which are of class H q has the structure of a Hilbert manifold, modelled on H q (S 1 ) (see [START_REF] Ebin | Groups of diffeomorphisms and the motion of an incompressible fluid[END_REF]). The manifold D q (S 1 ) is also a topological group but not a Lie group (composition and inversion in D q (S 1 ) are continuous but not differentiable). Note however, that, given ϕ ∈ D q (S 1 ),

u → R ϕ (u) := u • ϕ, H q (S 1 ) → H q (S 1 )
is a smooth map, and that

(u, ϕ) → u • ϕ, H q+k (S 1 ) × D q (S 1 ) → H q (S 1 ) is of class C k .
The Fréchet Lie group group Diff ∞ (S 1 ) may be viewed as an inverse limit of Hilbert manifolds (ILH) Diff ∞ (S 1 ) = q> 3 2 D q (S 1 ), and we call the scales of manifolds D q (S 1 )) q>3/2 , a Hilbert manifold approximation of Diff ∞ (S 1 ).

Remark 2.3. Note that the tangent bundle of the Hilbert manifold D q (S 1 ) is trivial. Indeed, let t : T S 1 → S 1 × R be a trivialisation of the tangent bundle of the circle. Then Ψ : T D q (S 1 ) → D q (S 1 ) × H q (S 1 ), ξ → t • ξ defines a smooth vector bundle isomorphism because t is smooth (see [9, Page 107]).

Within this framework, the case where the inertia operator A is a differential operator with constant coefficients has been extensively studied in the literature (see for instance [START_REF] Constantin | On the geometric approach to the motion of inertial mechanical systems[END_REF][START_REF] Constantin | Geodesic flow on the diffeomorphism group of the circle[END_REF][START_REF] Escher | The Degasperis-Procesi equation as a non-metric Euler equation[END_REF]). It is the aim of the present paper to extend these results when A is a general Fourier multiplier, that is, a continuous linear operator on C ∞ (S 1 ), which commutes with D := d/dx. In that case, we get

(Au)(x) = k∈Z a(k)û(k) exp(2iπkx),
where û(k) is the k-th Fourier coefficients of u (see lemma A.1). The sequence a : Z → C is called the symbol of A and we shall use the notation A = op (a(k)). When a(k) = O(|k| r ), the Fourier multiplier A = op (a(k)) extends, for each q ≥ r, to a bounded linear operator in L(H q (S 1 ), H q-r (S 1 )). It is said to be of order r.

Example. The inertia operator for the H s Sobolev metric (s ≥ 1/2), defined by Λ 2s := op (1 + k 2 s ) is of order 2s ≥ 1.

Smoothness of the extended metric and spray

The fact that a right-invariant metric, defined by (2), and its geodesic spray F , defined by (5) are smooth on Diff ∞ (S 1 ) is unfortunately useless to establish the well-posedness of the geodesic flow. What we need to do is to study under which conditions, the metric and its spray can be extended smoothly to the Hilbert approximation manifolds D q (S 1 ). In this section, we provide a criteria on the inertia operator A (satisfied by almost all known examples) which ensures the smoothness of the metric on the extended manifolds D q (S 1 ), for q large enough.

For general materials on Banach manifolds, we refer to [START_REF] Lang | Fundamentals of Differential Geometry[END_REF]. Let X be a Banach manifold modelled over a Banach space E. We recall that a Riemannian metric g on X is a smooth, symmetric, positive definite, covariant 2-tensor field on X. In other words, we have for each x ∈ X a symmetric, positive definite, bounded, bilinear form g(x) on T x X and, in any local chart U , the mapping x → g(x), U → L 2 sym (E, R) is smooth. Given any x ∈ X, we can then consider the bounded, linear operator h x : T x X → T * x X, called the flat map and defined by h x (ξ x ) = g(x)(ξ x , •). The metric is strong if h x is a topological linear isomorphism for all x ∈ X, whereas it is weak if h x is only injective for all x ∈ X.

Given A ∈ Isom(H q (S 1 ), H q-r (S 1 )), it induces a bounded inner product on each tangent space, T ϕ D q (S 1 ) for each ϕ ∈ D q (S 1 ), given by

v 1 , v 2 ϕ = S 1 v 1 (A ϕ v 2 )ϕ x dx, where A ϕ := R ϕ • A • R ϕ -1 .
To conclude, however, that the family •, • ϕ defines a (weak) Riemannian metric on the Banach manifold D q (S 1 ), we need to show that the mapping ϕ → ϕ x A ϕ , D q (S 1 ) → L 2 (H q (S 1 ), H -q (S 1 )), is smooth. Remark 3.1. Note that even when the flat map

à : (ϕ, v) → (ϕ, ϕ x A ϕ v), D q (S 1 ) × H q (S 1 ) → D q (S 1 ) × H -q (S 1 )
is smooth and defines an injective vector bundle morphism, its image

D q (S 1 ) × H q-r (S 1 )
is not a subbundle of T * D q (S 1 ) in the sense of [START_REF] Lang | Fundamentals of Differential Geometry[END_REF]III.3], because H q-r (S 1 ) is not a closed subspace of H -q (S 1 ).

For q > 3/2, the mappings ϕ → ϕ x and ϕ → 1/ϕ x are smooth from D q (S 1 ) → H q-1 (S 1 ). Thus, for r ≥ 1 and q -r ≥ 0, lemma B.1 shows that the metric is smooth if and only if ϕ → A ϕ , D q (S 1 ) → L(H q (S 1 ), H q-r (S 1 )) is smooth. If this holds, we can compute, for each n ≥ 1, the n-th Fréchet differential 1 ∂ n ϕ A ϕ ∈ L n+1 (H q (S 1 ), H q-r (S 1 )). which is itself smooth. Lemma 3.2. We have

(6) ∂ n ϕ A ϕ (v, δϕ 1 , . . . , δϕ n ) = R ϕ A n R -1 ϕ (v, δϕ 1 , . . . , δϕ n ), where A n := ∂ id A ϕ ∈ L n+1 (H q (S 1 ), H q-r (S 1 ))
is the (n + 1)-linear operator defined inductively by A 0 = A and

(7) A n+1 (u 0 , u 1 , . . . , u n+1 ) = ∇ u n+1 A n (u 0 , u 1 , . . . , u n ) - n k=0 A n (u 0 , . . . , ∇ u n+1 u k , . . . , u n ),
where ∇ is the canonical connection on the Lie group S 1 .

1 We have chosen to denote by ∂ the Fréchet differential to avoid the confusion with the already used notation D = d/dx.

Remark 3.3. For n = 1, we get

A 1 (u 0 , u 1 ) = [∇ u 1 , A]u 0 ,
and for n = 2, we get

A 2 (u 0 , u 1 , u 2 ) = [∇ u 2 , [∇ u 1 , A]] -[∇ ∇u 2 u 1 , A] u 0 .
Proof. We will make the computations for smooth functions. The general result follows from a density argument and the fact that the expressions are continuous. Formula ( 6) is trivially true for n = 0. Now suppose it is true for some n ∈ N, that is

∂ n ϕ A ϕ (v, δϕ 1 , . . . , δϕ n ) = R ϕ A n R -1 ϕ (v, δϕ 1 , . . . , δϕ n ), Let ϕ(s) be a smooth path in Diff ∞ (S 1 ) such that ϕ(0) = ϕ, ∂ s ϕ(s) s=0 = δϕ n+1 . Set u k = δϕ k • ϕ -1 , for 1 ≤ k ≤ n + 1 and u 0 = v • ϕ -1 . We compute first ∂ s R ϕ(s) w s=0 = R ϕ (u n+1 w x ) , for w ∈ C ∞ (S 1
), and

∂ s R -1 ϕ(s) w s=0 = -u n+1 R -1 ϕ w x , for w ∈ C ∞ (S 1 ). Hence ∂ s R ϕ A n R -1 ϕ (v, δϕ 1 , . . . , δϕ n ) s=0 = R ϕ {u n+1 (A n (u 0 , . . . , u n )) x } - n k=0 R ϕ A n (u 0 , . . . , u n+1 (u k ) x , . . . , u n ),
which gives the recurrence relation [START_REF] Constantin | Geodesic flow on the diffeomorphism group of the circle[END_REF], since

u n+1 (A n (u 0 , . . . , u n )) x = ∇ u n+1 A n (u 0 , u 1 , . . . , u n ),
and

A n (u 0 , . . . , u n+1 (u k ) x , . . . , u n ) = A n (u 0 , . . . , ∇ u n+1 u k , . . . , u n ).
We shall prove now the following necessary and sufficient condition for smoothness.

Theorem 3.4 (Smoothness Theorem). Let A : C ∞ (S 1 ) → C ∞ (S 1 ) be a continuous linear operator of order r ≥ 1. Let q > 3/2 with q -r ≥ 0. Then ϕ → A ϕ := R ϕ • A • R ϕ -1 , D q (S 1 ) → L(H q (S 1 ), H q-r (S 1 ))
is smooth, if and only if, each A n extends to a bounded (n+1)-linear operator in L n+1 (H q (S 1 ), H q-r (S 1 )).

The idea of the proof of theorem 3.4, which is inductive, is the following. First, we show that if A n is bounded, then the mapping

ϕ → A n,ϕ := R ϕ A n R ϕ -1 , D q (S 1 ) → L n+1 (H q (S 1 ), H q-r (S 1 ))
is locally bounded. Then, we prove that if A n+1,ϕ is locally bounded, then A n,ϕ is locally Lipschitz. Finally we show that if A n+1,ϕ is locally Lipschitz, then A n,ϕ is C 1 . The full detail proof, given below, requires the following two elementary lemmas, which will be stated without proof. Lemma 3.5. Let X be a topological space and E a Banach space. Let f : [0, 1] × X → E be a continuous mapping. Then the mapping

g(x) := 1 0 f (t, x) dt is continuous. Lemma 3.6. Let E, F be Banach spaces and U a convex, open set in E. Let α : U → L(E, F ) be a continuous mapping and f : U → F a mapping such that f (y) -f (x) = 1 0 α(ty + (1 -t)x)(y -x) dt, for all x, y ∈ U . Then f is C 1 on U and df = α.
Proof of theorem 3.4. Note first that for q > 3/2 and q -r ≥ 0, the mapping

ϕ → R ϕ , D q (S 1 ) → L(H q-r (S 1 ), H q-r (S 1 ))
is locally bounded (lemma B.2) and that the mapping

(ϕ, v) → v • ϕ, D q (S 1 ) × H q-r (S 1 ) → H q-r (S 1 )
is continuous (corollary B.3). Since, moreover, the mapping

ϕ → ϕ -1 , D q (S 1 ) → D q (S 1 )
is continuous for q > 3/2 (see [START_REF] Ebin | Groups of diffeomorphisms and the motion of an incompressible fluid[END_REF][START_REF] Inci | On the Regularity of the Composition of Diffeomorphisms[END_REF]), we conclude that

ϕ → R ϕ -1 , D q (S 1 ) → L(H q (S 1 ), H q (S 1 ))
is locally bounded and that

(ϕ, v) → v • ϕ -1 , D q (S 1 ) × H q (S 1 ) → H q (S 1 ) is continuous.
The fact that the boundedness of the A n is a necessary condition results from lemma 3.2. Conversely, suppose that each A n is bounded in L n+1 (H q (S 1 ), H q-r (S 1 )), then

A n,ϕ := R ϕ A n R ϕ -1 ∈ L n+1 (H q (S 1 ), H q-r (S 1 )), is a bounded (n + 1)-linear operator, with A n,ϕ L n+1 (H q ,H q-r ) ≤ R ϕ L(H q-r ,H q-r ) A n L n+1 (H q ,H q-r ) R ϕ -1 n+1
L(H q ,H q ) , and we conclude, thus, that

ϕ → A n,ϕ , D q (S 1 ) → L n+1 (H q (S 1 ), H q-r (S 1 )) is locally bounded, for each n ∈ N.
(a) We will first show that ϕ → A n,ϕ is locally Lipschitz continuous 2 , for each n ∈ N. Let ψ ∈ D q (S 1 ) be given. Because ϕ → A n+1,ϕ is locally 2 On D q (S 1 ), we did not introduce any distance compatible with the topology. The concept of a locally Lipschitz mapping f : M → E, from a Banach manifold M to a Banach vector space E does not require such an additional structure. It is defined using a local chart, and then shown to be independent of the choice of the particular chart.

bounded, it is possible to find a neighbourhood U of ψ and a positive constant

K such that A n+1,ϕ L n+1 (H q ,H q-r ) ≤ K, ∀ϕ ∈ U.
We can further assume (using a local chart) that U is a ball in H q (S 1 ). Pick now ϕ 0 and ϕ 1 in Diff ∞ (S 1 ) ∩ U and set ϕ(t

) := (1 -t)ϕ 0 + tϕ 1 for t ∈ [0, 1]. Choosing v 0 , . . . , v n ∈ C ∞ (S 1
) with v j H q ≤ 1, we obtain from lemma 3.2 that

A n,ϕ 1 (v 0 , . . . , v n ) -A n,ϕ 0 (v 0 , . . . , v n ) = 1 0 A n+1,ϕ(t) (v 0 , . . . , v n , ϕ 1 -ϕ 0 ) dt.
This implies

A n,ϕ 1 (v 0 , . . . , v n ) -A n,ϕ 0 (v 0 , . . . , v n ) H q-r ≤ K ϕ 1 -ϕ 0 H q , for all v 0 , . . . , v n ∈ C ∞ (S 1 ) with v j H q ≤ 1.
The assertion that A n,ϕ is Lipschitz continuous follows from the density of the embedding C ∞ (S 1 ) ֒→ H q (S 1 ), and continuity of the mapping

(ϕ, v 0 , . . . , v n ) → A n,ϕ (v 0 , . . . , v n ), D q (S 1 ) × H q (S 1 ) n+1 → H q-r (S 1
).

(b) We will now show by induction, that ϕ → A ϕ is of class C n for all n ∈ N, and that its n-th Fréchet derivative is A n,ϕ . For each n ≥ 1, let

α n : D q (S 1 ) → L H q (S 1 ), L n H q (S 1
), H q-r (S 1 ) , be the Lipschitz continuous mapping defined by

α n (ϕ) := δϕ n → A n,ϕ (•, . . . , •, δϕ n ) .
Let U be a local chart in D q (S 1 ), that we choose to be a convex open subset of H q (S 1 ). By its very definition, we have

A ϕ 1 (v) -A ϕ 0 (v) = 1 0 A 1,tϕ 1 +(1-t)ϕ 0 (v, ϕ 1 -ϕ 0 ) dt, for all ϕ 0 , ϕ 1 ∈ U ∩C ∞ (S 1 ) and v ∈ C ∞ (S 1
). But, the continuity of the mapping ϕ → A 1,ϕ , together with lemma 3.5, and the density of the embedding C ∞ (S 1 ) ֒→ H q (S 1 ), permit to conclude that this formula is still true for all ϕ 0 , ϕ 1 ∈ U and v ∈ H q (S 1 ). Therefore, we can write in L(H q (S 1 ), H q-r (S 1 ))

A ϕ 1 -A ϕ 0 = 1 0 α 1 (tϕ 1 + (1 -t)ϕ 0 )(ϕ 1 -ϕ 0 ) dt,
and, by virtue of lemma 3.6, we conclude that ϕ → A ϕ is C 1 and that DA ϕ = α 1 . A similar argument shows that, for each n ≥ 1, we have

α n (ϕ 1 ) -α n (ϕ 0 ) = 1 0 α n+1 (tϕ 1 + (1 -t)ϕ 0 )(ϕ 1 -ϕ 0 ) dt,
and hence that α n is C 1 with Dα n = α n+1 . This completes the proof.

When A is a Fourier multiplier, a criteria on the symbol a of A which ensures that all the A n are bounded and thus that the metric is smooth is given below. It's proof is a direct consequence of lemma A.4, lemma A.6 and corollary A.7 in Appendix A. Theorem 3.7. Let A = op (a(k)) be a Fourier multiplier of order r ≥ 1. Suppose that its symbol a extends to R and that for each n ≥ 1, the function

f n (ξ) := ξ n-1 a(ξ) is of class C n-1 , that f (n-1) n
is absolutely continuous and that there exists

C n > 0 such that (8) f (n) n (ξ) ≤ C n (1 + ξ 2 ) (r-1)/2 ,
almost everywhere. Then,

ϕ → A ϕ := R ϕ • A • R ϕ -1 , D q (S 1 ) → L(H q (S 1 ), H q-r (S 1 ))
is smooth for q > 3/2 and q -r ≥ 0.

Remark 3.8. This criteria is always satisfied when the symbol A = op (a(k)) belongs to the class S r , that is when a can be extended to a smooth function on R such that ( 9)

a (k) (ξ) = O(|ξ| r-k ), ∀k ∈ N.
This applies in particular to the inertia operator Λ 2s of the Sobolev metric H s , when s ≥ 1/2. Indeed, let a s (ξ) := 1 + ξ 2 s be the symbol of Λ 2s . One can check that

a (k) s (ξ) = p k (ξ) (1 + ξ 2 ) k a s (ξ),
for k ≥ 1, where p k is a polynomial function with d(p k ) ≤ k. Thus, ( 9) is true for a s , and we have the following result. Corollary 3.9. Let s ∈ R and Λ 2s := op 1 + n 2 s . If s ≥ 1/2 then the mapping

ϕ → R ϕ • Λ 2s • R ϕ -1 , D q (S 1 ) → L(H q (S 1 ), H q-2s (S 1 ))
is smooth for q > 3/2 and q -2s ≥ 0.

On a finite dimensional manifold, as soon as the metric is C k , the geodesic spray is C k-1 , because the components of the spray, in any local chart, involve the Christoffel symbols which depend on the first derivatives of the metric. We might, therefore, expect some kind of analog results to hold for a weak Riemannian metric on a Banach manifold, as soon as the spray exists. Theorem 3.10. Let A be a Fourier multiplier of order r ≥ 1 and let q > 3/2, with q -r ≥ 0. Suppose, moreover, that

ϕ → A ϕ = R ϕ • A • R ϕ -1 , D q (S 1 ) → Isom(H q (S 1 ), H q-r (S 1 )) is smooth. Then the geodesic spray (ϕ, v) → S ϕ (v) = R ϕ • S • R ϕ -1 (v),
where

S(u) = A -1 {[A, u]u x -2(Au)u x } .
extends smoothly to T D q (S 1 ) = D q (S 1 ) × H q (S 1 ).

Proof. Let P (u) := (Au)u x and Q(u

) := [A, u]u x . Then, S ϕ (v) = A -1 ϕ {Q ϕ (v) -2P ϕ (v)} .
The proof reduces to establish, using the chain rule, that the three mappings

(ϕ, v) → P ϕ (v), (ϕ, v) → Q ϕ (v), (ϕ, w) → A -1 ϕ (w) are smooth. (a) We have P ϕ (v) = A ϕ (v) D ϕ (v) . But (ϕ, v) → A ϕ (v), D q (S 1 ) × H q (S 1 ) → H q-r (S 1 )
is smooth by hypothesis, whereas

(ϕ, v) → D ϕ (v), D q (S 1 ) × H q (S 1 ) → H q-1 (S 1 ) is smooth since D ϕ (v) = v x /ϕ x and H q-1 (S 1
) is a multiplicative algebra for q > 3/2. To conclude that

(ϕ, v) → P ϕ (v), D q (S 1 ) × H q (S 1 ) → H q-r (S 1 )
is smooth, we use the fact that pointwise multiplication extends to a bounded bilinear mapping

H q-r (S 1 ) × H q-1 (S 1 ) → H q-r (S 1 ), if q -1 > 1/2 and 0 ≤ q -r ≤ q -1 (c.f. lemma B.1).
(b) By virtue of lemma 3.2, we have

∂ ϕ A ϕ (v, v) = A 1,ϕ (v, v) = -Q ϕ (v),
and therefore

(ϕ, v) → Q ϕ (v), D q (S 1 ) × H q (S 1 ) → H q-r (S 1 )
is smooth. (c) The set Isom(H q (S 1 ), H q-r (S 1 )) is open in L(H q (S 1 ), H q-r (S 1 )) and the mapping P → P -1 , Isom(H q (S 1 ), H q-r (S 1 )) → Isom(H q-r (S 1 ), H q (S 1 )) is smooth (it is even real analytic). Besides A ϕ ∈ Isom(H q (S 1 ), H q-r (S 1 )), for all ϕ ∈ D q (S 1 ), and the mapping

ϕ → A ϕ , D q (S 1 ) → Isom(H q (S 1 ), H q-r (S 1 )) is smooth. Thus (ϕ, w) → A -1 ϕ (w), H q-r (S 1 ) → H q (S 1 ) is smooth.
We have, in particular, the following corollary.

Corollary 3.11. (Smoothness of the H s metric and its spray) Let s ≥ 1/2 and assume that q > 3/2, q -2s ≥ 0. Then the right-invariant, weak Riemannian metric defined on Diff ∞ (S 1 ) by the inertia operator A = Λ 2s extends to a smooth weak Riemannian metric on the Banach manifold D q (S 1 ) with a smooth geodesic spray. Remark 3.12. When A is a differential operator, theorem 3.10 can be sharpened; in that case, we can conclude that the spray is smooth when q > 3/2 and q ≥ r -1. Indeed, we have

S 1 {[A, u]u x -2(Au)u x } dx = 0, so that [A, u]u x -2(Au)u x = DB(u)
where B is a quadratic, differential operator of order r -1 (see [START_REF] Olver | Applications of Lie Groups to Differential Equations[END_REF]Chapter 4]). Hence

(ϕ, v) → B ϕ (v), D q (S 1 ) × H q (S 1 ) → H q-r+1 (S 1 )
is smooth for q ≥ r -1. Moreover, the symbol a of A is a real, even polynomial (because A is L 2 -symmetric) with no real roots (because A is invertible). Therefore, A can be written as

A = A ′ (α -iD)(ᾱ -iD),
where α ∈ C \ R and A ′ is an invertible, differential operator of degree r -2. We have thus

A -1 D = (A ′ ) -1 1 2 α Im α (α -iD) -1 - ᾱ Im α (ᾱ -iD) -1 .
The conclusion follows now from the fact that

ϕ → A ′ ϕ , D q (S 1 ) → Isom(H q (S 1 ), H q-r+2 (S 1 )) is smooth if q > 3/2 and q ≥ r -2 and that ϕ → (α -iD) ϕ , (ᾱ -iD) ϕ D q (S 1 ) → Isom(H q (S 1 ), H q-1 (S 1 ))
are smooth if q > 3/2. This applies in particular for the Camassa-Holm equation where the spray is smooth for q > 3 2 (in [START_REF] Misio | Classical solutions of the periodic Camassa-Holm equation[END_REF], it was proved that the spray is of class C 1 for q > 3 2 ).

Well-posedness

In this section, we will prove local existence and uniqueness of the initial value problem for the geodesics of the right-invariant H s metric on the Fréchet-Lie group Diff ∞ (S 1 ), and more generally for any right-invariant weak Riemannian metric for which the inertia operator A is such that [START_REF] Escher | The Degasperis-Procesi equation as a non-metric Euler equation[END_REF] ϕ

→ A ϕ = R ϕ • A • R ϕ -1 , D q (S 1 ) → Isom(H q (S 1 ), H q-r (S 1 ))
is smooth, for q > 3/2 and q -r ≥ 0. Under these assumptions, the metric admits a smooth spray F q defined on T D q (S 1 ) (c.f. theorem 3.10) and we can apply the Picard-Lindelöf theorem. For each (ϕ 0 , v 0 ) ∈ T D q (S 1 ), there exists a unique non-extendable solution

(ϕ, v) ∈ C ∞ (J q (ϕ 0 , v 0 ), T D q (S 1 )),
of the Cauchy problem (11)

ϕ t = v, v t = S ϕ (v),
with ϕ(0) = ϕ 0 and v(0) = v 0 , defined on some maximal interval of existence J q (ϕ 0 , v 0 ), which is open and contains 0. Note that in general J q (ϕ 0 , v 0 ) = R, meaning that the solutions are not global.

To prove well-posedness of the Cauchy problem [START_REF] Escher | The geometry of a vorticity model equation[END_REF] on the smooth manifold T Diff ∞ (S 1 ), we need precise regularity properties of solutions to [START_REF] Escher | The geometry of a vorticity model equation[END_REF] on each Hilbert approximation manifold T D q (S 1 ). More precisely, assume that (ϕ 0 , v 0 ) ∈ T D q+1 (S 1 ). Then, we may solve [START_REF] Escher | The geometry of a vorticity model equation[END_REF] in T D q (S 1 ) and in T D q+1 (S 1 ). Since solutions on each level are non-extendable, we clearly have [START_REF] Escher | Restrictions on the geometry of the periodic vorticity equation[END_REF] J q+1 (ϕ 0 , v 0 ) ⊂ J q (ϕ 0 , v 0 ), which could lead to

q J q (ϕ 0 , v 0 ) = {0}.
The remarkable observation that the maximal interval of existence is independent of the parameter q, due to the right-invariance of the spray (cf. lemma 4.1) was pointed out in [START_REF] Ebin | Groups of diffeomorphisms and the motion of an incompressible fluid[END_REF]Theorem 12.1]. This makes it possible to avoid Nash-Moser type schemes to prove local existence of smooth geodesics. Lemma 4.1 (No loss, nor Gain). Given (ϕ 0 , v 0 ) ∈ T D q+1 (S 1 ), we have

J q+1 (ϕ 0 , v 0 ) = J q (ϕ 0 , v 0 ),
for q > (3/2) and q -r ≥ 0.

Proof. Let Φ q be the flow of the spray F q and R s be the (right) action of the rotation group S 1 on D q (S 1 ), defined by

R s • ϕ (x) := ϕ(x + s), ϕ ∈ D q (S 1 ), x ∈ S 1 .
This action induces an action on T D q (S 1 ) given by

R s • (ϕ, v) (x) := (ϕ(x + s), v(x + s)), (ϕ, v) ∈ T D q (S 1 ), x ∈ S 1 . Note that if (ϕ, v) ∈ T D q+1 (S 1 ), then 3 s → R s • (ϕ, v), S 1 → T D q (S 1 ) is a C 1 map, and that d ds R s • (ϕ, v) = (ϕ x , v x ).
Therefore, if (ϕ 0 , v 0 ) ∈ T D q+1 (S 1 ), we get

(13) d ds s=0 Φ q (t, R s • (ϕ 0 , v 0 )) = ∂ (ϕ,v) Φ q (t, (ϕ 0 , v 0 )).(ϕ 0,x , v 0,x ).
On the other hand, the spray F q is invariant under each right-invariant translation R ϕ where ϕ ∈ D q (S 1 ). The same is true for its flow Φ q , and hence

Φ q (t, R s • (ϕ 0 , v 0 )) = R s • Φ q (t, (ϕ 0 , v 0 )) for all t ∈ J q (ϕ 0 , v 0 ), s ∈ R.
We get thus

∂ (ϕ,v) Φ q (t, (ϕ 0 , v 0 )).(ϕ 0,x , v 0,x ) = (ϕ x (t), v x (t)).
But ∂ (ϕ,v) Φ q (t, (ϕ 0 , v 0 )).(ϕ 0,x , v 0,x ) belongs to H q (S 1 ) × H q (S 1 ), and hence (ϕ(t), v(t)) ∈ T D q+1 (S 1 ) for all t ∈ J q (ϕ 0 , v 0 ).

We conclude therefore that J q (ϕ 0 , v 0 ) = J q+1 (ϕ 0 , v 0 ), which completes the proof.

Remark 4.2. Lemma 4.1 states that there is no loss of spatial regularity during the evolution of [START_REF] Escher | The geometry of a vorticity model equation[END_REF]. By reversing the time direction, it follows from the unique solvability that there is also no gain of regularity in the following sense: Let (ϕ 0 , v 0 ) ∈ T D q (S 1 ) be given and assume that (ϕ(t 1 ), v(t 1 )) ∈ T D q+1 (S 1 ) for some t 1 ∈ J q (ϕ 0 , v 0 ). Then (ϕ 0 , v 0 ) ∈ T D q+1 (S 1 ).

We get therefore the following local existence result.

Theorem 4.3. Let (10) be satisfied and consider the geodesic flow on the tangent bundle T Diff ∞ (S 1 ) induced by the inertia operator A. Then, given any (ϕ 0 , v 0 ) ∈ T Diff ∞ (S 1 ), there exists a unique non-extendable solution

(ϕ, v) ∈ C ∞ (J, T Diff ∞ (S 1 ))
of (11) on the maximal interval of existence J, which is open and contains 0.

And we obtain well-posedness of the Euler equation.

u 0 ∈ C ∞ (S 1 ) a unique non-extendable smooth solution u ∈ C ∞ (J, C ∞ (S 1 )).
The maximal interval of existence J is open and contains 0.

It is known that the Euler equation induced by the inertia operator

A = op 1 + k 2
leads to the classical periodic Camassa-Holm equation ( 14)

u t -u txx + 3uu x = 2u x u xx + uu xxx , t > 0, x ∈ S 1 ,
cf. [START_REF] Constantin | Geodesic flow on the diffeomorphism group of the circle[END_REF]. It may be interesting to briefly discuss another possible option for A, namely A = op (|k| r + δ 0 (k)) . Observe that Theorem 3.7 is applicable provided r ≥ 1. In that case, the mapping

ϕ → R ϕ • A • R ϕ -1 , D q (S 1 ) → L(H q (S 1 ), H q-r (S 1 ))
is smooth for q > (3/2) and q-r ≥ 0. Since in addition, A is a topological linear isomorphism from H q (S 1 ) onto H q-r (S 1 ), the operator A satisfies clearly assumption [START_REF] Escher | The Degasperis-Procesi equation as a non-metric Euler equation[END_REF] and thus Theorem 4.3 guarantees the well-possedness, in the smooth category, of the corresponding Euler equation ( 15)

m t + um x + 2u x m = 0, m = µ(u) + (-∆) r/2 u
where (-∆) r/2 := op (|k| r ) and µ(u) := S 1 u. Note that S 1 m dx is a conserved quantity for the evolution under [START_REF] Gay-Balmaz | Infinite dimensional geodesic flows and the universal Teichmüller space[END_REF], since S 1 m t dx = 0. Equation ( 15) is of particular interest for the values r = 2 and r = 1, respectively. In the first case we get the so-called µ-Hunter-Saxton equation, cf. [START_REF] Lenells | Integrable evolution equations on spaces of tensor densities and their peakon solutions[END_REF][START_REF] Escher | Restrictions on the geometry of the periodic vorticity equation[END_REF] (16)

u txx + uu xxx + 2u x u xx -2µ(u)u x = 0, t > 0, x ∈ S 1 ,
In the case r = 1 we get the so-called generalized CLM equation, cf. [START_REF] Escher | Restrictions on the geometry of the periodic vorticity equation[END_REF] (17)

Hu tx + uHu xx + 2µ(u)u x + 2u x Hu x = 0, t > 0, x ∈ S 1 ,
where H = op (i sgn(k)) denotes the Hilbert transform, acting on the spatial variable x ∈ S 1 . Note that op (|k|) = H • D = (-∆) 1/2 .

Exponential map and minimisation problems

The geodesic flow Φ q on the Hilbert manifold T D q (S 1 ) satisfies the following remarkable property Φ q (t, ϕ, σv) = Φ q (σt, ϕ, v), σ > 0, which is a consequence of the quadratic nature of the geodesic spray [START_REF] Lang | Fundamentals of Differential Geometry[END_REF]Chapter 4]. Therefore, the time one map of the flow is defined on some open set W q of T D q (S 1 ). The exponential map exp q is defined as

exp q : (ϕ, v) → π • Φ q (1, ϕ, v), W q :→ D q (S 1 ),
where π : T D q (S 1 ) → D q (S 1 ) is the canonical projection. For each ϕ ∈ D q (S 1 ), we denote by exp q,ϕ , the restriction of exp q to the tangent space T ϕ D q (S 1 ). Thus exp q,ϕ : T ϕ D q (S 1 ) → D q (S 1 ).

If the spray F q is smooth, then exp q,id is a local diffeomorphism from a neighbourhood V q of the 0 ∈ T id D q (S 1 ) onto a neighbourhood U q of id ∈ D q (S 1 ) [26, Theorem 4.1].

This last assertion is in general no longer true on a Fréchet manifold and in particular on Diff ∞ (S 1 ). One may find useful to recall on this occasion that the group exponential of Diff ∞ (S 1 ) is not a local diffeomorphism [START_REF] Milnor | Remarks on infinite-dimensional Lie groups[END_REF]. Moreover, the Riemannian exponential map for the L 2 metric (Burgers equation) on Diff ∞ (S 1 ) is not a local C 1 -diffeomorphism near the origin [START_REF] Constantin | On the geometric approach to the motion of inertial mechanical systems[END_REF]. Nevertheless, it has been established in [START_REF] Constantin | On the geometric approach to the motion of inertial mechanical systems[END_REF], that for the Camassa-Holm equation -which corresponds to the Euler equation of the H 1 metric on Diff ∞ (S 1 )and more generally for H k metrics (k ≥ 1) (see [START_REF] Constantin | Geodesic flow on the diffeomorphism group of the circle[END_REF]), the Riemannian exponential map was in fact a smooth local diffeomorphism. This result is still true for H s right-invariant metrics on Diff ∞ (S 1 ) provided s ∈ [1/2, +∞).

Theorem 5.1. The exponential mapping exp id for the H s -metric on Diff ∞ (S 1 ) is a smooth local diffeomorphism from a neighbourhood V of 0 in Vect(S 1 ) onto a neighbourhood U of id in Diff ∞ (S 1 ) for each s ≥ 1/2. The proof of theorem 5.1 relies mainly on a linearized version of the no loss, no gain lemma 4.1, and is stated below. The full proof is similar to the one given for [10, Theorem 14] and will be omitted. Lemma 5.2. Let v ∈ V q ∩ H q+1 (S 1 ), we have T v exp q,id H q+1 (S 1 ) = H q+1 (S 1 ).

Proof. Let G q : W q → D q (S 1 ) × D q (S 1 ) be the smooth mapping defined by G q (ϕ, v) := (ϕ, exp q (ϕ, v)).

From the invariance of the flow Φ t q under the right action of Diff ∞ (S 1 ), we deduce that

G q (R s • (ϕ, v)) = R s • G q (ϕ, v)
where R s denotes the natural action on derived spaces, induced by the action of the rotation group S 1 on D q (S 1 )

R s • ϕ (x) := ϕ(x + s).
Therefore, we have

(18) T G q .(R s • (ϕ, v, δϕ, δv)) = R s • T G q .(ϕ, v, δϕ, δv). Now, if (ϕ, v, δϕ, δv) ∈ T 2 D q+1 (S 1 ), then s → R s • (ϕ, v, δϕ, δv), S 1 → T 2 D q (S 1 )
is a C 1 mapping and

d ds R s • (ϕ, v, δϕ, δv) = (ϕ x , v x , δϕ x , δv x ).
Therefore, taking derivatives in s, at s = 0, in equation ( 18), we get

T 2 G q .(ϕ, v, δϕ, δv, ϕ x , v x , δϕ x , δv x ) = ϕ, v, δϕ, δv, ϕ x , exp q (ϕ, v) x , δϕ x , T exp q .(ϕ, v, δϕ, δv) x .
Since the left hand-side of the preceding equation lies in T 2 (D q (S 1 )×D q (S 1 )), we deduce that T v exp q,ϕ H q+1 (S 1 ) ⊂ H q+1 (S 1 ) as soon as (ϕ, v) ∈ D q+1 (S 1 ) × H q+1 (S 1 ). Since a similar statement can be made for T v exp q,ϕ -1 , the proof of lemma 5.2 is complete.

Remark 5.3. Let U and V be the neighbourhoods introduced in theorem 5.1. We define

V := ϕ∈Diff ∞ (S 1 ) R ϕ V,
which is an open neighbourhood of the zero section in T Diff ∞ (S 1 ), and

U := (ϕ, ψ) ∈ Diff ∞ (S 1 ) × Diff ∞ (S 1 ); ψ • ϕ -1 ∈ U ,
which is an open neighbourhood of the diagonal in Diff ∞ (S 1 ) × Diff ∞ (S 1 ). One can deduce, from theorem 5.1, that the mapping

G : V → U , (ϕ, v) → (ϕ, exp ϕ (v))
is a smooth diffeomorphism.

The restriction of exp q,ϕ to V q defines a local chart around id in D q (S 1 ). On this chart, called a normal chart, we have local polar coordinates, defined as follows. Given ϕ ∈ U q -{id}, there is a v ∈ V q \ {0} such that ϕ = exp q,ϕ (v). Letting now

w := v/ v H s , ρ := v H s ,
we have that ϕ = exp(ρw) and (ρ, w) are called the polar coordinates of ϕ ∈ U -{id}. Note that (ρ, w) depend smoothly of ϕ and that ρ(ϕ) → 0 as ϕ → id.

As can be checked in [START_REF] Lang | Fundamentals of Differential Geometry[END_REF], the following result is valid not only for a strong Riemannian metric but also for a weak Riemannian metric, provided there exists a compatible, symmetric covariant derivative. Lemma 5.4. For a piecewise C 1 curve ϕ : [a, b] → U q -{id}, we have the inequality

(19) L s (γ) ≥ |ρ(b) -ρ(a)| ,
where

L s (γ) := b a R ϕ -1 ϕ t H s dt.
A consequence of lemma 5.4 is that the length of any path which lies inside the normal neighbourhood is bounded below by r := |ρ(b) -ρ(a)|. Note also that a path, of constant velocity norm, which minimizes locally the arc-length minimizes also the energy defined as

E s (γ) := 1 2 b a R ϕ -1 ϕ t 2 H s dt.
We get therefore the following theorem.

Theorem 5.5. Let s ≥ 1/2. Given two nearby diffeomorphisms ϕ, ψ ∈ Diff ∞ (S 1 ), there exists a unique geodesic for the right-invariant H s metric on Diff ∞ (S 1 ), joining them and which minimizes locally the arc-length and the energy.

On a strong Riemannian manifold, given two nearby points, there exists a unique geodesic, joining these two points, which minimizes globally the arc-length and the energy. Note however, that for a weak metric, this might not be true. Indeed, in lemma 5.4, the bound (19) might not be true for a path which leaves the normal neighbourhood before leaving the (weak ball) of radius r defined as

B s (id, r) := {ϕ ∈ U ; ρ(ϕ) ≤ r} .
This happens, in particular, for the critical exponent s = 1/2 as it follows from [START_REF] Bauer | Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group[END_REF]. Note also that, for the L 2 metric, the situation is even worse since the energy has no local minimum [START_REF] Bruveris | The energy functional on the Virasoro-Bott group with the L 2 -metric has no local minima[END_REF]. The problem wether the geodesic joining two nearby points is a global minimum for s > 1/2 seems to be still an open problem.

To make this clear we close this section by a remark concerning the geodesic semi-distance d s induced by the H s metric and defined as the greatest lower bound of path-lengths L s (ϕ), for piecewise C 1 paths ϕ(t) in Diff ∞ (S 1 ) joining ϕ 0 and ϕ 1 . It was first shown in [START_REF] Michor | Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms[END_REF], that this semi-distance vanishes identically for the L 2 right-invariant metric on the diffeomorphism group of any compact manifold. More recently, it was shown in [START_REF] Bauer | Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group[END_REF] that d s vanishes identically on Diff ∞ (S 1 ) if s ∈ [0, 1/2], whereas d s is a distance for s > 1/2

∀ϕ 0 , ϕ 1 ∈ Diff ∞ (S 1 ), ϕ 0 = ϕ 1 ⇒ d s (ϕ 0 , ϕ 1 ) > 0.
Anyway, it should be noted that lemma 5.4 does not imply that the geodesic semi-distance is in fact a distance.

Euler equations on homogeneous spaces

The theory of Euler equations on a homogeneous space G/K has been developed in [START_REF] Khesin | Euler equations on homogeneous spaces and Virasoro orbits[END_REF]. In that case, the geodesic flow for a right-invariant Riemannian metric on the homogeneous space G/K, can be reduced to the so called Euler-Poincaré equation ( 20)

m t = ad * u m, m ∈ g * ,
on the dual space g * of the Lie algebra of G (see [START_REF] Khesin | Euler equations on homogeneous spaces and Virasoro orbits[END_REF] or Poincaré's original paper [START_REF] Poincaré | Sur une nouvelle forme des équations de la mécanique[END_REF]). Unfortunately, there is no natural contravariant formulation of equation ( 20), leading to an Euler equation, as it is the case for a Lie group.

In that case, the Eulerian velocity (defined using a lift g(t) in G of a path x(t) in G/K) is only defined up to a path in K and the relation between u and m is not one-to-one (see [START_REF] Tiglay | Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications[END_REF] for a recent survey on this subject).

Another way to treat the problem is to introduce sub-Riemannian geometry on G (see [START_REF] Grong | Sub-Riemannian geometry on infinitedimensional manifolds[END_REF][START_REF] Grong | Sub-Riemannian structures corresponding to Kählerian metrics on the universal Teichmüller space and curve[END_REF] for a deep study of this approach for Diff ∞ (S 1 )). These difficulties clear away if K is a normal subgroup. Indeed, in that case, the coset manifold G/K is a Lie group equipped with a right-invariant Riemannian metric. But this special case is not very useful for our study, since Diff ∞ (S 1 ) is a simple group: it has no nontrivial normal subgroups (see [START_REF] Guieu | L'algèbre et le groupe de Virasoro[END_REF]).

Hopefully, there is another situation where a right-invariant Riemannian metric on a homogeneous space can be reduced to the ordinary theory of the Euler equation on a Lie group, namely when there exists a section of the projection map π : G → G/K onto a subgroup H of G. This situation occurs for Diff ∞ (S 1 )/Rot(S 1 ), in which case H := Diff ∞ 1 (S 1 ), the subgroup of diffeomorphisms which fix one point, and for Diff ∞ (S 1 )/PSL(2, R) in which case H := Diff ∞ 3 (S 1 ), the subgroup diffeomorphisms which fix three points. More precisely, we have the following. Lemma 6.1. Let G be a group and H, K some subgroups of G. Suppose that

(1) The restriction to H of the projection map π :

G → G/K is surjec- tive, (2) H ∩ K = {e}.
Then H acts simply and transitively on G/K. Remark 6.2. As a result, if the hypothesis of lemma 6.1 are satisfied, then G/K inherits a group structure. Note, however that the restriction of the projection π : H → G/K is a group morphism, if and only if, K is a normal subgroup of G.

Proof. By definition, the projection map π sends and element g ∈ G to the coset Kg. To show that the (right) action of H on G/K is transitive, it suffices to show that for any coset Kg we can find h ∈ H such that Kh = Kg. But this means precisely that π : H → G/K is surjective. Hence the transitivity of the action is equivalent to the surjectivity of π. To prove that the action is simple, it is enough to show that the only element h ∈ H which fixes the coset K is h = e, the unit element. But this means Kh = K, and thus h ∈ K ∩ H, which leads to h = e by condition [START_REF] Bauer | Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group[END_REF]. Note that this implies that π : H → G/K is injective.

The mentioned scenario is summarized in the following proposition. Proposition 6.3. Suppose that H and K are closed subgroups of a Lie group G such that H ∩ K = {e} and such that π : H → G/K is surjective. Let g, h, and k denote the Lie algebras of G, H and K, respectively. Let A : g → g * be the inertia operator of a (degenerate) non-negative inner product •, • on g, which satisfies the following conditions: [START_REF] Hunter | Dynamics of director fields[END_REF] ker A = k, and

(22) Ad k u, Ad k v = u, v , ∀k ∈ K, ∀u, v ∈ g,
where Ad is the adjoint action of G on its Lie algebra g. Then, A induces a right-invariant Riemannian metric γ on G/K and π : H → G/K is a Riemannian isometry between H, endowed with the right-invariant metric induced by A, and (G/K, γ).

Remark 6.4. In the situation described, we have g = k ⊕ h and h * can be identified with

k 0 = {m ∈ g * ; (m, w) = 0, ∀w ∈ k} . Now, condition (22) implies 4 (23) ad w u, v = -u, ad w v , ∀w ∈ k, ∀u, v ∈ g.
Therefore, we have

(ad * u A(v), w) = -(ad * v A(u), w) , ∀w ∈ k, ∀u, v ∈ g,
where ad * is the coadjoint action of g on g * , defined by

(ad * u m, v) = -(m, [u, v]), u, v ∈ g, m ∈ g * . Hence ad * u A(v) + ad * v A(u) ∈ k 0 = im A and Arnold's bilinear operator B(u, v) = 1 2 A -1 ad * u A(v) + ad * v A(u)
is well-defined as a mapping from h × h to h. The Euler equation on h is given by ( 24)

u t = -B(u, u) = -A -1 ad * u A(u) .
In the next subsections, we will extend our main theorems to some geodesic equations on Diff ∞ (S 1 )/Rot(S 1 ) and Diff ∞ (S 1 )/PSL(2, R). Since the proofs are very similar to what has been done so far, we will not exhibit all the details but only point out crucial changes.

Assume, in addition, that

ϕ → A ϕ = R ϕ • A • R ϕ -1 , D q (S 1 ) → L(H q (S 1 ), H q-r (S 1 ))
is smooth for q > 3/2 and q -r ≥ 0. Then, the induced right-invariant metric on D q 1 (S 1 ) is smooth and has a smooth spray. Moreover, given any

(ϕ 0 , v 0 ) ∈ T Diff ∞ 1 (S 1
), there exists a unique non-extendable solution

(ϕ, v) ∈ C ∞ (J, T Diff ∞ 1 (S 1
)) of the Cauchy problem for the associated geodesic spray on the maximal interval of existence J.

We briefly discuss two special instances, namely

A = op k 2
and A = op (|k|) .

In the first case, A = op k 2 , we get the periodic Hunter-Saxton equation (HS), see [START_REF] Hunter | Dynamics of director fields[END_REF][START_REF] Yin | On the structure of solutions to the periodic Hunter-Saxton equation[END_REF][START_REF] Bressan | Global solutions of the Hunter-Saxton equation[END_REF][START_REF] Lenells | The Hunter-Saxton equation describes the geodesic flow on a sphere[END_REF], [START_REF] Lang | Fundamentals of Differential Geometry[END_REF] u txx + 2u x u xx + uu xxx = 0.

When A = op (|k|), we get the Constantin-Lax-Majda equation (CLM), see [START_REF] Constantin | A simple one-dimensional model for the three-dimensional vorticity equation[END_REF][START_REF] Wunsch | On the geodesic flow on the group of diffeomorphisms of the circle with a fractional Sobolev right-invariant metric[END_REF][START_REF] Escher | The geometry of a vorticity model equation[END_REF], [START_REF] Lenells | The Hunter-Saxton equation describes the geodesic flow on a sphere[END_REF] ∂ t (Hu x ) + uHu xx + 2u x Hu x = 0, where H = op (i sgn(k)) denotes the Hilbert transform, acting on the spatial variable x ∈ S 1 . Clearly, both symbols (k 2 ) k∈Z and (|k|) k∈Z satisfy [START_REF] Kolev | Lie groups and mechanics: an introduction[END_REF]. Moreover, they also fulfill the hypotheses of theorem 3.7, so that theorem 6.5 is applicable to both [START_REF] Lang | Fundamentals of Differential Geometry[END_REF] and [START_REF] Lenells | The Hunter-Saxton equation describes the geodesic flow on a sphere[END_REF]. 6.2. Euler equations on the coadjoint orbit Diff ∞ (S 1 )/PSL(2, R). Let PSL(2, R) denotes the subgroup of all rigid Möbius transformations which preserves the circle S 1 and let Diff ∞ (S 1 )/PSL(2, R), be the corresponding homogeneous space of right cosets. Let Diff ∞ 3 (S 1 ) be the subgroup of Diff ∞ (S 1 ) consisting of all diffeomorphisms of S 1 which fix 3 arbitrary distinct points (say x 0 , x 1 , x 2 ). Then PSL(2, R) ∩ Diff ∞ 3 (S 1 ) = {e} and the canonical projection

Diff ∞ 3 (S 1 ) → Diff ∞ (S 1 )/PSL(2, R) is a bijection. The group Diff ∞ 3 (S 1
) is a Fréchet Lie group and we can use this Fréchet structure to endow the quotient space Diff ∞ (S 1 )/PSL(2, R) with a Fréchet manifold structure. In that way, the canonical projection becomes a diffeomorphism. The Lie algebras of Diff ∞ 3 (S 1 ) is given by C

∞ 3 (S 1 ) := {u ∈ C ∞ (S 1 ) ; u(x 0 ) = 0, u(x 1 ) = 0, u(x 2 ) = 0}, whereas the Lie algebra sl(2, R) ⊂ C ∞ (S 1 ) of PSL(2, R) is the 3-dimensional subalgebra of C ∞ (S 1
), generated by w 0 (x) := 1, w 1 (x) := cos(x), w -1 (x) := sin(x).

An ILH structure on Diff ∞ 3 (S 1 ) is given by the Hilbert manifolds

D q 3 (S 1 ) := ϕ ∈ D q (S 1 ); ϕ(x 0 ) = x 0 , ϕ(x 1 ) = x 1 , ϕ(x 2 ) = x 2 ,
An important application of Theorem 6.6 is the Euler-Weil-Petersson equation, which corresponds to the inertia operator

A := HD(D 2 + 1) = op |k| (k 2 -1) .
This equation has been related with the Weil-Petersson metric on the universal Teichmüller space T (1) in [START_REF] Nag | Teichmüller theory and the universal period mapping via quantum calculus and the H 1/2 space on the circle[END_REF][START_REF] Takhtajan | Weil-Petersson metric on the universal Teichmüller space[END_REF]. The corresponding geodesic flow has been extensively studied in [START_REF] Gay-Balmaz | Infinite dimensional geodesic flows and the universal Teichmüller space[END_REF]. One of the main results of this paper is that the inertia operator A defines on a suitable replacement 6 for the "diffeomorphism group of class H 3/2 ", a right-invariant strong Riemannian structure which is, moreover, geodesically complete.

Our point of view is completely different because we are interested in the geodesic flow for the right-invariant metric on the Fréchet Lie group Diff ∞ 3 (S 1 ) and its Hilbert approximation D s (S 1 ) for s > 3/2. The price to pay is the fact that the metric only defines a weak Riemannian structure. Nevertheless, theorem 6.6 applies in this case. Indeed, A satisfies the hypothesis of theorem 3.7 and all conditions of theorem 6.6. This proves short time existence of geodesics on Diff ∞ 3 (S 1 ), which doesn't seem to be a consequence of the results in [START_REF] Gay-Balmaz | Infinite dimensional geodesic flows and the universal Teichmüller space[END_REF].

Appendix A. Fourier multipliers

In this Appendix, we recall and establish some basic results about Fourier multipliers. Here and in the following, we use the notation e k (x) = exp(2πikx), for k ∈ Z and x ∈ S 1 .

Lemma A.1. Let A be a continuous linear operator on the Fréchet space C ∞ (S 1 , C). Then the following three conditions are equivalent:

(1) A commutes with all rotations R s .

(2) [A, D] = 0, where

D = d/dx. (3) For each k ∈ Z, there is a a(k) ∈ C such that Ae k = a(k)e k .
In that case, we say that A is a Fourier multiplier.

Since every smooth function on the unit circle S 1 can be represented by its Fourier series, we get that

(29) (Au)(x) = k∈Z a(k)û(k)e k (x),
for every Fourier multiplier A and every u ∈ C ∞ (S 1 ), where

û(k) := S 1 u(x)e k (x) dx,
stands for the k-th Fourier coefficients of u. The sequence a : Z → C is called the symbol of A. We use also the notation A := op (a(k)) for the Fourier multiplier induced by the sequence a.

Proof. Given s ∈ R and u ∈ C ∞ (S 1 ), let u s (x) := u(x + s). If A commutes with translations we have

(Au) s (x) = (Au s )(x).
Taking the derivative of both sides of this equation with respect to s at 0 and using the continuity of A, we get DAu = ADu which proves the implication (1) ⇒ (2). If [A, D] = 0, then both Ae k and e k are solutions of the linear differential equation u ′ = (-2πik)u and are therefore equal, up to a multiplicative constant a(k). This proves that (2) ⇒ (3).

If Ae k = a(k)e k , for each k ∈ Z and A is continuous, then we have representation [START_REF] Michor | Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms[END_REF]. Therefore

(Au) s (x) = k∈Z a(k)û(k)e k (x + s) = k∈Z a(k) u s (k)e k (x) = (Au s )(x),
which proves that (3) ⇒ (1).

Remark A.2. The space of Fourier multipliers is a commutative subalgebra of the algebra of linear operators on C ∞ (S 1 , C). It contains all linear differential operators with constant coefficients. Note that a Fourier multiplier A is L 2symmetric iff its symbol a is real.

Let I n := {1, . . . , n}. Given a function f and n ≥ 1, we introduce

S f,n (m 0 , m 1 , . . . , m n ) := m 0 n p=0 (-1) p J⊂In, |J|=p f n m 0 + j∈J m j ,
where f n (ξ) := ξ n-1 f (ξ) and m j ∈ R, for 0 ≤ j ≤ n.

Lemma A.3. For each n ≥ 1, we have

(30) S f,n+1 (m 0 , m 1 , . . . , m n+1 ) = n k=0 m k S f,n (m 0 , . . . , m n ) -S f,n (m 0 , . . . , m k + m n+1 , . . . , m n ) .
Proof. Using the definition of S f,n , the right hand-side of (30) writes as

n p=0 (-1) p J⊂In, |J|=p    m 2 0 f n m 0 + j∈J m j -m 0 (m 0 + m n+1 )f n m 0 + j∈J m j + n k=1 m 0 m k f n m 0 + j∈J m j -f n m 0 + j∈J m j + δ J (k) m n+1    ,
which can be recast as

m 0 n p=0 (-1) p J⊂In, |J|=p    m 0 + j∈J m j f n m 0 + j∈J m j -m 0 + m n+1 + j∈J m j f n m 0 + m n+1 + j∈J m j    ,
and which is equal to

m 0 n p=0 (-1) p J⊂In, |J|=p    f n+1 m 0 + j∈J m j -f n+1 m 0 + m n+1 + j∈J m j    , since f n+1 (ξ) = ξf n (ξ).
Therefore, the right hand-side of ( 30) is equal to

m 0 n p=0 (-1) p J⊂I n+1 , |J|=p, n+1 / ∈J f n+1 m 0 + j∈J m j + m 0 n p=0 (-1) p+1 J⊂I n+1 , |J|=p+1, n+1∈J f n+1 m 0 + j∈J m j ,
which is exactly S f,n+1 (m 0 , m 1 , . . . , m n+1 ).

Lemma A.4. Let A = op (a(k)) be a Fourier multiplier on C ∞ (S 1 ), and let (A n ) be the sequence of multilinear operators defined inductively in lemma 3.2. Then, for each n ≥ 1, we have [START_REF] Misio | A shallow water equation as a geodesic flow on the Bott-Virasoro group[END_REF] A n (e m 0 , . . . ,

e mn ) = a n (m 0 , m 1 , . . . , m n )e m 0 +m 1 •••+mn , where (32)
a n (m 0 , m 1 , . . . , m n ) = (2πi) n S a,n (m 0 , m 1 , . . . , m n ).
Remark A.5. For n = 1, we have

a 1 (m 0 , m 1 ) = (2πi)m 0 a(m 0 ) -a(m 0 + m 1 )
and for n = 2, we get

a 2 (m 0 , m 1 , m 2 ) = (2πi) 2 m 0 (m 0 + m 1 + m 2 )a(m 0 + m 1 + m 2 ) -(m 0 + m 1 )a(m 0 + m 1 ) -(m 0 + m 2 )a(m 0 + m 2 ) + m 0 a(m 0 ) .
Proof. From relation [START_REF] Constantin | Geodesic flow on the diffeomorphism group of the circle[END_REF], we obtain (31) by induction, where the sequence (a n ) satisfies

(33) a n+1 (m 0 , . . . , m n+1 ) = (2πi) n k=0 m k a n (m 0 , . . . , m n ) -a n (m 0 , . . . , m k + m n+1 , . . . , m n ) .
For n = 1, relation ( 32) is clear. Now, suppose inductively that (32) holds for some n ≥ 1. Then, using [START_REF] Nag | Teichmüller theory and the universal period mapping via quantum calculus and the H 1/2 space on the circle[END_REF], we get

a n+1 (m 0 , . . . , m n+1 ) = (2πi) n+1 n k=0 m k S a,n (m 0 , . . . , m n ) -S a,n (m 0 , . . . , m k + m n+1 , . . . , m n ) ,
which is equal to (2πi) n+1 S a,n+1 (m 0 , m 1 , . . . , m n+1 ), by virtue of lemma A.3. This achieves the proof.

Recall that a function f : R → R is absolutely continuous if f has a derivative almost everywhere, the derivative is locally Lebesgue integrable and

f (b) = f (a) + b a f ′ (τ ) dτ, for all a, b ∈ R. Lemma A.6. Let f : R → R and n ≥ 1. Suppose that f n (ξ) := ξ n-1 f (ξ) is of class C n-1 , that f (n-1) n
is absolutely continuous and that there exists C n > 0 and r ≥ 1 such that

(34) f (n) n (ξ) ≤ C n (1 + ξ 2 ) (r-1)/2 , almost everywhere. Then |S f,n (m 0 , m 1 , . . . , m n )| ≤ C n   n j=0 |m j |   J⊂In   1 +   m 0 + j∈J m j   2   (r-1)/2
, for all m 0 , m 1 , . . . , m n ∈ R.

Proof. Fix n ≥ 1 and m 0 , m 1 , . . . , m n ∈ R. Let g k be the sequence of functions defined inductively by

g 0 (ξ) = f n (ξ), g k+1 (ξ) = g k (ξ) -g k (ξ + m n-k ),
for k = 0, . . . , n -1. We have in particular S f,n (m 0 , m 1 , . . . , m n ) = m 0 g n (m 0 ).

Let K 0 = {m 0 } and for p = 1, . . . , n, let K p be the convex set generated by K p-1 and K p-1 + m p . Note that K n is the convex hull of the points m 0 + j∈J m j , for all subset J of {1, . . . , n}. Let

M := max ξ∈Kn (1 + ξ 2 ) (r-1)/2 .
By hypothesis, we have g (n) 0 (ξ) ≤ C n M almost everywhere on K n , and using the mean value theorem, we get inductively

g (n-k) k (ξ) ≤ C n M |m n | • • • |m n-k+1 | , ∀ξ ∈ K n-k ,
for k = 1, . . . n. In particular, we have

|g n (m 0 )| ≤ C n M n j=1 |m j | .
Let's now estimate M . For r ≥ 1, the function ξ → (1 + ξ 2 ) (r-1)/2 has no local maximum on R (or is constant). Thus, it attains its maximum on the compact, convex set K n at some extremal point m 0 + j∈J 0 m j and we get

M =   1 +   m 0 + j∈J 0 m j   2   (r-1)/2 ≤ J⊂In   1 +   m 0 + j∈J m j   2   (r-1)/2
, which achieves the proof.

Corollary A.7. Let A = op (a(k)) be a Fourier multiplier of order r ≥ 1.

Suppose that a satisfies the hypothesis of lemma A.6. Then, each A n extends to a bounded multilinear operator

A n ∈ L n+1 H q (S 1 ), H q-r (S 1 ) when q > 3/2 and q -r ≥ 0.

Proof. Given smooth functions u 0 , . . . , u n , we have

A n (u 0 , . . . , u n ) 2 H q-r = k∈Z (1 + k 2 ) q-r Ân (k) 2 ,
where

Ân (k) :=< A n (u 0 , . . . , u n ), e k > L 2 = m 0 +•••+mn=k a n (m 0 , . . . , m n ) u 0 (m 0 ) • • • u n (m n ).
If we assume, moreover, that a satisfies the hypothesis of lemma A.6, we get Ân (k)

2

J⊂In

Ân,J (k)

2
where Ân,J (k) :=

m 0 +•••+mn=k 1 + m 0 + j∈J m j 2 (r-1)/2 u ′ 0 (m 0 ) • • • u ′ n (m n ) .
Recall now, that, for smooth functions on C ∞ (S 1 ), we have

v 0 • • • v n (k) = m 0 +•••+mn=k v 0 (m 0 ) • • • v n (m n ).
Therefore, Ân,J (k) is the k-th Fourier coefficient of

Λ r-1   ũ0 j∈J ũj   j∈In\J ũj ,
where Λ s := op (1 + k 2 ) s/2 and ũj is the smooth function with Fourier coefficients ũj (k) = u ′ j (k) . Thus

A n (u 0 , . . . , u n ) 2 H q-r J⊂In Λ r-1   ũ0 j∈J ũj   j∈In\J ũj 2 H q-r
. Now, because q -1 > 1/2 and 0 ≤ q -r ≤ q -1, we deduce from lemma B.1 that

A n (u 0 , . . . , u n ) 2 H q-r J⊂In Λ r-1   ũ0 j∈J ũj   2 H q-r j∈In\J ũj 2 H q-1 J⊂In ũ0 j∈J ũj 2 H q-1 j∈In\J ũj 2 H q-1 ũ0 2 H q-1 • • • ũn 2 H q-1 , because H q-1 (S 1
) is a multiplicative algebra. Since, moreover, u ′ j and ũj have the same H q-1 norm, we obtain finally

A n (u 0 , . . . , u n ) H q-r u 0 H q • • • u n H q ,
which achieves the proof.

Appendix B. Boundedness properties of right translations

In this Appendix, we provide some explicit estimates for the right representation of D q (S 1 ) on L(H ρ (S 1 ), H ρ (S 1 )), when q > 3/2 and 0 ≤ ρ ≤ q. These results are certainly not new but we give here very precise estimates.

Let us recall first the Slobodeckij spaces W s,p (S 1 ). For m ∈ N and 1 ≤ p ≤ ∞, the W m,p -norm of a measurable functions on S 1 is defined by

u W m,p := m j=0 u (j) L p ,
where u (j) denotes the derivative of order j. When s > 0 is not an integer and p < ∞, the W s,p -norm is defined by

u W s,p := u W m,p + p σ,p (u (m) )
where m = [s] and s = m + σ, and the semi-norm p σ,p (0 < σ < 1) is defined by p σ,p (w) = In the following, we write p σ,2 = p σ , when there is no ambiguity. The Banach space W s,p (S 1 ) is by definition the completion of C ∞ (S 1 ) with respect to the W s,p -norm and when p = 2, we get the Hilbert space W s,2 (S 1 ) = H s (S 1 ).

Recall that for 1 ≤ p, q < ∞ and r, s ∈ R such that r ≥ s, and r -

1 p ≥ s - 1 q ,
we have the continuous Sobolev embeddings (see [START_REF] Triebel | Theory of Function Spaces[END_REF]) W r,p (S 1 ) ⊂ W s,q (S 1 ), and

H s (S 1 ) ⊂ W m,∞ (S 1 ), for s = m + σ, m ∈ N and σ > 1/2.
Recall that the space H s (S 1 ) is a multiplicative algebra for s > 1/2. We have, moreover, the following result which is a discrete version of [START_REF] Inci | On the Regularity of the Composition of Diffeomorphisms[END_REF]Lemma 2.3].

Lemma B.1. Let q > 1/2 and 0 ≤ ρ ≤ q. Then, pointwise multiplication in C ∞ (S 1 ) extends to a continuous bilinear mapping H q (S 1 ) × H ρ (S 1 ) → H ρ (S 1 ).

Proof. Given u, v ∈ C ∞ (S 1 ), we have

uv 2 H ρ = n∈Z (1 + n 2 ) ρ | uv(n)| 2 .
Let us introduce the sequences ũ(k) := (1 + k 2 ) q/2 |û(k)| , and ṽ(l) := (1 + l 2 ) ρ/2 |v(l)| , so that ũ l 2 = u H q , and ṽ l 2 = v H ρ . We have thus

(1 + n 2 ) ρ/2 | uv(n)| = (1 + n 2 ) ρ/2 k+l=n û(k)v(l) k+l=n |k|≤|l| 1 (1 + k 2 ) q/2 ũ(k)ṽ(l) + k+l=n |k|>|l| 1 (1 + l 2 ) q/2 ũ(k)ṽ(l) (λ q ũ * ṽ)(n) + (ũ * λ q ṽ)(n),
where λ q is the sequence defined by

λ q (k) := 1 (1 + k 2 ) q/2
and * stand for the convolution of sequences. By virtue of the Young inequality a * b l 2 a l 1 b l 2 , which is valid for any complex sequences a,b, we get λ q ũ * ṽ l 2 λ q ũ l 1 ṽ l 2 , ũ * λ q ṽ l 2 λ q ṽ l 1 ũ l 2 , and by Cauchy-Schwarz, we obtain finally

uv H ρ λ q l 2 ũ l 2 ṽ l 2 u H q v H ρ ,
which achieves the proof.

The main estimates which have been used in this paper concerning the right representation of D q (S 1 ) on L(H ρ (S 1 ), H ρ (S 1 )) are given below. Note that the cases overlap.

Lemma B.2. Let q > 3/2, ϕ ∈ D q (S 1 ) and 0 ≤ ρ ≤ q.

Case 1: For 0 ≤ ρ ≤ 1, we have

(35) R ϕ L(H ρ (S 1 ),H ρ (S 1 )) ≤ C ρ ( 1/ϕ x L ∞ , ϕ x L ∞ ) .
Case 2: For 0 ≤ ρ ≤ 2, we have

(36) R ϕ L(H ρ (S 1 ),H ρ (S 1 )) ≤ C ρ ( 1/ϕ x L ∞ , ϕ x H q-1 ) .
Case 3: For 3/2 < ρ ≤ 3, we have

(37) R ϕ L(H ρ (S 1 ),H ρ (S 1 )) ≤ C ρ ( 1/ϕ x L ∞ , ϕ x L ∞ ) ϕ x H ρ-1 .
Case 4: For ρ > 5/2, we have

(38) R ϕ L(H ρ (S 1 ),H ρ (S 1 )) ≤ C ρ ( 1/ϕ x L ∞ , ϕ x H ρ-2 ) ϕ x H ρ-1 .
In each case, C ρ is a positive, continuous function on (R + ) 2 , depending on ρ.

Proof. Let q > 3/2 and fix ϕ ∈ D q (S 1 ). Since the estimates involve only linear expressions of u and C ∞ (S 1 ) is dense in H ρ (S 1 ), it is enough to establish them for u ∈ C ∞ (S 1 ). Note first that a change of variables leads to

u • ϕ L 2 ≤ 1/ϕ x 1/2 L ∞ u L 2 , whereas p σ (u • ϕ) ≤ 1/ϕ x L ∞ ϕ x (1+2σ)/2 L ∞ p σ (u), for σ ∈ (0, 1). Moreover, we have (u • ϕ) (1) L 2 ϕ x L ∞ 1/ϕ x 1/2 L ∞ u x L 2 ,
which proves the first case. Note also that this proves also [START_REF] Poincaré | Sur une nouvelle forme des équations de la mécanique[END_REF] for 0 ≤ ρ ≤ 1, because ϕ x L ∞ ≤ ϕ x H q-1 . Suppose now that ρ = 1+σ, where 0 < σ ≤ 1. In this case, we have

u • ϕ H ρ u • ϕ H 1 + (u x • ϕ)ϕ x H σ C 1 ( 1/ϕ x L ∞ , ϕ x H q-1 ) u H 1 + C σ ( 1/ϕ x L ∞ , ϕ x H q-1 ) u x H σ ϕ x H q-1 ,
where we have used lemma B.1 and (36) for 0 ≤ ρ ≤ 1. This completes the proof of the second case. If ρ = 1 + σ with 1/2 < σ ≤ 1, we have using [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] and the fact that H σ (S 1 ) is a multiplicative algebra

u • ϕ H ρ u • ϕ H 1 + u x • ϕ H σ ϕ x H σ C 1 ( 1/ϕ x L ∞ , ϕ x L ∞ ) u H 1 + C σ ( 1/ϕ x L ∞ , ϕ x L ∞ ) u x H σ ϕ x H σ . Now, noting that 1 ≤ 1/ϕ x L ∞ ϕ x L ∞ ≤ 1/ϕ x L ∞ ϕ x H σ , we get u • ϕ H ρ C ρ ( 1/ϕ x L ∞ , ϕ x L ∞ ) ϕ x H ρ-1 u H ρ ,
where C ρ is a positive, continuous function on (R + ) 2 . If ρ = 2 + σ where 0 < σ ≤ 1, we have by virtue of lemma B.1 and ( 35)

(u • ϕ) (2) H σ u xx • ϕ H σ (ϕ x ) 2 H 1 + u x • ϕ H 1 ϕ xx H σ C σ ( 1/ϕ x L ∞ , ϕ x L ∞ ) u xx H σ ϕ x L ∞ ϕ x H 1 + C 1 ( 1/ϕ x L ∞ , ϕ x L ∞ ) u x H 1 ϕ xx H σ because (ϕ x ) 2 H 1 ϕ x L ∞ ϕ x H 1 . Therefore u • ϕ H ρ u • ϕ H 1 + (u • ϕ) (2) H σ C ρ ( 1/ϕ x L ∞ , ϕ x L ∞ ) ϕ x H ρ-1 u H ρ ,
where C ρ is a positive, continuous function. This proves the third case. If ρ = 2 + σ and 1/2 < σ ≤ 1 we get immediately [START_REF] Takhtajan | Weil-Petersson metric on the universal Teichmüller space[END_REF] from the preceding computation because then ϕ x L ∞ ϕ x H σ . Suppose now that ρ = m+σ, where m ≥ 3 and σ ∈ [0, 1). Given an integer n, we have

(u • ϕ) (n+1) = n k=0 u (k) x • ϕ W n,k (ϕ),
where W n,k (ϕ) is a homogeneous polynomial of degree k + 1 in the variables ϕ x , . . . , ϕ

(n-k) x
. The sequence W n,k is defined by

W n,n (ϕ) = ϕ n+1 x , W n,0 (ϕ) = ϕ (n) x , and
W n+1,k (ϕ) = W n,k-1 (ϕ)ϕ x + W n,k (ϕ) ′ , 1 ≤ k ≤ n.
In particular, for n ≥ 2, we have

W n,1 (ϕ) = (n + 1)ϕ x ϕ (n-1)
x + P n (ϕ x , . . . , ϕ (n-2)

x ),

where P n is a homogeneous polynomial of degree 2. Thus, for m ≥ 2, we get

(u • ϕ) (m+1) L 2 u x L ∞ ϕ (m) x L 2 + u xx • ϕ L 2 W m,1 (ϕ) L ∞ + m k=2 u (k) x • ϕ L 2 W m,k (ϕ) L ∞ 1 + 1/ϕ x L ∞ ϕ x L ∞ + m k=1 ϕ x k H m-1 × ϕ x H m-1 u H m , (39) because
W m,1 (ϕ) L ∞ ϕ x L ∞ ϕ x H m + ϕ x 2 H m-1 , and W m,k (ϕ) L ∞ ϕ x k+1 H m-1 , 2 ≤ k ≤ m.
Starting, with m = 2, we get first [START_REF] Takhtajan | Weil-Petersson metric on the universal Teichmüller space[END_REF] for ρ = 3 and the case when ρ is an integer m ≥ 3 is obtained by an inductive argument, using [START_REF] Tiglay | Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications[END_REF]. Now, using lemma B.1, we have for m ≥ 3 and 0 < σ < 1

(u • ϕ) (m) H σ u x • ϕ H 1 ϕ (m-1) x H σ + m-1 k=1 u (k) x • ϕ H σ W m-1,k (ϕ) H 1 . But, for 2 ≤ k ≤ m -1 we have W m-1,k (ϕ) H 1 ϕ x k+1
H m-2 . and for k = 1 and m -1 ≥ 2, we have

W m-1,1 (ϕ) H 1 ϕ x H m-2 ϕ x H m-1 .
We get therefore

(u • ϕ) (m) H σ 1/ϕ x 1/2 L ∞ + 1/ϕ x 3/2 L ∞ + 1/ϕ x 1/2 L ∞ m-1 k=1 ϕ x k H m-2 × ϕ x H ρ-1 u H ρ
which shows that estimate [START_REF] Takhtajan | Weil-Petersson metric on the universal Teichmüller space[END_REF] is also true when ρ > 5/2 is not necessary an integer. This proves the fourth case and achieves the proof.

Corollary B.3. Let q > 3/2 and 0 ≤ ρ ≤ q. Then, the mapping

(ϕ, u) → u • ϕ, D q (S 1 ) × H ρ (S 1 ) → H ρ (S 1 )
is continuous.

Proof. Lemma B.2 shows that the mapping

ϕ → R ϕ , D q (S 1 ) → L(H ρ (S 1 ), H ρ (S 1 ))
is locally bounded for q > 3/2 and 0 ≤ ρ ≤ q. To establish the continuity of the mapping

(ϕ, u) → u • ϕ, D q (S 1 ) × H ρ (S 1 ) → H ρ (S 1 ), it is thus sufficient to prove that ϕ → u • ϕ, D q (S 1 ) → H ρ (S 1 ).
is continuous for each fixed u ∈ H ρ (S 1 ). Let ϕ 0 ∈ D q (S 1 ) and ε > 0. Choose a neighbourhood V of ϕ 0 , that we may suppose to be a ball in a local chart of the Banach manifold D q (S 1 ), and on which

R ϕ L(H ρ (S 1 ),H ρ (S 1 )) < K, for some positive constant K. Since C ∞ (S 1 ) is dense in H ρ (S 1 ), we can find w ∈ C ∞ (S 1 ) such that u -w H ρ < ε/K. We have thus u • ϕ -u • ϕ 0 H ρ < w • ϕ -w • ϕ 0 H ρ + 2ε. Now, let ϕ(t) := tϕ + (1 -t)ϕ 0 . We have, pointwise w • ϕ -w • ϕ 0 = 1 0 (w x • ϕ(t))(ϕ -ϕ 0 ) dt,
and we deduce from lemma B.1, that

w • ϕ -w • ϕ 0 H ρ ≤ C 1 0 w x • ϕ(t) H ρ ϕ -ϕ 0 H q dt,
for some positive constant C, which depends only on q and ρ. Thus

w • ϕ -w • ϕ 0 H ρ ≤ CK w x H ρ ϕ -ϕ 0 H q ,
and therefore, if ϕ -ϕ 0 H q is small enough, we get that

u • ϕ -u • ϕ 0 H ρ < 3ε,
which achieves the proof.

Remark B.4. The mapping (ϕ, u) → R ϕ (u) := u • ϕ, D q (S 1 ) × H q (S 1 ) → H q (S 1 ).

is continuous for q > 3/2 (see [START_REF] Ebin | Groups of diffeomorphisms and the motion of an incompressible fluid[END_REF]), but that does not imply that the mapping ϕ → R ϕ , D q (S 1 ) → L(H q (S 1 ))

is continuous with respect to the operator norm on L(H q (S 1 )) (norm continuity). Norm continuity obviously implies continuity but the converse is false. Indeed, a general result in the theory of semigroups of linear operators states that a semigroup on a Banach space E is norm continuous at 0, if and only if, its infinitesimal generator is bounded on E, cf. [35, Theorem 1.2]. Let now q > 3/2 and let τ s be the rotation by the angle s on S 1 . Then the representation of the group {R τs ; s ∈ R} is continuous on H q (S 1 ). But it cannot be norm continuous, since its infinitesimal generator D is not bounded on H q (S 1 ). A direct argument, which shows that R τs -Id L(H q (S 1 )) is bounded away from 0 for all s near 0 is runs as follows: Let s ∈ (-1/2, 1/2) and u s be a periodic, bump function with support in (k -s/2, k + s/2) (k ∈ Z) with u s L 2 = 1. We have then R τs u s -u s 2 H q (S 1 ) = 2 u s 2 H q (S 1 ) , because u s and R τs u are H q (S 1 )-orthogonal and R τs is an H q (S 1 )-isometry. Hence R τs -Id L(H q (S 1 )) ≥ √ 2 for -1 2 < s < 1 2 , which proves that the representation ϕ → R ϕ is not norm continuous.

Nevertheless, we have the following result.

Corollary B.5. Let q > 3/2. Then, the mappings ϕ → R ϕ , D q (S 1 ) → L(H q (S 1 ), H q-1 (S 1 ))

and ϕ → R ϕ -1 , D q (S 1 ) → L(H q (S 1 ), H q-1 (S 1 ))

are continuous.

Proof. Note first that since ϕ → ϕ -1 from D q (S 1 ) to D q (S 1 ) is continuous (see for instance [START_REF] Inci | On the Regularity of the Composition of Diffeomorphisms[END_REF]), it is enough to show that ϕ → R ϕ , D q (S 1 ) → L(H q (S 1 ), H q-1 (S 1 )) is continuous. Let ϕ 0 ∈ D q (S 1 ). Choose a neighbourhood V of ϕ 0 , that we may suppose to be a ball in a local chart of the Banach manifold D q (S 1 ), and on which R ϕ L(H q-1 (S 1 ),H q-1 (S 1 )) < K, for some positive constant K. Now, let ϕ(t) := tϕ+(1-t)ϕ 0 and v ∈ H q (S 1 ). We have, pointwise

v • ϕ -v • ϕ 0 = 1 0 (v x • ϕ(t))(ϕ -ϕ 0 ) dt. Hence v • ϕ -v • ϕ 0 H q-1 ≤ C 1 0 v x • ϕ(t) H q-1 ϕ -ϕ 0 H q-1 dt,
for some positive constant C and from which we deduce that R ϕ -R ϕ 0 L(H q (S 1 ),H q-1 (S 1 )) ≤ CK ϕ -ϕ 0 H q .

This shows that ϕ → R ϕ , D q (S 1 ) → L(H q (S 1 ), H q-1 (S 1 ))

is locally Lipschitz continuous and completes the proof.

Another interpreting consequence of estimates B.2 is the following results, which extends [22, Theorem 1.2] beyond the critical exponent.

Corollary B.6. Let q > 3/2. Then, the mappings (ϕ, v) → v • ϕ, D q (S 1 ) × H q (S 1 ) → H q-1 (S 1 ) and (ϕ, v) → v • ϕ -1 , D q (S 1 ) × H q (S 1 ) → H q-1 (S 1 ) are C 1 .

Proof. We are going to show that (ϕ, v) → v • ϕ -1 , D q (S 1 ) × H q (S 1 ) → H q-1 (S 1 ) is C 1 . The proof that the first mapping is C 1 is similar and easier. Observe first that if ϕ(s) and v(s) are smooth paths in Diff ∞ (S 1 ) and C ∞ (S 1 ) respectively with ϕ(0) = ϕ and v(0) = v, we have

∂ s v(s) • ϕ(s) -1 s=0 = δv • ϕ -1 - v x • ϕ -1 ϕ x • ϕ -1 δϕ • ϕ -1 ,
where δϕ = ∂ s ϕ(s)| s=0 and δv = ∂ s v(s)| s=0 . Let U be a local chart in D q (S 1 ), that we assume to be a ball in H q (S 1 ). Given ϕ 0 , ϕ 1 in Diff ∞ (S 1 )∩U and v 0 , v 1 ∈ C ∞ (S 1 ), we set ϕ(t) := (1 -t)ϕ 0 + tϕ 1 , and v(t) := (1 -t)v 0 + tv 1 , for t ∈ [0, 1]. We have therefore

(40) v 1 • ϕ -1 1 -v 0 • ϕ -1 0 = 1 0 (v 1 -v 0 ) • ϕ(t) -1 dt - 1 0 v x (t) • ϕ(t) -1 ϕ x (t) • ϕ(t) -1 (ϕ 1 -ϕ 0 ) • ϕ(t) -1 dt.
Now, using lemma 3.5, we observe that both sides of ( 40) are continuous expressions of ϕ k , v k (k = 0, 1). Using a density argument, we conclude therefore, that the equality is still true in H q-1 (S 1 ) if we take ϕ 0 , ϕ 1 ∈ D q (S 1 ) and v 0 , v 1 ∈ D q (S 1 ). Furthermore, the mapping

(ϕ, v) → (δϕ, δv) → R ϕ -1 δv - v x • ϕ -1 ϕ x • ϕ -1 R ϕ -1 δϕ
from D q (S 1 ) × H q (S 1 ) to L H q (S 1 ) × H q (S 1 ), H q-1 (S 1 ) is continuous by corollary B.5 and the fact that H q-1 (S 1 ) is a multiplicative algebra for q > 3/2. We conclude the proof using lemma 3.6.

To conclude this Appendix, we provide an estimate for the norm of (ϕ -1 ) x which might be useful, on its own. Lemma B.7. Let q > 3/2. Then (ϕ -1 ) x H q-1 C q ( 1/ϕ x L ∞ , ϕ x H q-1 )

where C q is a positive, continuous function on (R + ) 2 .

Proof. Given σ ∈ (0, 1), a change of variables leads to the estimate

w • ϕ -1 H σ ϕ x 1/2 L ∞ + ϕ x L ∞ 1/ϕ x (1+2σ)/2 L ∞ w H σ ,
for any w ∈ H σ (S 1 ) and any C 1 diffeomorphism ϕ, whereas w • ϕ -1

H 1 ϕ x 1/2 L ∞ + ϕ x 3/2 L ∞ 1/ϕ x L ∞ w H 1 ,
for any w ∈ H 1 (S 1 ) and any C 1 diffeomorphism ϕ.

1) Suppose first that q = 1 + σ and thus σ > 1/2. We get

(ϕ -1 ) x H σ = 1 ϕ x • ϕ -1 H σ ≤ ϕ x 1/2 L ∞ + ϕ x L ∞ 1/ϕ x (1+2σ)/2 L ∞ 1/ϕ x H σ .
But a direct computation shows that

1/ϕ x H σ 1/ϕ x 2 L ∞ ϕ x H σ ,
which finishes the proof of the lemma for q < 2, since ϕ x L ∞ ϕ x H σ . 2) Suppose now that q = m + σ, where m ≥ 2 and σ ∈ [0, 1). Given an integer n, we have

1 ϕ x • ϕ -1 (n) = 1 (ϕ x • ϕ -1 ) 2n+1 P n (ϕ x • ϕ -1 , . . . , ϕ (n) x • ϕ -1 ),
where P n is a homogeneous polynomial of degree n, which is of partial degree at most one in ϕ

(n)

x • ϕ -1 . We have therefore • ϕ -1 . Therefore, we have

1 ϕ x • ϕ -1 H m-2 m-2 k=0 1/ϕ x 2k+1 L ∞ P k L ∞ m-2 k=0 1/ϕ x 2k+1 L ∞ ϕ x k H m-1 , whereas 1 ϕ x • ϕ -1 (m-1) H σ 1/ϕ x 2m-1 H 1 P m-
P m-1 H σ a 0 (ϕ) H 1 + a 1 (ϕ) H 1 ϕ (m-1) x • ϕ -1 H σ , where a j (ϕ) H 1 ϕ x 1/2 L ∞ + ϕ x 3/2 L ∞ m-1-j 1/ϕ x m-1-j L ∞ ϕ x m-1-j H m-1 , and
ϕ (m-1) x • ϕ -1 H σ ϕ x 1/2 L ∞ + ϕ x L ∞ 1/ϕ x (1+2σ)/2 L ∞ ϕ x H q-1
which ends the proof.

Corollary 4 . 4 .

 44 The corresponding Euler equation has for any initial data

S 1 S 1

 1 |w(x) -w(y)| p |x -y| 1+pσ dx dy 1/p .

  1 H σ , by virtue of lemma B.1. ButP m-1 = a 0 (ϕ) + a 1 (ϕ) ϕ (m-1) x • ϕ -1 ,where a j (ϕ) is a homogeneous polynomial of degree m-1-j in the variables ϕ x • ϕ -1 , . . . , ϕ (m-2) x

We will avoid to write T Rs, T (T Rs), . . . and simply keep the notation Rs.

If the subgroup K is connected, (22) and (23) are equivalent.

If E, F are Banach spaces and Λ : E → F is a continuous linear map, which is injective, then we say that Λ splits if Λ(E) is closed and complemented in F (i.e there exists a closed subspace G of F such that F = Λ(E) ⊕ G). Note that if F is a Hilbert space, then every injective, continuous linear map with closed range, splits.

D s (S 1 ), the space of homeomorphisms of class H s as well as their inverse is a topological group, if and only if, s > 3/2.

Acknowledgments

The authors wish to express their gratitude to the Erwin Schrödinger International Institute for Mathematical Physics for providing an excellent research environment.

Finally it is a pleasure to thank Anders Melin and Elmar Schrohe for helpful discussions.

Acknowledgments References 6.1. Euler equations on the coadjoint orbit Diff ∞ (S 1 )/Rot(S 1 ). Let Rot(S 1 ) denotes the subgroup of all rigid rotations of S 1 and Diff ∞ (S 1 )/Rot(S 1 ), be the corresponding homogeneous space of right cosets. Let Diff ∞ 1 (S 1 ) be the subgroup of Diff ∞ (S 1 ) consisting of all diffeomorphisms of S 1 which fix one arbitrarily point (say x 0 ). It is easy to check that the conditions of lemma 6.1 are satisfied and hence that the canonical projection

) is a Fréchet Lie group and we can use the Fréchet manifold structure of Diff ∞ 1 (S 1 ) to endow the quotient space Diff ∞ (S 1 )/Rot(S 1 ) with a Fréchet manifold structure, so that the canonical projection becomes a diffeomorphism. The Lie algebras of Diff ∞ 1 (S 1 ) and Rot(S 1 ) are given by C ∞ 1 (S 1 ) := {u ∈ C ∞ (S 1 ) ; u(x 0 ) = 0} and R • w 0 , respectively, where w 0 stands for the constant function with value 1.

Diff ∞ 1 (S 1 ) is an ILH space; a Hilbert approximation being given by the Hilbert manifolds

) is a closed submanifold of the Hilbert manifold D q (S 1 ) and a closed topological subgroup of D q (S 1 ), for q > 3/2.

Let A = op (p(k)) be a L 2 -symmetric, Fourier multiplier on C ∞ (S 1 ) and assume that its symbol satisfies

which is equivalent to ker A = R • w 0 . We have ad w 0 = -D, so that condition ( 23) is satisfied. Since Rot(S 1 ) is connected, hypotheses of proposition 6.3 are fulfilled. If A is of order r ≥ 1, then A extends to H q 1 (S 1 ), for all q > 3/2, and A ∈ Isom(H q 1 (S 1 ), Ĥq-r

where Ĥq-r 1 (S 1 ) := m ∈ H q-r (S 1 ) ; m(0) = 0 . Then, for each ϕ ∈ D q 1 (S 1 ), A induces a positive inner product on each tangent space, T ϕ D q 1 (S 1 ), with flat map Ãϕ = ϕ x A ϕ ∈ Isom(H q 1 (S 1 ), Ĥq-r 1 (S 1 )). Note that Ã(T D q 1 (S 1 )) = D q 1 (S 1 ) × Ĥq-r 1 (S 1 ) and the proof of theorem 6.5 is similar to that of theorem 3.10 and will be omitted. Theorem 6.5. Let A = op (p(k)) be a L 2 -symmetric, non negative, Fourier multiplier of order r ≥ 1, satisfying [START_REF] Kolev | Lie groups and mechanics: an introduction[END_REF] p(k) = 0 ⇐⇒ k = 0. modelled on the Hilbert spaces H q 3 (S 1 ) := {u ∈ H q (S 1 ) ; u(x 0 ) = 0, u(x 1 ) = 0, u(x 2 ) = 0}. Note that D q 3 (S 1 ) is a closed submanifold of the Hilbert manifold D q (S 1 ) and a topological subgroup of D q (S 1 ), for q > 3/2. Theorem 6.6. Let A = op (p(k)) be a L 2 -symmetric, non negative, Fourier multiplier of order r ≥ 1, satisfying

). Assume, in addition, that

is smooth for q > 3/2 and q -r ≥ 0. Then, the induced right-invariant metric on D q 3 (S 1 ) is smooth and has a smooth spray. Moreover, given any

), there exists a unique non-extendable solution

)) of the Cauchy problem for the associated geodesic spray on the maximal interval of existence J.

Proof. The proof is similar to that of theorem 3.10, except for point (c). Note that A extends to H q 3 (S 1 ), for all q > 3/2, and that A ∈ Isom(H q 1 (S 1 ), Ĥq-r

where Ĥq-r 3 (S 1 ) := m ∈ H q-r (S 1 ) ; m(0) = 0, m(1) = 0, m(-1) = 0 . Let à : D q 3 (S 1 ) × H q 3 (S 1 ) → D q 3 (S 1 ) × H q-r (S 1 ), (ϕ, v) → (ϕ, ϕ x A ϕ v). Given ϕ ∈ D q 3 (S 1 ), we have Ãϕ (H q 3 (S 1 )) = m ∈ H q-r (S 1 ); ϕ x (m • ϕ) ∈ Ĥq-r 3 (S 1 ) , so we cannot conclude immediately that Ã(T D q 3 (S 1 )) is a trivial bundle, as this was the case for Ã(T D q (S 1 )) and Ã(T D q 1 (S 1 )). To overcome this difficulty, we first remark that à is a vector bundle morphism. Moreover, the continuous linear map Ãϕ : H q 3 (S 1 ) → H q-r (S 1 ) is injective and splits 5 because Ãϕ (H q 3 (S 1 )) is a closed subspace of H q-r (S 1 ), for every ϕ ∈ D q 3 (S 1 ). Then according to [START_REF] Lang | Fundamentals of Differential Geometry[END_REF]Proposition 3.1], Ã(T D q 3 (S 1 )) is a subbundle of D q 3 (S 1 ) × H q-r (S 1 ) which is isomorphic to D q 3 (S 1 ) × H q-r 3 (S 1 ). Finally, an argument similar to the one given in point (c) of the proof of theorem 3.10 does apply and achieves the proof.