The geometry of a vorticity model equation

We provide rigorous evidence of the fact that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics describes the geodesic flow on the subgroup of orientation-preserving diffeomorphisms fixing one point, with respect to right-invariant metric induced by the homogeneous Sobolev norm $H^{1/2}$ and show the local existence of the geodesics in the extended group of diffeomorphisms of Sobolev class $H^{k}$ with $k\ge 2$.


Introduction
Despite their derivation by Leonhard Euler as early as in 1757 [14], the eponymous equations governing inviscid fluid flow still pose highly challenging mathematical problems. For example, our knowledge about the propagation of regularity of solutions to the three-dimensional Euler equations remains fragmentary. One way of attacking this and related question is to resort to simpler, lower-dimensional, differential equations whose solutions share features with those observed for (ideal) fluids. This approach was chosen by Constantin, Lax, and Majda [4] in their derivation of a model equation for the three-dimensional vorticity equation (the Euler equation for the vorticity ω = curl u of the velocity vector u): (1) ∂ t ω = Hω ω, t > 0, x ∈ R; the original equation being This reduction occurs if the convective derivative ∂ t +u·∇ is replaced by the temporal derivative, and it can be justified by the identical properties, in one or three space dimensions, respectively, of the singular integral operators D, given by the Biot-Savart law, and H, the Hilbert transform [23]. A tremendous drawback of the vorticity model (1), however -despite its having solutions exhibiting finite-time blow-up -is the paradox that its viscous extension, first explored by Schochet [26], has solutions which can break down earlier than in the inviscid regime. 1 In an attempt to circumvent this anomaly, De Gregorio [7] proposed another model for the vorticity equation: (2) ∂ t ω + uω x = Hω ω, u x = Hω, t > 0, x ∈ R.
In this convective perturbation of the CLM equation, he chose the "velocity" u to be the antiderivative of the Hilbert transform of the "vorticity" ω. Numerical studies [22] lead to the conjecture that De Gregorio's model equation has global solutions -a fact that has yet to be verified mathematically. Also in [22], a more general model equation of hydrodynamic type was presented: (3) ∂ t ω + α uω x = Hω ω, u x = Hω; α being an arbitrary real parameter. Three cases of which have been studied before: • α = −1 corresponds to the model for the quasi-geostrophic equations of [5,6]; • α = 0 reduces the model equation to the CLM equation; • α = 1 becomes the equation proposed by De Gregorio. Recently, it has been shown that for any α lying in the negative halfline, there are solutions which blow up in finite time [1]. However, the description of the asymptotic behavior in the case α > 0 remains open to further scrutiny.
Note that the time scaling t → t/α transforms (3) into the following equation (4) ∂ t ω + uω x + a u x ω = 0, ω = Hu x , with a := −1/α. This equation is known as the modified CLM equation, cf. [22] for a recent study of (4). It was observed by Wunsch [30] that in the case a = 2 equation (4) admits a geometric interpretation as the description of the geodesic flow of the homogeneousḢ 1/2 (S) right-invariant metric on the homogeneous space Diff ∞ (S)/Rot(S) of orientation-preserving diffeomorphisms of the circle modulo the subgroup of rotations Rot(S), putting it "midway" between the Burgers equation (giving the geodesic flow of the L 2 (S) metric) and the Hunter-Saxton equation (describing the geodesic flow of the homogeneousḢ 1 metric).
In this work, we will show that the periodic modified CLM can be realized as the geodesic flow of a symmetric linear connection on the subgroup Diff ∞ 1 (S) of orientation-preserving diffeomorphisms ϕ ∈ Diff ∞ (S) such that ϕ(1) = 1, which is canonically diffeomorphic to the coset manifold Diff ∞ (S)/Rot(S). For the value a = 2 this connection is compatible with a Riemannian metric. In fact in this case, the metric is induced by the inertia operator Λ := HD (with respect to the L 2 inner product on the tangent bundle).
Moreover, we present a thorough study of the regularity properties of the geodesic flow on a suitable Banach approximation of the Fréchet manifold Diff ∞ 1 (S). Introducing furthermore Lagrangian coordinates, these regularity results allow us to prove the well-posedness of the modified CLM equation on large phase spaces. An immediate application of our main results Theorem 4.2 and Corollary 4.4, in combination with Lemma A.2 gives the following conclusion: Theorem 1.1. Let a ∈ R and k ≥ 2 be given. Then there exist δ k > 0 and T k > 0 such for each ω 0 ∈ H k−1 (S) with spatial mean zero and ω 0 H k−1 < δ k there exists a unique solution to the modified CLM equation (4) with initial condition ω(0) = ω 0 .
Let us briefly outline the plan of our paper. In Section 2 we recall the construction of Euler-Poincaré equations on Lie groups and provide basic facts on Fourier multipliers on S, which is the class of inertia operators we are interested in. In Section 3 we realize the modified CLM equation as an Euler equation on the Fréchet Lie group Diff ∞ 1 (S). Introducing Lagrangian coordinates it is possible to re-formulate the geodesic flow on Diff ∞ 1 (S) as a system a first order ordinary differential equations on the tangent bundle of the Banach manifold D k 1 (S), consisting in all orientation-preserving diffeomorphisms ϕ of Sobolev class H k with k ≥ 2 such that ϕ(1) = 1. In Section 4 we study the regularity of the vector field induced by the above mentioned dynamical system. This section contains also our main results. The proof of several continuity properties of the composition mapping and some technical estimates for operators on Sobolev spaces are postponed to the Appendix.

Settings
A right-invariant Riemannian metric on a Lie group G is defined by its value at the unit element, that is by an inner product on the Lie algebra g of the group. If this inner product is represented by an invertible operator A : g → g * , for historical reasons, going back to the work of Euler on the motion of the rigid body, this inner product is called the inertia operator. The Levi-Civita connection of such a Riemannian metric is itself right-invariant and given by where ξ u is the right-invariant vector field on G generated by u ∈ g and B is the right-invariant tensor field on G, generated by the bilinear operator where u, v ∈ g and (ad u ) * is the adjoint (relatively to the inertia operator A) of the natural action of the Lie algebra on itself given by Given a smooth path g(t) in G, we define its Eulerian velocity, which lies in the Lie algebra g, by u(t) = R g −1 (t)ġ (t) where R g stands for the right translation in G. It can then be shown (see [11] for instance) that g(t) is a geodesic if and only if its Eulerian velocity u satisfies the first order equation (7) u t = −B(u, u).
known as the Euler equation induced by A.
It was noticed in [11] that the concept of Euler equation does not necessarily require the linear connection ∇ to be Riemannian -there may not exist a Riemannian metric which is preserved by this connection. We have therefore called such an equation a non-metric Euler equation. With this extended definition, every quadratic evolution equation on g corresponds to the reduced geodesic equation (Euler equation) of a rightinvariant symmetric linear connection on G.
The theory of Euler equations on a homogeneous space G/K has been developed in [18] in the metric case and more generally for Hamiltonian systems. It corresponds to a special case of the Hamiltonian reduction with respect to the subgroup K action. Let A : g → g * , be the inertia operator of a degenerate symmetric bilinear form < ·, · > on g, such that ker A = k, the Lie algebra of K. If moreover, the inner product < ·, · > is Ad K -invariant, that is < Ad k u, Ad k v >=< u, v >, for all k ∈ K and all u, v ∈ g, then A induces a right G-invariant pseudo-Riemannian metric on the space G/K of right cosets (Kg, g ∈ G). In that case, the Euler equation, which corresponds to the inertia operator A and describes the geodesic flow on the homogeneous space G/K, has the following Hamiltonian form (see [18]): it is the quotient with respect to the K-action of the restriction to L = Im A ⊂ g * of the following Hamiltonian equation on g * m t = −ad * A −1 m m for m ∈ L.
However, it appears difficult to work easily with a contravariant formulation of this equation similar to equation (7) in this more general situation. Indeed, in that case, the Eulerian velocity is only defined up to a path in K (see [28] for a recent survey on the subject). Moreover, it is not clear how this formalism can be generalized to non-metric Euler equations.
Fortunately, in the case we consider in this paper, these difficulties can be avoided because of a prolific structure. More precisely, in the situation we consider, there exists a closed subgroup H of G, such that the restriction to H of the canonical right action of G on G/K is transitive and without fixed points. In that case, g = k ⊕ h, where k is the Lie algebra of K and h is the Lie algebra of H and the study of a degenerate, Ad K -invariant inner product on g with kernel k can be reduced to an Euler equation on the Lie group H, where h * has be identified with Example. Let E(3) be the Lie group of direct euclidean motions in 3-space. The homogeneous space E(3)/SO(3) ≃ R 3 satisfies the hypothesis of our framework: the subgroup of translations T (3) ≃ R 3 acts transitively and without fixed points on the quotient space. Notice however, that in this particular example, the subgroup T (3) is a normal subgroup of E(3), something we do not assume explicitly in our more general framework.
Remark 2.1. We emphasize, that contrary to what one might expect at first glance, the system of free motions of a rod (degenerate rigid body) does not enter into this framework. Indeed, the configuration space of a rigid rod can be realized as the homogenous space SO(3)/SO(2) ≃ S 2 which is not diffeomorphic to any Lie group, otherwise, its tangent bundle would be trivial, which is not the case. Nevertheless, the geometric framework of Diff ∞ (S)/Rot(S) suits perfectly well for the study of the Hunter-Saxton equation (see [21] for instance) and other hydrodynamical models we shall consider in this article.
Let Diff ∞ (S) be the Fréchet Lie group of smooth and orientation preserving diffeomorphisms of the unit circle S ≃ R/Z and Diff ∞ (S)/Rot(S) be the homogeneous space of right cosets The restriction of this action to the subgroup Diff ∞ 1 (S) of diffeomorphisms ϕ ∈ Diff ∞ (S) such that ϕ(1) = 1 is transitive and simple (without fixed point). Therefore, the restriction of the projection map ϕ → [ϕ] to Diff ∞ 1 (S) defines a bijection between Diff ∞ 1 (S) and Diff ∞ (S)/Rot(S). The inverse map is given by [ϕ] → ϕ · ϕ(1) −1 . Notice however that the restriction to Diff ∞ 1 (S) of the projection map is not a group morphism. Otherwise Rot(S) would be a normal subgroup of Diff ∞ (S), which is not the case: Diff ∞ (S) is simple, it has no (non trivial) normal subgroup [16].
The Fréchet manifold structure on Diff ∞ 1 (S) is obtained by the existence of the global chart (8) U which is an open set in the closed hyperplane id Remark 2.2. To summarize, there is a prolific structure in the special framework we consider in this paper and which simplifies our work. This structure is essentially due to the fact that there exists a smooth section (not a group morphism however) In particular the Fréchet Lie group Diff ∞ (S) is diffeomorphic to the product manifold Rot(S) × Diff ∞ 1 (S). Notice that the particular choice of the fixed point in the definition of Diff ∞ 1 (S) does not affect the general structure. Consider now a non-negative bilinear form on C ∞ (S) which can be written as Then, A induces a weak Riemannian metric on the homogeneous space is well-defined on C ∞ 0 (S) and the associated symmetric, right-invariant, linear connection on Diff ∞ 1 (S) is compatible with the metric. The corresponding Euler equation on C ∞ 0 (S) is given by Remark 2.4. Notice that condition (3) is equivalent to the property for the degenerate inner product on C ∞ (S) defined by A to be Ad Rot(S) -invariant.
Remark 2.5. Observe that the topology induced by the pre-Hilbertian structure on each tangent space of the Fréchet manifold Diff ∞ (S)/Rot(S) is weaker than the usual Fréchet topology. For this reason such a structure is called a weak Riemannian metric. On a Fréchet manifold, only covariant derivatives along curves are meaningful. As expounded in [11], the general expression of a right-invariant, covariant derivative of a vector field where u = ϕ t • ϕ −1 and B is a symmetric bilinear operator on C ∞ 0 (S). However, and contrary to the finite dimensional case, the existence of a symmetric, linear connection on a Fréchet manifold, compatible with a weak Riemannian metric, that is is far from being granted.
Proof of proposition 2.3. If conditions (1) and (3) of proposition 2.3 are fulfilled, A induces a pre-Hilbertian structure on each tangent space of the homogeneous space Diff ∞ (S)/Rot(S), identified with Diff ∞ 1 (S). This inner product is given by This family of pre-Hilbertian structures, indexed by ϕ ∈ Diff ∞ 1 (S), is smooth because composition and inversion are smooth on the Fréchet Lie group Diff ∞ 1 (S). This way we obtain a right-invariant, weak Riemannian metric on Diff ∞ 1 (S). Formula (6) cannot be used directly to define a connection compatible with the metric because the adjoint operators ad t u (relatively to the pre-Hilbertian structure) are not well-defined. Indeed, given u, v, w ∈ C ∞ 0 (S), we have x u belongs to Im A, the space of smooth functions of mean value zero. One can check that this not the case in general. However, the expression has mean value zero, and belongs to Im A (condition (2)), provided A commutes with D. This is true by virtue of lemma 2.6, if A commutes with all rotations (condition (3)). Therefore, one can define and check that the associated right-invariant, symmetric linear connection on Diff ∞ 1 (S) is compatible with the metric. More generally, for each a ∈ R, the equation is the (non-metric) Euler equation of a well-defined symmetric, right-invariant, linear connection on Diff ∞ 1 (S). The special case where A is a differential operator with constant coefficients has been extensively studied (see, e.g., [2,3,11]). In this paper, we will need to extend the theory when A is a Fourier multiplier. For later reference, let us first give a useful characterization of Fourier multipliers. Here and in the following we use the notation e n (x) = exp(2πinx), n ∈ Z, x ∈ S. Lemma 2.6. Let P a continuous linear operator on the Fréchet space C ∞ (S). Then the following three conditions are equivalent: (1) P commutes with all rotations R s .
(3) For each n ∈ N, there is a p(n) ∈ C such that P e n = p(n)e n . In that case, we say that P is a Fourier multiplier.
Since every smooth function on the unit circle S can be represented by its Fourier series, we get that for every Fourier multiplier P and every u ∈ C ∞ (S), wherê stands for the k-th Fourier coefficients of u. The sequence p : Z → C is called the symbol of P .
Proof. Given s ∈ R and u ∈ C ∞ (S), let u s (x) := u(x + s). If P commutes with translations we have Taking the derivative of both sides of this equation with respect to s at 0 and using the continuity of P , we get DP u = P Du which proves the implication (1) ⇒ (2). If [P, D] = 0, then both P e n and e n are solutions of the linear differential equation u ′ = (−2πin)u and are therefore equal up to a multiplicative constant p(n). This proves that (2) ⇒ (3).
If P e n = p(n)e n , for each n ∈ N and P is continuous, then we have representation (12). Therefore Remark 2.7. Notice that the space of Fourier multipliers is a commutative subalgebra of the algebra of linear operators on C ∞ (S) which contains all linear differential operators with constant coefficients.
A Fourier multiplier P with symbol p is said to be of order s ∈ N if there exists a constant C > 0 such that for every m = 0. In that case, for each k ≥ s, the operator P extends to a bounded linear operator from H k (S) into H k−s (S). We express this fact by the notation P ∈ L(H k (S), H k−s (S)).

The modified CLM equation as an Euler equation
The homogeneousḢ 1/2 norm defined on C ∞ 0 (S) is introduced by means of Fourier series. We let for u ∈ C ∞ (S). The corresponding inner product on C ∞ 0 (S) can be written as In this formula, D = d/dx and H is the Hilbert transform, defined either as a Cauchy principal value, cf. [23] (Hu or, equivalently, as the Fourier multiplier with symbol h(k) = −i sgn(k), The convention sgn(0) = 0 permits to extend H on C ∞ (S). Notice that and that H defines a complex structure on this space. Moreover, H is an isometry for the L 2 inner product of function equivalence classes having zero mean value.
Since [Λ, D] = 0 and the inertia operator Λ is an isomorphism from C ∞ 0 (S) onto C ∞ 0 (S) * := {u ∈ C ∞ (S);û(0) = 0} , the space of smooth functions of mean value zero, the existence of a linear connection compatible with the metric is granted and the Euler equation is defined.
Proof. It suffices to replace A by the expression Λ = H • D in formula (11) and to use the definition ω = Λu, to get the general assertion.  (14) is equivalent to the Euler equation To the best of our knowledge, theorem 3.1 is the first time the model equation for the 2D quasi-geostrophic and the Birkhoff-Rott equations studied in [5,6] has been identified as a non-metric Euler equation on Diff ∞ 1 (S). (b) We recall that the connection is Riemannian if a = 2. Moreover, it follows from [13] that for there is no inertia operator of Fourier multiplier type such that (14) can be realized as the geodesic flow with respect to the corresponding metric. These results extend similar statements for the b-equation [19,12].
(c) It is also possible to consider evolution equations on Diff ∞ 1 (S), related to the inertia operator Λ 2 = −D 2 . Given a ∈ R, one may study the family The most prominent equations in this family are the Hunter-Saxton equation (16) m t + um + 2u x m = 0, and the Proudman-Johnson equation respectively. The Hunter-Saxton equation is closely related to the Camassa-Holm Equation and the geometric picture of (16) is in fact fairly good understood, cf. [20]. In particular, (16) is the geodesic flow on Diff ∞ 1 (S) with respect to the inertia operator −D 2 , i.e. with respect to the homogeneouṡ H 1 -metric on C ∞ 0 (S). In contrast, (17) is a non-metric Euler equation, cf. [13]. 4. The geodesic flow on D k 1 (S) In this section, we will study the regularity of the geodesic flow on suitable Banach approximations of the Fréchet manifold Diff ∞ 1 (S). More precisely, let D k (S) be the Banach manifold of orientation-preserving diffeomorphisms ϕ of Sobolev class H k (defined for some integer k ≥ 2). This Banach manifold is a topological group with respect to composition of diffeomorphisms but it is not a Lie group. Indeed, on D k (S), right translation R ϕ : ψ → ψ • ϕ is linear, hence smooth; whereas left translation L ϕ : ψ → ϕ • ψ is only continuous but not differentiable (see [10,17]).
Remark 4.1. The space of homeomorphisms of the circle of Sobolev class H 1 (as well as their inverse) is not a group. Indeed, let 1/2 < α < 1/ √ 2. Then F : x → x α is an increasing homeomorphism of [0, 1] which induces an homeomorphism of the circle. One can check that F as well as F −1 are of class Analogous to Diff ∞ 1 (S), the codimension one Banach submanifold D k 1 (S) of diffeomorphisms ϕ ∈ D k (S) such that ϕ(1) = 1 is covered by the global chart which is an open set in the closed hyperplane id + H k 0 (S), where H k 0 (S) is the closed linear subspace Notice that the restriction of D to H k 0 (S) is a bounded isomorphism ontô Its inverse is given by Since the Hilbert transform H, which is an isometry for the L 2 product for zero-mean periodic functions, commutes with D, its restrictions toĤ k 0 (S) is an isometry ofĤ k 0 (S) (for the H k inner product). The Euler equation (15) is not an ODE on H k 0 (S) because the secondorder term Λu x = Hu xx is not regularized by the inverse of the first order Fourier multiplier Λ. By introducing Lagrangian coordinates, however, one can get around this impediment and it is possible to re-formulate (15) as a well defined vector field on the Banach manifold D k 1 (S) × H k 0 (S). Theorem 4.2. Let a ∈ R and k ∈ N with k ≥ 2 be given. The timedependent vector field u ∈ H k 0 (S) is a solution to the modified CLM equation if and only if (ϕ, v) is a solution to Moreover, the second order vector field The proof of the second part of the theorem, i.e. the smoothness of the spray Φ consists of several reductions, some of them being true for general Fourier multipliers. We outline these reductions in the remainder of this section. Some technicalities will be postponed to Appendix B.
Before entering into the details of the proof, let us state that the above result allows us to apply the Picard-Lindelöf theorem, which immediately yields: Corollary 4.4. Let a ∈ R and k ≥ 2 be given. Then there exist δ k > 0 and (19) such that ϕ(0) = id and v(0) = u 0 .
Let us start with the first reduction. If we assume that the conjugation A ϕ of the inertia operator A is of class C m then the spray Φ is of class C m−1 .
Proposition 4.5. Let m ≥ 1, a ∈ R, s ≥ 1 and k ≥ s + 1. Let A be a Fourier multiplier of order s. Suppose that where the subscript ϕ indicates the conjugacy by the right translation R ϕ in D k 1 (S). Although P and Q are smooth operators, these results do not carry over when conjugated with translation in D k 1 (S) since for k ≥ 2 these sets only form topological groups: neither composition nor inversion are differentiable.
Given an operator K, we introduce the following notatioñ Also H k−s (S) is a Banach algebra because k − s ≥ 1. Hence the fact that P ϕ (v) ∈ H k−s (S) and our assumption ensure that is of class C m .
2) Since (S), we conclude, using the inverse mapping theorem on Banach spaces, thatÃ 3) Taking P = A and δϕ 1 = v = u • ϕ in Proposition 4.6 when ϕ, v are smooth, we get Now since smooth maps are dense in Sobolev spaces, this relation is still valid for ϕ ∈ D k 1 (S) and v ∈ H k 0 (S) and thereforẽ is of class C m−1 . The assertion now follows from the chain rule.
Next we show that the conjugation of an inertia operator of Fourier multiplier type is in fact smooth. In order to do so we first consider operators in the smooth category and extend them in s second step to Sobolev spaces.
Let (ϕ, v) → P ϕ (v) be a smooth map on the Fréchet manifold Diff ∞ (S) × C ∞ (S), where P is linear in v. The partial Gâteaux derivative of P in the first variable ϕ and in the direction δϕ 1 ∈ C ∞ (S) is a smooth map which is linear both in v and δϕ 1 and that we will denote by (20) ∂ ϕ P ϕ (v, δϕ 1 ).
Therefore, the partial Gâteaux derivative of P in the variable ϕ is a map of three independent variables : ϕ, v, δϕ 1 . The second partial derivative of P is directions δϕ 1 , δϕ 2 ∈ C ∞ (S) is the partial Gâteaux derivative of (20) in the variable ϕ and in the direction δϕ 2 . We will denoted it by It can be checked that this expression is symmetric in δϕ 1 , δϕ 2 (see [17]). Inductively, we define this way the n-th partial derivative of P in directions δϕ 1 , . . . , δϕ n and we write it as ∂ n ϕ P ϕ (v, δϕ 1 , . . . , δϕ n ). The space of linear operators on a Fréchet space is a locally convex topological vector space, but in general is not a Fréchet space (see [17]). For this reason, we will avoid taking limits and derivatives of linear operators. In the sequel, if such equalities appear for notational simplicity, it just means equality of operators. Proposition 4.6. Let P be a continuous, linear operator on C ∞ (S) and let where ϕ ∈ Diff ∞ (S). Then, given n ∈ N, we have where u i = δϕ i • ϕ −1 and P n is the multilinear operator defined inductively by P 0 = P and Proof. Formula (21) is trivially true for n = 0. Now suppose it is true for some n ∈ N, that is is a family of linear operator on C ∞ (S) indexed by ϕ and which depend on ϕ only through the u i . Let ϕ(s) be a smooth path in Diff ∞ (S) such that and let u n+1 = δϕ n+1 • ϕ −1 . We compute firsṫ Finally, we have (simplifying the notation P n for P n (u 1 , . . . , u n )) which gives the recurrence relation (22), since ∂ n+1 ϕ P ϕ (v, δϕ 1 , . . . , δϕ n+1 ) = ∂ s R ϕ P n (u 1 , . . . , u n )R −1 ϕ (v) s=0 , the proof, the proof is complete. Proposition 4.6 is the core of the following result, which ensures smoothness of the inertia operator Λ ϕ (v) in both variables with respect to suitable Sobolev norms. To avoid too much technicalities here, we postpone its proof to Appendix B.
In contrast to finite dimensional Riemannian geometry the topology of the fibre of the tangent bundle is fundamental importance in the infinite dimensional case. It is clear that in the smooth category the pre-Hilbertian structure defined by (10) will not induce the Fréchet topology of the tangent space C ∞ 0 (S). The very same is true if we complete the tangent space with respect to a general Banach norm. Therefore we call the metric induced by (10) a weak Riemannian metric. defined on the diffeomorphism group of the circle which fixes the three points −1, 0, 1. This metric has been related with the Weil-Petersson metric on the universal Teichmüller space T (1) in [27]. The corresponding geodesic flow has been extensively studied in [15]. Recall first that D s (S), the space of homeomorphisms of class H s as well as their inverse is a topological group only for s > 3/2 and that 3/2 is therefore a critical exponent. One of the main results in [15] is that, the inertia operator A defines on a suitable replacement for the "H 3/2 diffeomorphism group", a right-invariant strong Riemannian structure which is moreover complete (geodesics are defined globally). Our point of view in this paper is completely different in the sense that we work on a well defined topological group D s (S) for s > 3/2 equipped with a Banach manifold structure 2 . The price to pay for this nice structure is the fact that the metric only defines a weak Riemannian structure. Nevertheless, we have been able to show local existence of the geodesics, also in this context.

Appendix A. Continuity lemmas
In this section we provide some continuity properties of the composition mapping in Sobolev spaces. Given Fréchet spaces X and Y , let L(X, Y ) denote the space of all continuous linear operators from X into Y .
Lemma A.1. Let X, Y be Fréchet spaces and let G be a metric space. Given F : G × X → Y , assume that Then F ∈ C(G × X, Y ).
Proof. Fix (g 0 , x 0 ) ∈ G × X and pick a sequence (g n , x n ) in G × X such that lim n (g n , x n ) = (g 0 , x 0 ). Let further V denote a neighbourhood of F (g 0 , x 0 ) in Y . We set B n := F (g n , ·) ∈ L(X, Y ), n ∈ N. Then, given x ∈ X, we have Hence {B n (x) ; n ∈ N} is bounded in Y . Invoking the uniform boundedness principle in Fréchet spaces (see [9,Theorem II.11]), we deduce that the family {B n ; n ∈ N} is equicontinuous. In particular there is a neighbourhood U of x 0 in X such that B n (U ) ⊂ V for all n ∈ N. But lim n x n = x 0 . Hence there is a n 0 ∈ N such that x n ∈ U for all n ≥ n 0 . This implies that B n (x) = F (g n , x n ) ∈ V for all n ≥ n 0 .
Thus F is continuous in (g 0 , x 0 ).
(b) Let now v ∈ H 1 (S) be fixed. We are going to show that For this pick ϕ 0 ∈ D 2 (S) and ε > 0. By Sobolev's embedding theorem, the function v is uniformly continuous. Thus there is a δ > 0 such that |v(x) − v(y)| < ε for all |x − y| < δ.
Next let j denote the embedding constant of H l (S) ֒→ C(S) for l = 1, 2 and choose ϕ ∈ D 2 (S) such that ϕ 0 − ϕ H 2 < δ/j. Then Thus we get To estimate D(v •ϕ 0 − v •ϕ) in L 2 , we first remark that it is no restriction to assume that δ ∈ (0, 1]. Writing now K := j ϕ 0 H 2 + 1 and we have that Note also that by shrinking δ > 0, we may assume that We now proceed as follows. First we have For the first term of the right-hand side of (28), we find To estimate the second term in (28), choose w ∈ C 2 (S) such that where we also employed the mean value theorem and (27) to derive the last estimate. Invoking (26) and (30), we get for all ϕ ∈ B 2 (δ). Combining (25), (28), (29), and (31), we arrive at the following estimate (c) Let k ≥ 2 be given. Then it follows from the considerations from [10, page 108] that F (ϕ, ·) ∈ L(H k (S), H k (S)), for all ϕ ∈ D k (S) and that for all v ∈ H k (S). Hence, again by lemma A.1, we conclude that The last assertion is now obvious.
Remark A.3. (a) For simplicity we treated here the case s ∈ N. Using an intrinsic representation of the Sobolev norm for s ∈ R with s ≥ 1, it is possible to extend the results of Lemma A.2 to non-integer values of s ≥ 1.
(b) A similar result to (24) has recently been established in [8]. However, on the one hand, Corollary 3 in [8] fits not precisely into our setting, and on the other hand our scale of Sobolev spaces is simpler than the one in [8]. Therefore we decided to present a self-contained proof of (24).
(c) The higher the spatial regularity in the group D k (S) and the Lie algebra H k (S), the better the regularity of the mapping F in lemma A.2, cf. [10]. However, we are not aware of better regularity of F than (24). Finally, we remark that the continuity of F is sufficient for our purposes.
Appendix B. Proof of Proposition 4.8 In this section we provide the completion of the proof of the smoothness of the inertia operator Λ ϕ (v) with respect to suitable Sobolev norms.
Lemma B.1. Let P be a Fourier multiplier on C ∞ (S), and let P n be the multilinear operator defined in Proposition (4.6) for some n ∈ N. Then we have and achieves the proof.
Lemma B.3. Let P be a Fourier multiplier of order s ∈ N and k ≥ s+1. Let P n be the (n + 1)-multilinear operator defined by the recurrence relation (22) with P 0 = P . Suppose that there exists a constant C n > 0, such that (38) |p n (m 0 , . . . , m n )| ≤ C n |m 0 | s · · · |m n | s for all m j ∈ Z \ {0}. Then P n extends to a bounded multilinear operator Proof. By virtue of Proposition B.1, we have for any smooth functions u 0 , u 1 , . . . , u n , since (e l ) l∈Z is an orthogonal system for the H k−s inner product. Therefore, if relation (38) is satisfied, we get Observe now that, given smooth functions v 0 , v 1 , . . . , v n , we have v 0 · · · v n (l) = m 0 +···+mn=lv 0 (m 0 ) · · ·v n (m n ).
In addition H k−s (S) is a Banach algebra, since k − s ≥ 1. Consequently there exists a constant C ′ n,k,s such that H k · · · u n 2 H k , which achieves the proof.
Corollary B.4. Let P be a Fourier multiplier of order s. Let r ∈ N and k ≥ s + 1. Suppose that the operators P n , defined in Proposition (4.6), extend to bounded multilinear operators P n : n+1 H k (S) × · · · × H k (S) → H k−s (S).
for 0 ≤ n ≤ r. Then is of class C r from D k (S) × H k (S) to H k−s (S).
Proof. Notice first that if P 0 = P is bounded, then (ϕ, v) → P ϕ (v) is continuous from D k (S) × H k (S) into H k−s (S), by virtue of lemma A.2. Suppose now that the bilinear operator P 1 is bounded and let for smooth maps ϕ, u, v. But, since both sides of (39) are continuous in all the variables, we deduce, using a density argument, that this relation is still true for ϕ ∈ D k (S), u, v ∈ H k (S). We conclude therefore that (ϕ, v) → P ϕ (v) is a C 1 map from D k (S) × H k (S) into H k−s (S). An inductive argument using the same reasoning for P n shows that (ϕ, v) → P ϕ (v) is a C n map from D k (S) × H k (S) into H k−s (S), for each n ≤ r.
Finally, we need the following elementary lemma. for all t ∈ R and all m j ∈ R.
Proof. Let g k be the sequence of functions defined inductively by Then, we have On the other hand, the Lipschitz condition on the (n − 1) derivative of f leads to g (n−1) 1 (t) ≤ K |m 1 | , ∀t ∈ R. Now, using inductively the mean value theorem, we get g (n−k) k (t) ≤ K |m 1 | · · · |m k | , ∀t ∈ R.
In particular, for k = n, we have |g n (t)| ≤ K n j=1 |m j | , ∀t ∈ R, which achieves the proof.
Proof of Proposition 4.8. For each n ≥ 1, let f n (t) = t n−1 |t|. Then f n is of class C n−1 on R and f