Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle

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


Introduction
It is a fundamental observation due to Arnold [2] that the hydrodynamics of an ideal fluid on a compact Riemannian manifold M can be recast as the geodesic flow on the Lie group of volume preserving smooth diffeomorphisms of M . In this general picture there is a great latitude in the choice of an inertia operator which generates a weak right-invariant Riemannian metric on the Lie diffeomorphism group of M .
In the classical papers [2,11] L 2 -metrics have been used. This determination particularly allows to treat Euler's equation of hydrodynamics as geodesic flows. In the sequel several other equations of physical relevance have been found to arise in this manner [7,17,19,21,26,27,13]. Among these studies Sobolev metrics on the tangent bundle of type H k with k ∈ N and k ≥ 1 have been investigated in [7,19,21,27]. In [13] an inertia operator on the diffeomorphism group of the circle of the form HD, where H denotes the Hilbert transform and D the spatial derivative, has been considered.
It is the aim of this paper both to unify and to sharpen earlier results for differential operators as inertia operators on the diffeomorphism group of the circle. The essence of the method is to use a Lagrangian formalism for the geodesic flow that leads to evolution equations on the tangent bundle with a smooth propagator which is in addition bounded in the Sobolev norm H q on bounded subsets of H q for large q.
Our main result concerns the smoothness of the conjugation of Fourier multiplication operators with right translations in suitable Banach manifold approximation of Diff ∞ (S 1 ), the group of all smooth and orientation preserving diffeomorphisms on the circle. More precisely, let q ∈ R with q > 3/2 be given and let D q (S 1 ) denote the set of all orientation preserving homeomorphisms ϕ of the circle S 1 such that both ϕ and ϕ −1 belong to H q (S 1 ). Then D q (S 1 ) is a Banach manifold and a topological group but it is not a Lie group. Indeed, on D q (S 1 ), right translation R ϕ : ψ → ψ • ϕ is linear, continuous and hence smooth (for fixed ϕ); whereas left translation L ϕ : ψ → ϕ • ψ is only continuous but not in general not differentiable (see [11,16]). From an analytic point of view, Diff ∞ (S 1 ) may be viewed as an inverse limit of Banach manifolds The scales of space D q (S 1 ) q>3/2 is called a Banach manifold approximation of Diff ∞ (S 1 ).
Let P denotes a general Fourier multiplier of order r ≥ 1. It extends to a bounded operator from H q (S 1 ) to H q−r (S 1 ) for q ≥ 1. Given ϕ ∈ D q (S 1 ), we consider conjugation of P with right translations R ϕ (v) := v • ϕ for v ∈ H q (S 1 ). Notice that the map is smooth. However, it is not at all obvious that the extension to Sobolev spaces remains smooth.
Problem. Under which conditions on P is the mapping smooth?
It is a well-known fact that when P is a differential operator, P ϕ (v) is a rational expression of v, ϕ x and their derivatives (see [11,12] for instance). In that case, (ϕ, v) → P ϕ (v) is smooth for q large enough. However for a general Fourier multiplier we are not aware of any results in this direction. In our main theorem below, we give a condition on the symbol of P which ensures that this map is smooth. This answers a question raised in [11,Appendix A], at least in the case of the diffeomorphisms group of the circle. Up to the authors knowledge, these results are new. Theorem 1. Let P = op (p(k)) be a Fourier multiplier of order r ≥ 1. Suppose that its symbol p extends to R and that for each n ≥ 1, the function f n (ξ) := ξ n−1 p(ξ) is of class C n−1 , that f (n−1) is absolutely continuous 1 and that there exists C n > 0 such that (1) f (n) (ξ) ≤ C n (1 + ξ 2 ) (r−1)/2 , almost everywhere. Then the map is smooth for each q ∈ ( 3 2 + r, ∞) ∪ {1 + r}. This condition is satisfied by the inertia operator Λ 2s of the Sobolev metric H s on Diff ∞ (S 1 ) for s ∈ R and s ≥ 1/2. Corollary 2. Let s ∈ R and Λ 2s := op 1 + n 2 s . If s ≥ 1/2 then the mapping is smooth for any q ∈ ( 3 2 + 2s, ∞) ∪ {1 + 2s}. This allows us to prove that the corresponding weak Riemannian metric and its geodesic spray are smooth on each Banach manifold approximation D q (S 1 ) for sufficiently large q ∈ R. As a corollary and using an argument introduced by Ebin-Marsden in [11], we are able to prove local existence of geodesics on Diff ∞ (S 1 ). We shall prove the following well-posedness result: 1 A real function f is said to be absolutely continuous on R if f has a derivative almost everywhere, the derivative is locally Lebesgue integrable and for all a, b ∈ R. Theorem 3. Let s ≥ 1/2 be given and consider the right-invariant Sobolev H s -metric on Diff ∞ (S 1 ). Then the corresponding geodesic equation has for any initial data in the tangent bundle T Diff ∞ (S 1 ) a unique non-extendable smooth solution (ϕ, v) ∈ C ∞ (J, T Diff ∞ (S 1 )). The maximal interval of existence J is open and contains 0. Moreover, if J = (t − , t + ) and either It is known that for weak Riemannian metrics the exponential mapping fails in general to be a local diffeomorphism, cf. [7]. We clarify the picture to some extend by proving the following result.
Theorem 4. The exponential mapping exp at the unit element id for the H smetric on Diff ∞ (S 1 ) is a smooth local diffeomorphism from a neighbourhood of zero in Vect(S 1 ) to a neighbourhood of id on Diff ∞ (S 1 ) for each s ≥ 1/2.
Finally in applications, it is of some interest to study geodesic flows not only on the full group of diffeomorphism but to allow a normalization of solutions by fixing their value at one point. In our setting this means to consider the operatorΛ 2s of the homogeneous Sobolev metricḢ s on Diff ∞ (S 1 )/Rot(S 1 ).
Corollary 5. Let s ∈ R andΛ 2s := op |n| 2s . If s ≥ 1/2 then the mapping is smooth for any q ∈ ( 3 2 + 2s, ∞) ∪ {1 + 2s}. The plan of the paper is as follows. In Section 2 we recall some well-known facts on the geometry of the Euler equations. Some basic material related to weak Riemannian metrics on the diffeomorphisms group of the circle is collected in Section 3. In Section 4 we provide the proofs of our main results: Theorem 1, Corollary 2 and Corollary 5. Section 5 is devoted to the study of the smoothness of the metric and the geodesic spray on the extended Banach manifolds D q (S 1 ). In Section 6 we prove local existence and uniqueness of the initial value problem for the geodesics of the right-invariant H s metric on Diff ∞ (S 1 ) and discuss the blow-up problem, while in Section 7 we deal with the Riemannian exponential mapping and discuss the problem of geodesic distance. In Section 8 we extend our study to the homogeneous space Diff ∞ (S 1 )/Rot(S 1 ) and prove local existence result for the homogeneous metricḢ s . Technical lemmas on Fourier multiplier and continuity lemmas are collected in Appendix A and Appendix B, respectively.

Geometric context
2.1. Euler equation on a Lie group. 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 v is the right-invariant vector field on G generated by v ∈ g and B is the right-invariant tensor field on G, generated by the bilinear operator where v, w ∈ g and ad ⊤ v 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 where R g stands for the right translation 2 in G. It can then be shown, see e.g. [12] that g(t) is a geodesic if and only if its Eulerian velocity u satisfies the first order equation This equation for the velocities is known as the Euler equation.

2.2.
Euler equation on a homogeneous space. The theory of Euler equations on a homogeneous space G/K has been developed in [20]. Consider a non-negative degenerate inner product ·, · on g and let be the corresponding inertia operator. Suppose moreover that ker A = k, where k is the Lie algebra of K, and that the duality pairing is Ad K -invariant, that is for all k ∈ K and u, v ∈ g. Then A induces a right G-invariant Riemannian metric on the space G/K of right cosets (Kg, g ∈ G).
Remark 7. If the subgroup K is connected, which we suppose below, the condition (5) is equivalent to the following for all w ∈ k and all u, v ∈ g.
In that case, an "Euler equation" describing the geodesic flow on the homogeneous space G/K, can be defined as a special case of the Hamiltonian reduction with respect to the subgroup K action (see [20]). But, unfortunately, there is no useful contravariant formulation of this equation similar to the "genuine Euler equation on a Lie group" (4). Indeed, in this case, the Eulerian velocity is only defined up to a path in K (see [29] for a recent survey on the subject).
These difficulties clear away if the coset manifold G/K possesses an additional structure in the following sense. Assume that there is a closed subgroup H of G such that the restriction to H of the canonical projection G → G/K is a diffeomorphism. Then we may identify the Lie group H, endowed with the right-invariant metric with the Riemannian coset manifold G/K. If such a subgroup H exists, the study of a degenerate, Ad K -invariant inner product on g with kernel k reduces to the study of an Euler equation on the Lie algebra h of H.
Remark 8. Notice that the restriction of the projection H → G/K is a group morphism if and only if K is a normal subgroup of G.
Proposition 9. Suppose that H and K are closed subgroups of G such that the canonical projection H → G/K is a diffeomorphism. Let g, h, and k denote the Lie algebras of G, H, and K, respectively. Assume further that A : g → gg * satisfies the following hypotheses: Then, A induces a right-invariant, Riemannian metric on the homogeneous space G/K, identified with H. Moreover, the bilinear operator Its restriction to h × h induces the Riemannian connection of the metric on H. The corresponding Euler equation on h is given by . Proof of Proposition 9. In the situation described, we have g = k ⊕ h and h * can be identified with

The inertia operator induces an isomorphism
so that the restriction of the associated inner product on h induces a Riemannian structure on H. Now condition (ii) (and the symmetry of A) leads to , w for all u, v ∈ g and all w ∈ k. Hence, and the bilinear operator B is well defined. It can be checked directly that the corresponding right-invariant symmetric linear connection defined by B is compatible with the metric on H. This achieves the proof.
3. Right-invariant metrics on Diff ∞ (S 1 ) Let Diff ∞ (S 1 ) denote the group of all smooth and orientation preserving diffeomorphisms on the circle. This group is naturally equipped with a Fréchet manifold structure. More precisely, we can cover Diff ∞ (S 1 ) with charts taking values in the Fréchet vector space C ∞ (S 1 ) and in such a way that the change of charts are smooth maps (see [7] or [11] for more details). Since the composition and the inverse are smooth maps for this structure we say that Diff ∞ (S 1 ) is a Fréchet-Lie group, cf. [16]. Its Lie algebra Notice that Diff ∞ (S 1 ) (like any Lie group) is parallelizable 3.1. Weak Riemannian metrics on Diff ∞ (S 1 ). To define a right-invariant metric on Diff ∞ (S 1 ), we introduce an inner product on the Lie algebra Vect(S 1 ) = C ∞ (S 1 ). In the present paper, we assume that this inner product is given by where A : C ∞ (S 1 ) → C ∞ (S 1 ) is an invertible Fourier multiplier (see Appendix A for precise definitions). By translating the above inner product, we obtain an inner product on each tangent space T ϕ Diff ∞ (S 1 ) where η, ξ ∈ T ϕ Diff ∞ (S 1 ) and A ϕ = R ϕ • A • R ϕ −1 . This family of pre-Hilbertian structures, indexed by ϕ ∈ Diff ∞ (S 1 ), is smooth because composition and inversion are smooth on the Fréchet Lie group Diff ∞ (S 1 ). This way we obtain a right-invariant, weak Riemannian metric on Diff ∞ (S 1 ). In contrast to finite dimensional Riemannian geometry the topology of the fibre of the tangent bundle is of fundamental importance in the case of infinite dimensional manifolds. It is clear that in the smooth category the pre-Hilbertian structure defined by (8) will not induce the Fréchet topology of the tangent space C ∞ (S 1 ). This is why it is called weak, because the corresponding topology induced on each tangent space of the Fréchet manifold Diff ∞ (S 1 ) is weaker than the usual Fréchet topology.
The very same is still true if we complete the tangent space with respect to a Sobolev norm. Worse, the weak Riemannian metric on Diff ∞ (S 1 ) defined by (8), extends (a priori ) only to a continuous family of pre-Hilbertian structures on the bundle T D q (S 1 ), provided A is a Fourier multiplier of order r ≥ 1 and q ≥ r.
On a Fréchet manifold, only covariant derivatives along curves are meaningful and in general, the existence of a symmetric, linear connection, compatible with a weak Riemannian metric, that is is far from being granted. Nevertheless, in the situation we consider, the map ad ⊤ u is well defined and given by and check that the expression is a vector field defined along the curve ϕ(t) ∈ Diff ∞ (S 1 ) and u = ϕ t • ϕ −1 , defines a right-invariant, symmetric linear connection on Diff ∞ (S 1 ) which is compatible with the metric induced by A.
The corresponding Euler equation on Diff ∞ (S 1 ) is given by 3.2. The geodesic spray. Note that the right hand side of (10) is bilinear in u and of order 1 in the sense that if u ∈ H q (S 1 ) ∪ C k (S 1 ) then Hence the Euler equation (10) cannot be realized as a dynamical system on any of the Banach spaces H q (S 1 ). Of course it can be realized as an ordinary differential equation on Diff ∞ (S 1 ), but it is not for free to change from Banach spaces to Fréchet spaces, bearing in mind that there is no good practical direct theory for ordinary differential equation on Fréchet spaces. This is why we would wish to work on the Banach approximations spaces D q (S 1 ), rather than on Diff ∞ (S 1 ). It is however quite surprising that in Lagrangian coordinates the propagator of evolution equation of the geodesic flow possesses better mapping properties, provided that (1) A and D commute, i.e. A is a Fourier multiplier , cf. Lemma 23 in the Appendix A, (2) the order of A is not less than 1. In fact, assume that u(t) is a solution to (10) on some time interval J and let ϕ(t) the flow associated with u(t), i.e. ϕ solves Letting now v := u • ϕ, we clearly have But by assumption u solves the Euler equation (10). Therefore we find where we used that AD = DA to obtain the second identity. Introducing now the operator The second order vector field, defined in a local chart by is called the geodesic spray, cf. [22]. Conversely, assume that (ϕ, v) solves is a solution on J to the Euler equation (10).
Let us now have a closer inspection of S(u), the spray at ϕ = id. Since we presupposed that A be of order r ≥ 1, the second term Although the first term A −1 [A, u]u x of the spray is formally still of order 1, we may hope for some smoothing coming from the commutator [A, u]. The case when A is a differential operator justifies this hope. Actually, this turns out to be true not only for differential operators 3 but even for general Fourier multipliers, cf. again Lemma 14.
Summarizing, in Lagrangian coordinates the geodesic flow with respect to an inertia operator which is Fourier multiplier of order not less than 1 can be realized as an ordinary differential equation on T Diff ∞ (S 1 ) and it propagator is smooth and bounded in H q (S 1 ) on bounded subsets of H q (S 1 ), provided q is large enough.

Proof of the main results
Before entering the details of the proof of Theorem 1, let us recall some basic definitions and notations. Given two Banach spaces E and F and a m-linear mapping U from the m-fold Cartesian product E m of E into F . The mapping U is bounded iff there is a constant c > 0 such that Moreover, the space 3 The special case where A is a differential operator with constant coefficients has been extensively studied in [6,7,12]. endowed with the operator norm According to Lemma 26, the n-th Gâteaux ϕ-partial derivative of the map and P n is n-multilinear. We will write P n,ϕ := R ϕ P n R −1 ϕ , which is a (n+1)-multilinear operator form C ∞ (S 1 ) to C ∞ (S 1 ). Our first task will be to show that if each P n extends to a bounded multilinear operator from H q (S 1 ) to H q−r (S 1 ) , then the mapping is smooth.
Lemma 10. Let P be a Fourier multiplier of order r ≥ 1 and assume that for some n ∈ N, where the operators P n are defined in Lemma 26. Then is locally Lipschitz continuous.
Proof. (a) Let ϕ ∈ D q (S 1 ) be given and choose v 0 , . . . , v n ∈ H q (S 1 ) with v j H q ≤ 1. By Lemma 34 there exist increasing continuous functions C 1 and C 2 such that (14) we find that

With this notation and
H q ). 4 The above notation is consistent in the case m = 1, i.e. L 1 (E, F ) = L(E, F ). To further disburden our notation, we set L(E) := L(E, E).
Choosing now v 0 , . . . , v n ∈ C ∞ (S 1 ) with v j H q ≤ 1, we obtain from Lemma 26 that

This implies
Similarly, we can find a constant C > 0, depending only on the norms of ϕ 0 and ϕ 1 in and the assertion follows from the density of the embedding C ∞ (S 1 ) ֒→ H q (S 1 ).
Proof. (a) We first treat the case n = 1.
for some positive constants c 1 and c 2 which depend on P , ϕ, and v. Given ε > 0, Lemma 10 ensures the existence of a δ > 0 such that and Letting ϕ(t) := ϕ + tδϕ 1 and invoking Lemma 26, we find Thus we find This shows that (ϕ, v) → P ϕ (v) is Fréchet differentiable and the derivative is given by Invoking again Lemma 10, we see that, given ε > 0, the expressions on the right hand side of the last equality can be estimated in H q−r (S 1 ) from above by provided δϕ n+1 H q is small enough. This completes the proof.
We are now ready to prove our main results : Theorem 1, Corollary 2 and Corollary 5.
Proof of Theorem 1. The condition on the symbol p of the Fourier multiplier P ensures that for q ∈ ( 3 2 + r, ∞) ∪ {1 + r} and n ∈ N (see Appendix A for the details). Now, Theorem 1 results from Corollary 11 of Lemma 10 given below.
Using the Newton formula for the n-th derivative of f n , it is sufficient to show that is bounded for k = 1, . . . , n. This can be checked easily using the fact that for k ≥ 1, where p k is a polynomial function of degree less than k.

5.
Smoothness of the the metric and the spray on D q (S 1 ) For a strong Riemannian metric on a Banach manifold M , there exists always a unique symmetric covariant derivative compatible with the metric (see [22]). The equations for the geodesics of this covariant derivative (curves which minimize the energy functional) correspond to a smooth quadratic second order vector field called a spray. Conversely, given a smooth spray on such a manifold, it induces a symmetric covariant derivative on the Banach manifold, which however may not be metric, in the sense that it may not be compatible with any Riemannian metric on M . This is no longer true for a weak Riemannian metric on M , in general. The standard proof of the existence of a symmetric compatible covariant derivative involves the Riesz lemma, which is not available for weak metrics.
Nevertheless, we shall prove in this section that the geodesic spray (12) of a right-invariant weak Riemannian metric on Diff ∞ (S 1 ) extends to a smooth spray on the extended Banach manifold D q (S 1 ) provided that the conjugates of the inertia operator A are smooth. The very same definition of the covariant derivative associated to a spray [22] on a Banach manifold and the origin of this spray ensures then that this covariant derivative is compatible with the weak metric on D q (S 1 ).
where the subscript ϕ indicates the conjugacy by the right translation R ϕ in D p (S 1 ). Although P and Q are smooth operators, these results do not carry over when conjugated with translation in D q (S 1 ) since for q > 3/2 these sets only form topological groups: neither composition nor inversion are differentiable.
Given an operator K, we introduce the following notatioñ is a Banach algebra because q − r > 1/2. Hence the fact that P ϕ (v) ∈ H q−r (S 1 ) and our assumption ensure that is a bounded, linear, invertible operator from H q (S 1 ) × H q (S 1 ) to H q (S 1 ) × H q−r (S 1 ), we conclude, using the inverse mapping theorem on Banach spaces, is of class C m . (c) Taking P = A and δϕ 1 = v = u • ϕ in Lemma 26 when ϕ, v are smooth, we get Due to the density of the embedding C ∞ (S 1 ) ֒→ H q (S 1 ), this relation is still valid for ϕ ∈ D q (S 1 ) and v ∈ H q (S 1 ) and thereforẽ is of class C m−1 . The assertion now follows from the chain rule.
Corollary 13. (Smoothness of the H s metric and its spray) Let s ≥ 1/2 be given and assume that q ∈ ( 3 2 + r, ∞) ∪ {1 + r}. 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.
Proof. The smoothness of the metric results at once from Corollary 2 and formula (8). The smoothness of the spray (12) is a consequence of Proposition 12.
For further purpose, we will also state the following interesting property of the spray.
Proof. Let first B ⊂ H q (S 1 ) be bounded. In addition, we use the notation P (u) := (Au)u x , Q(u) := [A, u]u x , and S(u) := P (u) + Q(u), introduced in Proposition 12. Since H q−r (S 1 ) is a Banach algebra, we conclude that Observing that A ∈ L(H q (S 1 ), H q−r (S 1 )) and using the fact that r ≥ 1, we find a positive constant M 1 such that P (u) H q−r ≤ M 1 for all u ∈ B. By our assumption on A ϕ (v) relation (17) holds true, from which we conclude at ϕ = id that Thus there is a positive constant M 2 such that Q(u) H q−r ≤ M 2 for all u ∈ B. Combining this with the fact that A −1 ∈ L(H q−r (S 1 ), H q (S 1 )), we find an M > 0 such that The assertion is now a consequence of (41) and Lemma 36, 6. Existence results for geodesics on Diff ∞ (S 1 ) 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 (and more generally for any right-invariant weak Riemannian metric for which the inertia operator satisfies the hypothesis of Theorem 1 on the Fréchet-Lie group Diff ∞ (S 1 ). For this we shall use the Banach approximation {D q (S 1 )} q>3/2 of Diff ∞ (S 1 ) and the corresponding results of the previous section. Here the remarkable observation that the maximal interval of existence is independent of the parameter q (cf. Lemma 15) in the approximation is most helpful. It is the essential ingredient which makes it possible to avoid Nash-Moser type schemes but to use a simple direct argument, cf. [12,Lemma 8] to prove the main result Theorem 17. We note that Lemma 15 is inspired by [11,Theorem 12.1].
In what follows, we start with a right-invariant metric on Diff ∞ (S 1 ) and we assume that its inertia operator A is a Fourier multiplier of order r ≥ 1. We suppose further that, for all q ∈ ( 3 2 + r, ∞) ∪ {1 + r}, A induces an isomorphism from H q (S 1 ) onto H q−r (S 1 ) and that the mapping is smooth. Under these assumptions Proposition 12 and Lemma 14 are available for any n ∈ N. We recall the notation introduced in the previous sections to describe the spray in a local chart: By Proposition 12 and the Picard-Lindelöf theorem there is, given any This solution can be continued to a unique non-extendable solution on the maximal interval of existence cf. Section 10 in [1]. The mapping Φ q is called the flow on T D q (S 1 ), induced by the vector field (v, S ϕ (v)) and dom (q) is its maximal domain of definition.
Our aim is to prove well-posedness of the Cauchy problem (19) on the smooth manifold T Diff ∞ (S 1 ). In order to do so, we aim to use a Banach manifold approximation of T Diff ∞ (S 1 ) based on the fact that q≥0 H q (S 1 ) = C ∞ (S 1 ).
To follow this approach we need precise regularity properties of solutions to (19) on each level H q (S 1 ) with q > (3/2) + r. More precisely, assume that (ϕ 0 , v 0 ) ∈ T D q+1 (S 1 ). Then we may solve (19) 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 To rule out the possibility that J q+1 (ϕ 0 , v 0 ) is a proper subset of J q (ϕ 0 , v 0 ), which could lead to ∩ q J q (ϕ 0 , v 0 ) = {0}, we need the following auxiliary considerations. Given σ ≥ 0, let {T (s) ; s ∈ R} denote the translation group in H σ (S 1 ), i.e.
It is known that {T (s) ; s ∈ R} is a strongly continuous group in L(H σ (S 1 )). Its infinitesimal generator is given by ∂ x with domain of definition H σ+1 (S 1 ). This means in particular that, given v 0 ∈ H σ+1 (S 1 ), we have that There is also a one parameter group of right translations in D q (S 1 ) for which we use the same notation T (s)ϕ 0 (x) := ϕ 0 (x+s) for ϕ 0 ∈ D σ (S 1 ) and x ∈ S 1 . Finally, in the following auxiliary result, we resort on the notation, introduced in (21).
Thus the left hand side of (22) equals , v x (t)).
Combining these observations, we get But (20) reveals that the left hand side of the latter identity belongs to H q (S 1 ) × H q (S 1 ), which in turn implies that By the unique solvability of (19), we conclude that Invoking (21), the proof is completed.
Remark 16. Lemma 15 states that there is no loss of spatial regularity during the evolution of (19). 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 ).
Theorem 17. Let (18) 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 Proof. The result follows from (20), Lemma 15 and [12, Lemma 8], cf. the proof of Theorem 12 in [12].
Corollary 18. Let s ≥ 1/2 be given and consider the right-invariant Sobolev H s -metric on Diff ∞ (S 1 ). Then the corresponding geodesic equation has for any initial data in the tangent bundle T Diff ∞ (S 1 ) a unique non-extendable smooth solution (ϕ, v) ∈ C ∞ (J, T Diff ∞ (S 1 )). The maximal interval of existence J is open and contains 0. Moreover, if J = (t − , t + ) and either Proof. Let s ≥ 1/2 be given. Then Corollary 2 ensures that the smoothness assertion in (18) is satisfied for op 1 + k 2 s . The hypothesis on the isomorphy in (18) is obvious in this case. Thus the result follows from Theorem 17.
It is worth emphasizing that the blow up result in Corollary 18 only represents a necessary condition. For particular options of the inertia operator A it is known that the blow up occurs in weaker norms than H 1+2s (S 1 ). Indeed, for A = op 1 + k 2 , which leads to the periodic Camass-Holm equation, cf. [7], the precise blow up scenario is known: a classical solution u blows up in finite time if and only if cf. [5], which is considerably weaker than blow up in H 3 (S 1 ).
On the other hand there are several evolution equations, different from the Camassa-Holm equation, e.g. the Constantin-Lax-Majda equation, briefly discussed in Section 8, for which the blow up mechanism is much less understood and so far no sharper results than blow up in H 1+2s (S 1 ) seems to be known. 7. Exponential mapping and geodesic distance on Diff ∞ (S 1 ) The geodesic flow of a smooth spray on a Banach manifold M satisfies the following remarkable property which is a consequence of the quadratic nature of the geodesic equation [22]. Therefore, the exponential mapping exp x 0 , defined as the time one of the flow is well defined in a neighbourhood of 0 in T x 0 M for each point x 0 . It is moreover a local diffeomorphism from a neighbourhood V of 0 in T x 0 M onto a neighbourhood U (x 0 ) of x 0 in M [22]. This last assertion may be false 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 [25]. Moreover, the Riemannian exponential map for the L 2 metric (Burgers equation) on Diff ∞ (S 1 ) is not a local C 1diffeomorphism near the origin [6]. Nevertheless, it has been established in [6], 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 [7]), 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, +∞). The proof of Theorem 4 is similar to the one given in [12] and will be omitted. It requires only the smoothness of the spray on T D q (S 1 ) for all q large enough.
We will finish this section by a remark concerning the geodesic semidistance 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 [24], that this semi-distance vanishes identically for the L 2 right-invariant metric on the diffeomorphisms group of any compact manifold. More recently, it was shown in [3] that d s vanishes identically on Diff ∞ (S 1 ) if s ∈ [0, 1/2], whereas d s is a distance for s > 1/2 Since the geodesic spray of the weak H s right-invariant Riemannian metric on D q (S 1 ) is smooth for q ≥ 1 + 2s and s ≥ 1/2, its exponential mapping on D q (S 1 ) is a diffeomorphism from a neighbourhood V of 0 in H q (S 1 ) to a neighbourhood U of the identity in D q (S 1 ). This leads to the existence of local polar coordinates in the normal chart U . These coordinates are defined as follows. Given ϕ ∈ U − {id}, there is a v ∈ V \ {0} such that ϕ = exp(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}. Notice that (ρ, w) depend smoothly of ϕ and that ρ(ϕ) → 0 as ϕ → id.
As can be checked in [22], 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.
However, it should be noticed that Lemma 19 does not imply that the geodesic semi-distance is in fact a distance. What Lemma 19 says, is that the length of any path which lies inside the normal neighbourhood is bounded below by r := |ρ(b) − ρ(a)|. However for a path which leaves the normal neighbourhood, this might not be true. Such a path could leave the normal neighbourhood before leaving the (weak ball) of radius r defined as In fact this happens for the critical exponent s = 1/2 as it follows from [3]. 8. Euler equations on the homogeneous space Diff ∞ (S 1 )/Rot(S 1 ) In this section, we will apply our main theorems to some geodesic equations on the homogeneous space Diff ∞ (S 1 )/Rot(S 1 ). Since the proof are very similar to what has been done so far, we will not give all the details but only point out new difficulties that may arise.
Let Rot(S 1 ) denote the subgroup of all rigid rotations of S 1 and be the corresponding homogeneous space of right cosets. It is not difficult to verify that the restriction of the canonical projection to the subgroup Diff ∞ 1 (S 1 ) of Diff ∞ (S 1 ) consisting of all diffeomorphisms of S 1 ≃ R/Z which fixes one arbitrarily point (which we can take to be 0) is a diffeomorphism.
Remark 20. Since Diff ∞ (S 1 ) is simple: it possesses no nontrivial normal subgroups, cf. [15], the above identification Diff ∞ 1 (S 1 ) with Diff ∞ (S 1 )/Rot(S 1 ) is possible but the restriction of the canonical projection to Diff ∞ 1 (S 1 ) is not a group morphism.
Then ker A = R · w 0 . Furthermore ad w 0 = −D and ad * w 0 = −D. Thus hypothesis (ii) of Proposition 9 is satisfied. Therefore, A induces a rightinvariant Riemannian metric on Diff ∞ 1 (S 1 ) and the corresponding Euler equation is given by To solve this evolution equation, we also need a suitable Banach space approximation of Diff ∞ 1 (S 1 ). For this fix an arbitrary point x 0 ∈ S 1 and set D q 1 (S 1 ) := {ϕ ∈ D q (S 1 ) ; ϕ(x 0 ) = x 0 } for q > 3/2. Then D q 1 (S 1 ) is a Banach manifold modeled on the Banach space H q 0 (S 1 ) := {u ∈ H q (S 1 ) ; u(x 0 ) = 0}. and a topological group. Let for σ ≥ 0. If A = op (p(k)) is a Fourier multiplier of order r ≥ 1, satisfying (23), then A extends to H q 0 (S 1 ) such that (25) A ∈ Isom(H q 0 (S 1 ),Ĥ q−r 0 (S 1 )), for all q ∈ (3/2 + r, ∞) ∪ {r + 1}. From this we conclude that the map is of class C m from D q (S 1 ) × H q (S 1 ) to H q−r (S 1 ) then its restriction to D q 1 (S 1 ) × H q 0 (S 1 ) is also of class C m . Theorem 21. Let A = op (p(k)) be a Fourier multiplier of order r ≥ 1 with a real symbol p, satisfying Assume that in addition that ) of the Cauchy problem for the associated geodesic spray on the maximal interval of existence J = (t − , t + ). If either t − > −∞ or Sketch of proof. The proof requires Proposition 12 but the argument there has to be slightly modified to work in the setting here. Indeed, given ϕ ∈ D q 1 (S 1 ), A ϕ takes values in the vector space which depends on ϕ, which makes it difficult to apply the inverse mapping theorem in this setting as it is required in part (b) of the proof of Proposition 12. To overcome this difficulty, we replace the operator in the proof of Proposition 12 bỹ and we notice that m → ϕ x m is a linear isomorphism from R ϕ (Ĥ q−r 0 (S 1 )) ontoĤ q−r 0 (S 1 ).
We briefly discuss two special options of inertia operators A, namely A = op k 2 and A = op (|k|). In the first case A = op k 2 the Euler equation reads as (27) u and is known as the periodic Hunter-Saxton equation, cf. [18,33,4,23]. For the inertia operator A = op (|k|) we get the so called CLM equation, cf. [8,32,13].
where H = op (i sgn(k)) denotes the Hilbert transform, acting on the spatial variable x ∈ S 1 . Note that op (|k|) = H • D.
Remark 22. To conclude this section, it could be worth to bring together the present work with the right-invariant metric defined by the inertia operator A := HD(D 2 + 1) 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 [28]. The corresponding geodesic flow has been extensively studied in [14]. Recall first that D s (S 1 ), 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 [14] 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 geodesically complete (i.e., geodesics are defined for all times).
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 1 ) for s > 3/2 equipped with a Banach manifold structure. 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. Fourier multipliers
Here and in the following we use the notation e n (x) = exp(2πinx), for n ∈ Z and x ∈ S 1 . Lemma 23. Let P a continuous linear operator on the Fréchet space C ∞ (S 1 ). Then the following three conditions are equivalent: (1) P commutes with all rotations R s .
(3) For each n ∈ Z, 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 1 can be represented by its Fourier series, we get that for every Fourier multiplier P and every u ∈ C ∞ (S 1 ), wherê stands for the k-th Fourier coefficients of u. The sequence p : Z → C is called the symbol of P . We use also the notation P := op (p(k)) for the Fourier multiplier induced by the sequence p.
Proof. Given s ∈ R and u ∈ C ∞ (S 1 ), 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 ∈ Z and P is continuous, then we have representation (29). Therefore Remark 24. Notice that the space of Fourier multipliers is a commutative subalgebra of the algebra of linear operators on C ∞ (S 1 ) which contains all linear differential operators with constant coefficients. Notice also that each Fourier multiplier defines an L 2 -symmetric operator Remark 25. Notice that a Fourier multiplier P is L 2 -symmetric iff its symbol p is real.
A Fourier multiplier P = op (p(k)) with symbol p is said to be of order r ∈ R if there exists a constant C > 0 such that |p(k)| ≤ C 1 + k 2 r/2 , for every k ∈ Z. In that case, for each q ≥ r, the operator P extends 6 to a bounded linear operator from H q (S 1 ) into H q−r (S 1 ). We express this fact by the notation P ∈ L(H q (S 1 ), H q−r (S 1 )). 6 Throughout this paper we consider Fourier operators of order r ≥ 1, since our main results Theorem 1 is only true for this class of operators. It is nevertheless worth to mention that several results in this section remain true for operators of any positive order.
Let (ϕ, v) → P ϕ (v) be a smooth mapping on the Fréchet manifold Diff ∞ (S 1 )× C ∞ (S 1 ), 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 1 ) is a smooth map which is linear both in v and δϕ 1 and that we will denote by (30) ∂ ϕ P ϕ (v, δϕ 1 ).
Therefore, the partial Gâteaux derivative of P in the variable ϕ is a mapping of three independent variables : ϕ, v, δϕ 1 . The second partial derivative of P is directions δϕ 1 , δϕ 2 ∈ C ∞ (S 1 ) is the partial Gâteaux derivative of (30) 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 [16]). 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 [16]). 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 mappings.
Let P denote a general Fourier multiplier on C ∞ (S 1 ). We will now study conjugation of P with right translations R ϕ , where ϕ ∈ Diff ∞ (S 1 ). We will derive a recursion formula for the n−th derivative with respect to ϕ of such operators. In addition, we provide a sufficient criterion on the symbol of the original operator P , which ensures that the n−th derivative ∂ n ϕ P ϕ extends to an (n + 1)−linear mapping on suitable Sobolev spaces.
Lemma 28. Let P be a Fourier multiplier on C ∞ (S 1 ), and let P n be the multilinear operator defined in Lemma 26 for some n ∈ N. Then we have using the fact that f n+1 (t) = tf n (t), we have therefore which achieves the proof.
In addition H q−r (S 1 ) is a Banach algebra, since q − r ≥ 1, cf. [ for all smooth functions v 0 , v 1 , . . . , v n . Putting noŵ v j (m j ) := (1 + m 2 j ) r/2 |û j (m j )| , j = 0, . . . , n in this last inequality and using the fact that the functions with Fourier coefficientv(m) and |v(m)| have the same H q−r norm, we obtain P n (u 1 , . . . , u n )u 0 H q−r ≤ C n C ′ n,q−r u 0 H q · · · u n H q , which achieves the proof.
Finally, we will need to define a condition on the symbol of the Fourier multiplier P in order that the operators P n are bounded. For this purpose, the following lemma will be useful.
Lemma 32. Let f : R → R be a function of class C n−1 with n ≥ 1. Suppose that f (n−1) is absolutely continuous and that there exists C > 0 and α ≥ 1 such that for all m 0 , m 1 , . . . , m n ∈ R.
Proof. Let g k be the sequence of functions defined inductively by Then, we have (1 + m 2 j ) (α−1)/2 .

Appendix B. 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 33. 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 [10,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 n ) = F (g n , x n ) ∈ V for all n ≥ n 0 .
Thus F is continuous in (g 0 , x 0 ).
We now proceed as follows. First we have For the first term of the right-hand side of (45), we find To estimate the second term in (45), choose w ∈ C 2 (S 1 ) such that where we also employed the mean value theorem and (44) to derive the last estimate. Invoking (43) and (47), we get for all ϕ ∈ B q (δ). Combining (42), (45), (46), and (48), we arrive at the following estimate for all ϕ ∈ B q (δ). Shrinking δ > 0, we get from (49) that for all ϕ ∈ B q (δ). Thus F (·, v) is continuous in ϕ 0 ∈ D q (S 1 ). Invoking Lemma 33, we find that F ∈ C(D q (S 1 ) × H 1 (S 1 ), H 1 (S 1 )). This completes the proof of part (i).
(c) To prove the second assertion, fix first ϕ ∈ D q (S 1 ) and observe that F (ϕ, ·) = R ϕ . Given k ∈ N with 1 ≤ k ≤ q, a direct calculation and an application of Sobolev's embedding theorem shows that there exists a continuous, increasing function C k : R + → R + , such that R ϕ L(H k (S 1 )) ≤ C k ϕ H q (S 1 ) .
The last assertion F ∈ C(D q (S 1 ) × H q (S 1 ), H q (S 1 )) is now clear.
Remarks 35. (a) Observe that there is a gap in the range of admissible Sobolev norms in Lemma 34 between assertion (i) and the second assertion (ii).
(b) Continuity properties of composition operators in low Sobolev spaces have recently been investigated in [9]. However the focus of the studies in [9] is so to speak opposite to that of (40), since more regularity of the first factor v in v • ϕ is assumed, whereas we impose additional regularity on the second factor ϕ.
(c) The higher the spatial regularity in the group D q (S 1 ) and the Lie algebra H q (S 1 ), the better the regularity of the mapping F in Lemma 34, cf. [11]. However, we are not aware of better regularity of F than (40). Finally, we remark that the continuity of F is sufficient for our purposes.
We conclude this Appendix by an auxiliary result on the boundedness of the inverse of the right translation in H q (S 1 ).
Lemma 36. Let q > 3/2 be given and assume that B ⊂ D q (S 1 ) is bounded in H q (S 1 ). Then Proof. By the uniform boundedness principle it suffices to show that If q > 3/2 is an integer this follows by a direct calculation and an application of Sobolev's embedding theorem. If q is not an integer, let k ∈ N and σ ∈ (0, 1) such that q = k + σ. Again a direct calculation shows that it suffices to show that sup ϕ∈B (∂ k v) • ϕ −1 H σ < ∞ for all v ∈ H q (S 1 ), which follows from the fact that ∂ k v ∈ H σ (S 1 ) and by using the intrinsic norm |w(x) − w(y)| 2 |x − y| 1+2σ dx dy 1/2 for w ∈ H σ (S 1 ), cf. Section 2.2.2 in [30].