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Geodesic Completeness for Sobolev Hs-Metrics on the Diffeomorphisms Group of the Circle Joachim Escher, Boris Kolev

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Geodesic Completeness for Sobolev Hs-Metrics on the Diffeomorphisms Group of the Circle Joachim Escher, Boris Kolev Geodesic Completeness for Sobolev Hs-metrics on the Diffeomorphisms Group of the Circle Joachim Escher, Boris Kolev To cite this version: Joachim Escher, Boris Kolev. Geodesic Completeness for Sobolev Hs-metrics on the Diffeomorphisms Group of the Circle. 2013. hal-00851606v1 HAL Id: hal-00851606 https://hal.archives-ouvertes.fr/hal-00851606v1 Preprint submitted on 15 Aug 2013 (v1), last revised 24 May 2014 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. GEODESIC COMPLETENESS FOR SOBOLEV Hs-METRICS ON THE DIFFEOMORPHISMS GROUP OF THE CIRCLE JOACHIM ESCHER AND BORIS KOLEV Abstract. We prove that the weak Riemannian metric induced by the fractional Sobolev norm H s on the diffeomorphisms group of the circle is geodesically complete, provided s > 3/2. 1. Introduction The interest in right-invariant metrics on the diffeomorphism group of the circle started when it was discovered by Kouranbaeva [17] that the Camassa– Holm equation [3] could be recast as the Euler equation of the right-invariant metric on Diff∞(S1) induced by the H1 Sobolev inner product on the corre- sponding Lie algebra C∞(S1). The well-posedness of the geodesics flow for the right-invariant metric induced by the Hk inner product was obtained by Constantin and Kolev [5], for k ∈ N, k ≥ 1, following the pioneering work of Ebin and Marsden [8]. The well-posedness in the case of fractional or- s ders (H , s ∈ R+) has been established by Escher and Kolev [10], provided s ≥ 1/2. The way to establish local existence of geodesics is to extend the metric and its spray to the Hilbert approximation Dq(S1) (the Hilbert man- ifold of diffeomorphisms of class Hq) and then to show that the (extended) spray is smooth. This was proved to work in [10] for s ≥ 1/2 provided we choose1 q > 2s + 3/2. The well-posedness on Diff∞(S1) follows from a result on no loss and no gain of spatial regularity for the geodesic flow. Note that the extended metric defines only a weak Riemannian metric on the Hilbert manifold Dq(S1). A Riemannian metric is strong if at each point it induces a topological isomorphism between the tangent space and the cotangent space. It is weak if it defines only an injective linear map between the tangent space and the cotangent space. On a Banach manifold equipped with a strong metric, the geodesic semi-distance induced by the metric is a distance (i.e if ϕ1 = ϕ2, then d(ϕ1, ϕ2) > 0) [18]. This is no longer true for a weak metric, in general. It was shown by Bauer, Bruveris, Harms, and Michor [2] that the geodesic semi-distance identically vanishes for the Hs metric if 0 ≤ s ≤ 1/2, whereas it is a distance for s > 1/2. This distance is nevertheless probably never complete on T Dq(S1). Moreover, although for a strong metric topological completeness implies geodesic completeness, this is generally not true for a weak metric. Finally, we recall that the metric induced by the H1-norm (or equivalently by A = id−D2) is not geodesically Date: August 15, 2013. 2010 Mathematics Subject Classification. 58D05, 35Q53. 1This bound can be sharpen to q > 2s − 1/2 if s is an integer. 1 2 JOACHIMESCHERANDBORISKOLEV complete, as was shown by Constantin and Escher [4]. The main result of this paper is the following. Theorem 1.1. Let s > 3/2 be given. Then the geodesic flow on Dq(S1) for q > 2s + 3/2 and on Diff∞(S1), respectively, is complete for the weak Riemannian metric induced by the Hs(S1)-inner product. Note that we could be lead to think that for s > 3/2 the right-invariant Hs metric on the Banach manifold Ds(S1) is a strong Riemannian metric. However it is not clear at all that this metric is smooth on Ds(S1). In [10], we were only able to show that the Hs metric is smooth on the Banach manifold Dq(S1) for q > 2s + 3/2. Let us briefly give an outline of the paper. In Section 2, we introduce basic facts on right-invariant metrics on Diff∞(S1) and we recall a well-posedness result for related geodesic flows. In Section 3, we introduce a complete metric structure on suitable Banach approximations Dq(S1) of Diff∞(S1), which allows us to describe the precise blow-up mechanism of finite time geodesics. This is the contents of Section 4. In section 5, we prove our main result, theorem 1.1. In Appendix A, we recall the material on Friedrichs mollifier that have been used throughout the paper. 2. Right-invariant metrics on Diff∞(S1) Let Diff∞(S1) be the group of all smooth and orientation preserving dif- feomorphisms on the circle. This group is naturally equipped with a Fr´echet manifold structure; it can be covered by charts taking values in the Fr´echet vector space C∞(S1) and in such a way that the change of charts are smooth mappings (a smooth atlas with only two charts may be constructed, see for instance [14]). Since the composition and the inverse mapping are smooth for this struc- ture, we say that Diff∞(S1) is a Fr´echet-Lie group, c.f. [15]. Its Lie algebra, Vect(S1), is the space of smooth vector fields on the circle. It is isomorphic to C∞(S1), with the Lie bracket given by [u v] = uxv − uvx. From an analytic point of view, the Fr´echet Lie group Diff∞(S1) may be viewed as an inverse limit of Hilbert manifolds. More precisely, recall that the Sobolev space Hq(S1) is defined as the completion of C∞(S1) for the norm 1/2 2 q 2 uHq(S1) := (1 + n ) |uˆn| , ∈Z ! nX + where q ∈ R , and whereu ˆn stands for the n−th Fourier coefficient of u ∈ L2(S1). Let Dq(S1) denote the set of all orientation preserving home- omorphisms ϕ of the circle S1, such that both ϕ and ϕ−1 belong to the fractional Sobolev space Hq(S1). For q > 3/2, Dq(S1) is a Hilbert manifold and a topological group, as was shown by Ebin and Marsden [8]. It is how- ever not a Lie group because neither composition, nor inversion in Dq(S1) GEODESICCOMPLETENESS 3 are differentiable [8]. We have Diff∞(S1) = Dq(S1). 3 q>\2 Remark 2.1. Like any Lie group, Diff∞(S1) is a parallelizable manifold: T Diff∞(S1) ∼ Diff∞(S1) × C∞(S1). What is less obvious, however, is that T Dq(S1) is also a trivial bundle. In- deed, let t : T S1 → S1 × R be a smooth trivialisation of the tangent bundle of S1. Then T Dq(S1) → Dq(S1) × Hq(S1), ξ → t ◦ ξ defines a smooth vector bundle isomorphism (see [8, p. 107]). A right-invariant metric on Diff∞(S1) is defined by an inner product on the Lie algebra Vect(S1) = C∞(S1). In this paper, we assume that this inner product is given by u v = (Au)v dx, S1 Z where A :C∞(S1) → C∞(S1) is a L2-symmetric, positive definite, invertible Fourier multiplier (i.e. a continuous linear operator on C∞(S1) which com- mutes with D := d/dx). For historical reasons going back to Euler’s work [12], A is called the inertia operator. By translating the above inner product, we obtain one on each tangent ∞ 1 space TϕDiff (S ), which is given by −1 −1 (2.1) η, ξϕ = η ◦ ϕ , ξ ◦ ϕ id = η(Aϕξ)ϕx dx, S1 Z ∞ S1 − where η, ξ ∈ TϕDiff ( ), Aϕ = Rϕ ◦ A ◦ Rϕ 1 , and Rϕ(v) := v ◦ ϕ. This defines a smooth weak Riemannian metric on Diff∞(S1). This weak Riemannian metric admits the following geodesic spray2 (2.2) F :(ϕ, v) → (ϕ, v, v, Sϕ(v)) , where Sϕ(v) := Rϕ ◦ S ◦ Rϕ−1 (v), and −1 S(u) := A {[A, u]ux − 2(Au)ux} . A geodesic is an integral curve of this second order vector field, that is a solution (ϕ, v) of ϕt = v, (2.3) v = S (v), t ϕ Given a geodesic (ϕ, v), we define its Eulerian velocity as u := v ◦ ϕ−1. 2A Riemannian metric on a manifold M defines a smooth function on TM, given by half the square norm of a tangent vector. The corresponding Hamiltonian vector field on TM, relatively to the pullback of the canonical symplectic structure on T ∗M is called the geodesic spray. 4 JOACHIMESCHERANDBORISKOLEV Then u solves −1 (2.4) ut = −A [u(Au)x + 2(Au)ux] , called the Euler equation defined by the inertia operator A. Remark 2.2. When A is a differential operator of order r ≥ 1 then the quadratic operator −1 S(u) := A {[A, u]ux − 2(Au)ux} is of order 0 because the commutator [A, u] is of order not higher than r − 1. One might expect, that for a larger class of operators A, the quadratic operator S to be of order 0 and consequently that the second order system (2.3) may be viewed as an ODE on T Dq(S1). Definition 2.3. A Fourier multiplier A = op (a(k)) with symbol a is of order r ∈ R if there exists a constant C > 0 such that r/2 |a(k)| ≤ C 1 + k2 , Z for every k ∈ .
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