Two-Point Boundary Value Problems on Diffeomorphism Groups

Abstract

We consider a variety of geodesic equations on Sobolev diffeomorphism groups, including the equations of ideal hydrodynamics. We prove that solutions of the corresponding two-point boundary value problems are precisely as smooth as their boundary conditions. We further utilise this regularity property to construct continuously differentiable exponential maps in the Frechét setting.

1 Introduction

The modern formulation of hydrodynamics goes all the way back to Euler in 1757 from whom we acquire the following equations describing the motion of an ideal (incompressible and inviscid) fluid in a closed |$n$|-dimensional Riemannian manifold |$M$|⁠:

$$ \begin{align}& \begin{cases} \partial_t u + \nabla_uu = -\nabla p \\ \text{div}(u) = 0.\\ \end{cases} \end{align}$$

(1)

If we further impose the condition that |$u(0)=u_0$|⁠, we obtain the Cauchy problem for (1) whose rigorous studies date back to the 1920s with Günther and Lichtenstein, the 1930s with Wolibner, and the 1960s with Yudovich and Kato. The list goes on. Many of these references can be found in the monographs of Majda and Bertozzi [22], Arnold and Khesin [3], and Bahouri et al. [4].

In Arnold’s seminal paper [2], he observed that solutions of (1) can be viewed as geodesics of a right-invariant |$L^2$| metric on the group |$\mathscr {D}_\mu (M)$| of volume-preserving diffeomorphisms (sometimes referred to as volumorphisms). In essence, this approach showcases the natural framework in which to tackle (1) from the Lagrangian viewpoint. In their celebrated paper, Ebin and Marsden [12] provided the formulation of the above in the |$H^s$| Sobolev setting. Among other things, they proved that for |$s> \frac {n}{2}+1$|⁠, the space of volumorphisms can be given the structure of a smooth, infinite dimensional Hilbert manifold. They showed that when equipped with Arnold’s |$L^2$| metric, the geodesic equation on this manifold is a smooth ordinary differential equation. They then applied the classic iteration method of Picard to obtain existence, uniqueness, and smooth dependence on initial conditions. In particular, the last property allows one to define a smooth exponential map on |$\mathscr {D}^s_\mu (M)$| in analogy with the classical construction in finite dimensional geometry. Hence, the work of Arnold, Ebin, and Marsden enables us to explore the Riemannian geometry of fluid motion.

One of the central results in [12] says: let |$\gamma (t) \in \mathscr {D}^{s}_\mu (M)$| be an |$L^2$| geodesic. If |$\gamma (0) \in \mathscr {D}^{s+1}_\mu (M)$| and |$\dot {\gamma }(0) \in T_e\mathscr {D}^{s+1}_\mu (M)$|⁠, then |$\gamma (t) \in \mathscr {D}^{s+1}_\mu (M)$| for all |$t$| for which it was defined in |$\mathscr {D}^s_\mu (M)$|⁠, cf. [12, Theorem 12.1], that is, roughly speaking, if one considers (1) in Lagrangian coordinates as a 2nd-order ODE in |$\mathscr {D}^s_\mu (M)$|⁠, any solution of the corresponding Cauchy problem (an |$L^2$| geodesic) is as smooth as its initial conditions. A natural question is if a similar regularity result holds when the geodesic equation is framed as a two-point boundary value problem. More precisely, given an |$L^2$| geodesic |$\gamma (t)$| emanating from the identity in |$\mathscr {D}^{s}_\mu (M)$| and, at some later time |$t_0>0$|⁠, passing through |$\eta \in \mathscr {D}^{s+k}_\mu (M)$|⁠, can it be shown that |$\gamma (t)$| evolves entirely in |$\mathscr {D}^{s+k}_\mu (M)$|⁠, that is, is an |$L^2$| geodesic as smooth as its boundary conditions? We provide an affirmative answer to this question in the setting of 2D ideal fluids, 3D axisymmetric fluids with zero swirl, 1D equations such as the |$\mu $|CH, and Hunter–Saxton equations, as well as the Euler-|$\alpha $| equations, the symplectic Euler equations and other higher-order Euler–Arnold equations arising from |$H^r$| metrics with |$r>0$|⁠.

The earliest results of this nature, to the best of the author’s knowledge, are due to Constantin and Kolev [9, 10]. Here, they used the above property to show the existence of a continuously Frechét differentiable exponential map for right-invariant |$H^{r}$| metrics, |$r \in {\mathbb {Z}}_{\geq 1}$|⁠, on the group of |$C^\infty $| diffeomorphisms of the circle. This result was improved upon by Kappeler et al. [17] who showed that these exponential maps are in fact Frechét bianalytic near zero.

This regularity property was further explored and used as a key ingredient by the latter authors in their study [18] of the group of orientation-preserving diffeomorphisms of the 2-torus, equipped with various right-invariant |$H^r$| metrics, again for |$r \in {\mathbb {Z}}_{\geq 1}$|⁠. As in [10] and [9], their goal was to show the existence of a continuously Frechét differentiable exponential map on the group of |$C^{\infty }$| diffeomorphisms of the 2-torus.

More recent work includes that of Bruveris [8], who obtained a similar result to the above on the full diffeomorphism group of an arbitrary closed manifold. He shows, for a smooth right-invariant metric on |$\mathscr {D}^s(M)$| with smooth exponential map, a |$H^r$| geodesic is as smooth as its boundary points, provided that they are not conjugate.

As far as the author is aware, our results are the 1st of their nature pertaining to the Euler equations of ideal hydrodynamics. We record some examples of these here.

 

Theorem 1.1.

Let |$s> 6$|⁠, and consider the group of volume-preserving diffeomorphisms of |${\mathbb {T}}^2$| equipped with a right-invariant |$L^2$| metric. Given |$u_0 \in T_e\mathscr {D}^s_\mu $|⁠, let |$\gamma (t)$| denote the corresponding geodesic of the weak |$L^2$| metric. If at time |$t=1$|⁠, |$\gamma $| passes through a point |$\eta \in \mathscr {D}^{s+1}_\mu $|⁠, then we have |$u_0 \in T_e\mathscr {D}^{s+1}_\mu $| and consequently |$\gamma (t)$| evolves entirely in |$\mathscr {D}^{s+1}_\mu $|⁠.

 

Theorem 1.2.

Let |$s>\frac {13}{2}$|⁠, and consider the group of axisymmetric diffeomorphisms of |${\mathbb {T}}^3$| with respect to any of the Killing fields |$K=\partial _i$|⁠, equipped with a right invariant |$L^2$| metric. Given |$u_0 \in T_e\mathscr {A}^s_{\mu ,0}$|⁠, let |$\gamma (t)$| denote the corresponding geodesic of the weak |$L^2$| metric. If at time |$t=1$|⁠, |$\gamma $| passes through a point |$\eta \in \mathscr {A}^{s+1}_{\mu ,0}$|⁠, then we have |$u_0 \in T_e\mathscr {A}^{s+1}_{\mu , 0}$| and consequently |$\gamma (t)$| evolves entirely in |$\mathscr {A}^{s+1}_{\mu ,0}$|⁠.

We further use these regularity properties to construct continuously Frechét differentiable exponential maps in the smooth setting.

 

Theorem 1.3.

The Frechét manifold |$\mathscr {D}_\mu ({\mathbb {T}}^2)$| of smooth diffeomorphisms of |${\mathbb {T}}^2$| equipped with the |$L^2$| metric admits a well-defined exponential map that is a local |$C^1_F$|-diffeomorphism at the identity.

 

Theorem 1.4.

The Frechét manifold |$\mathscr {A}_{\mu ,0}({\mathbb {T}}^3)$| of smooth axisymmetric swirl-free diffeomorphisms of |${\mathbb {T}}^3$| with respect to any of the Killing Fields |$K=\partial _i$| equipped with the |$L^2$| metric admits a well-defined exponential map that is a local |$C^1_F$|-diffeomorphism at the identity.

Later sections will provide more precise statements of these results and notations.

Theorem 1.1 improves upon a similar result of Omori [24] that is infinitesimal in character. His method involves constructing a connection on the full diffeomorphism group that turns the volume-preserving diffeomorphisms into a totally geodesic submanifold. In the proof of Theorem 1.2, we make use of the recent work of Lichtenfelz et al. [21] on the Euler equations in three dimensions with axisymmetric swirl-free initial data.

Throughout the arguments, we will consider multiple configuration spaces: the full diffeomorphism group, the space of volumorphisms and its subspace of axisymmetric diffeomorphisms, and the space of symplectomorphisms, all equipped with various right-invariant metrics. In each setting, we will consider Sobolev diffeomorphisms of class |$H^s$| with varying requirements on the value of |$s$| and we may write |$\mathscr {G}^s(M)$| to refer to any of the aforementioned spaces. Similarly, we use |$\mathscr {G}(M)$| to refer to the corresponding smooth settings.

The paper is organised as follows: Sections 2 and 3 contain the necessary background information for the spaces we will be considering. Section 4 contains the main results. We begin in Section 4.1 with the setting of compressible fluids on |${\mathbb {T}}^n$|⁠. The theorems presented here generalise to arbitrary dimensions the results from one and two dimensions contained in [9, 10, 18] without any restrictions on whether or not the endpoints are conjugate, unlike the result contained in [8]. We then proceed to the Euler equations on |${\mathbb {T}}^2$| in Section 4.2, where we also cover the cases of the Euler-|$\alpha $| and higher-order Euler–Arnold equations mentioned above. This is followed by the axisymmetric results for 3D fluids in Section 4.3. The final case of the symplectic Euler equations on |${\mathbb {T}}^{2k}$| is covered in Section 4.4. In Section 5, we use our results from the Section 4 to construct continuously differentiable exponential maps in the Frechét settings.

2 Manifold Structure of Sobolev Diffeomorphisms

Here, we gather some basic facts about diffeomorphism groups. Further, details concerning these spaces can be found in Ebin and Marsden [12], Arnold and Khesin [3], Ebin [11], Omori [25], Misiołek and Preston [23], and Inci et al. [16].

2.1 Sobolev diffeomorphisms

Let |$M$| be a compact, |$n$|-dimensional Riemannian manifold without boundary, with metric |$g$| and volume form |$\mu $|⁠. Let |$g^\flat $| and |$g^\sharp $| denote the usual “musical isomorphisms”. We define the Hodge Laplacian on |$k$|-forms by |$\Delta = d\delta + \delta d$|⁠, where |$d$| is the exterior derivative, |$\delta =(-1)^{n(k+1)+1}\star d \star $| is its |$L^2$| dual, and |$\star $| is the Hodge Star operator.

We define |$H^s(TM)$| to be the space of vector fields on |$M$| with |$L^2$| derivatives up to order |$s$|⁠. We equip |$H^s(TM)$| with a |$H^s$| inner product via

$$ \begin{align*} &\langle u,v \rangle_s = \sum_{k \leq s} \langle u,(1 + \Delta)^s v \rangle_{L^2}.\end{align*}$$

In this paper, we will concern ourselves with the case |$s \in {\mathbb {Z}}_{\geq 0}$|⁠, but many of the constructions can be extended to the fractional case.

Recall that if |$s> \frac {n}{2}$|⁠, then |$H^s(M,M)$| is a Hilbert manifold modelled on |$H^s(TM)$| equipped with |$\langle \, \ \rangle _s$|⁠. If we further require |$s> \frac {n}{2} + 1$|⁠, then |$\mathscr {D}^s(M)$| inherits a smooth submanifold structure as an open subset of |$H^s(M,M)$|⁠. Its tangent space at the identity is |$T_e\mathscr {D}^s(M) = H^s(TM)$| whose dual enjoys an |$L^2$|-orthogonal decomposition by the Hodge theorem:

$$ \begin{align}& H^s(T^*M) = \mathcal{H} \oplus d \delta H^{s+2}(T^*M) \oplus \delta d H^{s+2}(T^*M), \end{align}$$

(2)

where |$\mathcal {H}$| denotes the finite dimensional subspace of harmonic one-forms on |$M$|⁠.

In what follows, it will be convenient to define the projections:

$$ \begin{align}& \pi_0: T_e\mathscr{D}^s \rightarrow d \delta H^{s+2} \oplus \delta d H^{s+2} \quad \text{and} \quad \pi_{\mathcal{H}}: T_e\mathscr{D}^s \rightarrow \mathcal{H}. \end{align}$$

(3)

2.2 Volume-preserving diffeomorphisms

The volumorphism group is defined in terms of the Riemannian volume form |$\mathscr {D}^s_{\mu }(M) = \{ \eta \in \mathscr {D}^s(M) \ \vert \ \eta ^*\mu = \mu \}$|⁠. It is a smooth Hilbert submanifold of |$\mathscr {D}^s(M)$| and the tangent space at the identity consists of all divergence-free |$H^s$| vector fields |$T_e\mathscr {D}^s_{\mu }(M) = \{ u \in T_e\mathscr {D}^s(M) \ \vert \ \text {div} \ u = 0 \} = g^\sharp \{\mathcal {H} \oplus \delta d H^{s+2}(T^*M)\}$|⁠, that is, the 1st and 3rd summands of the Hodge decomposition above.

2.3 Axisymmetric diffeomorphisms

Let |$M$| be a 3D manifold equipped with a smooth Killing field |$K$|⁠. Following [21], we define a divergence-free vector field |$u$| on |$M$| to be axisymmetric if it commutes with the Killing field: |$[K, u]=0$|⁠. We denote the set of |$H^s$| axisymmetric vector fields by |$T_e\mathscr {A}_\mu ^{s}(M)$|⁠.

A volume-preserving diffeomorphism of |$M$| is said to be axisymmetric if it commutes with the flow of the Killing field |$K$|⁠. The set of all such |$H^s$| volumorphisms, |$\mathscr {A}_\mu ^s(M)$| is a topological group, as well as a smooth totally geodesic Hilbert submanifold of |$\mathscr {D}_\mu ^s(M)$|⁠; cf. [21, Section 3].

Axisymmetric fluid flows are of great interest and their behaviour might be informally described as |$2\frac {1}{2}$|-dimensional fluids; cf. [30].

2.4 Symplectomorphisms

Let |$M$| be a symplectic manifold of dimension |$2k$|⁠, and let |$\omega ^\flat $| and |$\omega ^\sharp $| denote the standard “|$\omega $|-musical isomorphisms”. Analogously to volumorphisms, the symplectomorphism group |$\mathscr {D}^s_\omega (M)$| is a closed Hilbert submanifold of |$\mathscr {D}^s(M)$| consisting of those diffeomorphisms that preserve the symplectic form |$\omega $| under pullback. The tangent space at the identity is |$T_e\mathscr {D}^s_{\omega }(M) = \{ u \in T_e\mathscr {D}^s(M) \ \vert \ \ d\omega ^\flat u = 0 \} = \omega ^\sharp \big ( \mathcal {H} \oplus d \delta H^{s+2}(T^*M) \big )$|⁠; cf. (2).

For our purposes, we will further require that |$g$| and |$\omega $| are compatible, that is, the map |$J: = g^\sharp \omega ^\flat : TM \rightarrow TM$| satisfies |$J^2 = -Id$|⁠. In this case, |$J$| is said to give |$M$| an almost complex structure; cf. [6, 11, 12, 26].

3 Lie Group Structure and Geodesics on Diffeomorphism Groups

Here, we develop an infinite dimensional Lie group framework for |$\mathscr {D}^s(M)$| and its submanifolds of interest. (It is important to note that, strictly speaking, these Hilbert manifolds of finite regularity mappings are not Lie Groups. However, they are topological groups and possess sufficient geometric structure for our purposes. See Remark 3.2 for more details.) We use this to present geodesic equations on diffeomorphism groups in a general formulation for a variety of metrics. The construction follows closely that of [12] and [23].

Throughout the various arguments in Section 4, we will deal with products of |$H^s$| Sobolev functions and compositions with elements of |$\mathscr {D}^s(M)$|⁠, so it is crucial that we have some control over the regularity of these objects. For this reason, we recall the following results; cf. [1, 12, 16].

 

Lemma 3.1.

For any |$s> \frac {n}{2} + 1$|⁠, |$\lvert m\rvert \leq s$| and |$k \geq 0$| we have

1. |$H^{s}(M,{\mathbb {R}}) \times H^{m}(M, {\mathbb {R}}) \rightarrow H^{m}(M, {\mathbb {R}}) \ ; \ (u,v) \mapsto uv$| is bounded;

2. |$H^{s+k}(M,{\mathbb {R}}^d) \times \mathscr {D}^{s}(M) \rightarrow H^{s}(M, {\mathbb {R}}^d) \ ; \ (v,\phi ) \mapsto v\circ \phi $| is |$C^k$|-smooth;

3. |$\mathscr {D}^{s+k}(M) \rightarrow \mathscr {D}^{s}(M) \ ; \ \phi \mapsto \phi ^{-1}$| is |$C^k$|-smooth.

In particular, this gives us that |$\mathscr {D}^s(M)$| is a topological group (one can readily show that |$\mathscr {D}^s_\mu (M)$|⁠, |$\mathscr {A}^s_\mu (M)$|⁠, and |$\mathscr {D}^s_\omega (M)$| are all subgroups), where right translation |$R_\eta $| is smooth and left translation |$L_\eta $| is continuous (although not even Lipschitz continuous) in the |$H^s$| topology.

Again, using |$\mathscr {G}^s(M)$| to refer to any of the aforementioned manifolds, we denote the (almost) Lie algebra by |${\mathfrak {g}}^s = T_e\mathscr {G}^s(M)$|⁠, we recall the group adjoint:

$$ \begin{align*} {\operatorname{Ad}}_\eta v &= d_{\eta^{-1}}L_\eta d_eR_{\eta^{-1}} v \\ &= \big(D\eta \cdot v \big) \circ \eta^{-1} \end{align*}$$

and the Lie algebra adjoint:

$$ \begin{align*} &{\operatorname{ad}}_u v = -[u,v],\end{align*}$$

where |$\eta (t)$| is any curve in |$\mathscr {G}^s(M)$| with |$\eta (0) = e$| and |$\dot {\eta }(0) = u$|⁠. Observe that if |$u,v \in T_e\mathscr {G}^{s}$|⁠, their commutator is a priori only of Sobolev class |$H^{s-1}$|⁠.

 

Remark 3.2.

The derivative loss coming from left translation and the Lie bracket are examples of why these diffeomorphism groups are not Lie Groups. However, the group structure they possess is sufficient for our purposes.

We now equip |$\mathscr {G}^s(M)$| with a (weak) right-invariant, Riemannian metric |$\langle \cdot , \cdot \rangle $| (we say a metric on |$\mathscr {G}^s$| is weak if it induces a weaker topology than the inherent |$H^s$| topology). Note that the definitions of |${\operatorname {Ad}}_\eta $| and |${\operatorname {ad}}_u$| depend only on the group structure and not on the choice of the metric. Hence, the geometry is, in some sense, encoded in the coadjoint operators defined as follows:

$$ \begin{align*} &\langle {\operatorname{Ad}}^*_\eta u, v \rangle = \langle u, {\operatorname{Ad}}_\eta v \rangle \quad \text{and} \quad \langle {\operatorname{ad}}_u^*v, w \rangle = \langle v, {\operatorname{ad}}_u w \rangle\end{align*}$$

for all |$u, v, w \in {\mathfrak {g}}^s$|⁠.

Throughout this paper, * will always refer to the adjoint of an operator with respect to the relevant metric and configuration space, which should be clear from the context.

We are specifically interested in the case where we equip |$\mathscr {G}^s(M)$| with a right-invariant |$H^r$| metric with |$r\geq 0$| defined at the identity by

$$ \begin{align}& \langle u, v \rangle_{H^r}:= \langle A^ru,v \rangle_{L^2} \qquad \quad u, v \in T_e\mathscr{G}^s, \end{align}$$

(4)

where |$A^r$| denotes an invertible elliptic pseudo-differential operator of order |$2r$|⁠, for example, |$A^r = (1 + g^\sharp \Delta g^\flat )^r$|⁠; cf. Taylor [29]. We refer to such an |$A^r$| as an inertia operator and denote its inverse by |$A^{-r}$|⁠. We will always assume that |$A^r$| commutes with both |$d$| and |$\delta $| and that we have at least |$s> \frac {n}{2} + 2r + 1$| to guarantee a baseline level of control over the regularity of the vector fields involved in the later calculations. We will use |$*_r$| to denote the adjoint of an operator with respect to such a |$H^r$| metric.

We define a geodesic on |$\big ( \mathscr {G}^s(M), \langle \cdot , \cdot \rangle \big )$| to be a critical path for the energy functional induced by the metric |$\langle \cdot , \cdot \rangle $| and recall two important lemmas pertaining to geodesics on Lie groups with right-invariant metrics.

 

Lemma 3.3

(cf. [23, Theorem 3.2]).

If |$\mathscr {G}$| is a Lie group equipped with a (possibly weak) right-invariant metric |$\langle \cdot ,\cdot \rangle $|⁠, then a curve |$\sigma (t)$| is a geodesic if and only if the curve |$u(t)$| in the Lie algebra given by the flow equation:

$$ \begin{align*}& \dot{\sigma}(t) = d_{e}R_{\sigma(t)} u(t) \end{align*}$$

solves the Euler–Arnold equation:

$$ \begin{align}& \partial_t u (t) = - {\operatorname{ad}}^*_{u(t)}u(t). \end{align}$$

(5)

 

Lemma 3.4

(cf. [23, Corollary 3.3]).

If |$\sigma (t)$| is a curve in |$\mathscr {G}$| with |$u(t) = d_{\sigma (t)}R_{\sigma (t)^{-1}}(\dot {\sigma }(t))$| satisfying (5) with initial conditions |$\sigma (0) = e$| and |$u(0)=u_0$|⁠, then we have the following conservation law:

$$ \begin{align}& {\operatorname{Ad}}^{*}_{\sigma(t)}u(t) = u_0, \end{align}$$

(6)

and hence, we can rewrite the flow equation as

$$ \begin{align*}& \dot{\sigma}(t) = d_eR_{\sigma(t)} {\operatorname{Ad}}^{*}_{\sigma(t)^{-1}}u_0 =d_eL^{*}_{\sigma(t)^{-1}}u_0. \end{align*}$$

 

Remark 3.5.

It is important to note at this point that Lemmas 3.3 and 3.4 do not apply seamlessly to our setting of |$H^s$| Sobolev diffeomorphism groups. As mentioned earlier, |$\mathscr {D}^{s}(M)$| and its subgroups of interest are not Lie groups, on account of left translation only being continuous, etc. However, in any of the settings we consider in this paper, analogues of (5) and (6) will hold; cf. Section 3 of [23].

Notable examples of Euler–Arnold equations (5) in the setting of diffeomorphism groups include the incompressible Euler equations in two and three dimensions, Burgers’ equation, the Hunter–Saxton equation, the Camassa–Holm equation, the |$\mu $|CH equation, as well as the Euler-|$\alpha $| equations, and the symplectic Euler equations. The associated Riemannian geometry of these equations as well as the existence of a |$C^{\infty }$| exponential map and its properties has been studied extensively in the literature; cf. [9–13, 18, 20, 23, 25, 27, 28] and many others.

4 Main Results

Let |$\mathscr {G}^{s}$| be a group of diffeomorphisms of |$M$| equipped with a (weak) Riemannian metric |$\langle \cdot , \cdot \rangle $| and an associated exponential map where we assume that the results stated in Lemmas 3.3 and 3.4 hold. Our main focus is the following question: if a geodesic |$\gamma (t)$| emanating from the identity in |$\mathscr {G}^{s}$| at some later time |$t_0>0$| passes through |$\eta \in \mathscr {G}^{s+k}$|⁠, can it be shown that |$\gamma (t)$| evolves entirely in |$\mathscr {G}^{s+k}$|? For our purposes, it will be sufficient to assume that both |$t_0=1$| and |$k=1$|⁠. References for the various commutator estimates involved in the calculations include Kato and Ponce [19] and Taylor [29].

As mentioned in the introduction, previous results in this vein are due to Constantin and Kolev [9, 10] and Kappeler et al. [18], who worked with compressible equations on the circle and the 2-torus, respectively, and Bruveris [8]. At the heart of our method lie the various conservation laws motivated by Lemma 3.4.

In this paper, we will concern ourselves primarily with the flat case |$M = {\mathbb {T}}^n$|⁠. However, many of the constructions can be extended to the setting of curved spaces by collecting any lower-order terms arising in the various calculations due to derivatives of the components of the metric and its Christoffel symbols on |$M$| into a single term that will be negligible for our purposes. It should be noted that we do not believe our results are sharp. We require varying conditions on the parameter |$s$| due to the method of proof; all of which can be explained by Lemma 3.1.

We begin the groundwork for the main results. We aim to establish an explicit relationship between the regularity of the end configuration |$\eta $| and the regularity of the initial velocity |$u_0$|⁠. To do this, we will make use of (6), suitably modified to each case considered below.

 

Lemma 4.1.

Let |$s>\frac {n}{2}+3$|⁠, and let |$\gamma (t)$| be a smooth curve in |$\mathscr {G}^s({\mathbb {T}}^n)$| with |$v(t) = d_{\gamma (t)} R_{\gamma (t)^{-1}}\big ( \dot {\gamma }(t) \big ) $|⁠, |$\gamma (0) = e$| and |$\gamma (1)=\eta $|⁠. We have the following identity:

$$ \begin{align}& \Delta \eta = D\eta \int_{0}^{1} D\gamma(t)^{-1} (P_\lambda v)\circ \gamma (t) \ \text{d}t + G, \end{align}$$

(7)

where |$G = G(\gamma )$| is of class |$H^{s-1}$|⁠, |$\Delta \eta $| is defined by considering |$\eta : {\mathbb {T}}^n \rightarrow {\mathbb {T}}^n$| in coordinates and applying |$\Delta $| to each component and, for any |$\lambda \in {\mathbb {R}}$|⁠, |$P_\lambda $| is a differential operator acting component-wise given by

$$ \begin{align}& P_\lambda(t) = \lambda - \sum_{i,j=1}^{n}p_{ij}(t)\partial_i\partial_j \quad \text{with} \quad p_{ij}(t) = \big( D\gamma D\gamma^{\top} \big)_{ij} \circ \gamma^{-1}(t) \end{align}$$

(8)

with |$D\gamma ^{\top }$| denoting the pointwise adjoint of |$D\gamma $|⁠.

 

Proof.

Applying the Laplacian |$\Delta $| to the tangent vector to the curve |$\gamma (t)$|⁠, we get

$$ \begin{align*} \Delta \big(\frac{d \gamma}{dt}\big) &= \Delta(v\circ \gamma) = (Dv\circ \gamma ) \Delta \gamma - \lambda v \circ \gamma + \big(P_\lambda v\big) \circ \gamma. \end{align*}$$

Rearranging, we get

$$ \begin{align}& \begin{cases} \dfrac{d}{dt}\big(\Delta \gamma\big) - (Dv \circ \gamma)(\Delta \gamma) = (P_\lambda v ) \circ \gamma - \lambda v \circ \gamma\\ \Delta \gamma (0) = 0. \\ \end{cases} \end{align}$$

(9)

On the other hand, differentiating the flow equation |$d \gamma /d t = v \circ \gamma $| in the spatial variables gives

$$ \begin{align}& \begin{cases} \dfrac{d}{dt}(D\gamma) - Dv \circ \gamma D\gamma = 0 \\ D\gamma(0)= \textrm{Id}.\\ \end{cases} \end{align}$$

(10)

Using (10) and the Duhamel formula, we can now rewrite (9) as an integral equation in the form

$$ \begin{align*} & \Delta \gamma(t) = D\gamma(t) \int_{0}^{t} D\gamma(\tau)^{-1} (P_\lambda v )\circ \gamma (\tau) \ \text{d}\tau + G(t), \end{align*}$$

where

$$ \begin{align*} &G(t) = -\lambda D\gamma(t) \int_{0}^{t} D\gamma(\tau)^{-1}(v\circ \gamma)(\tau) \ \text{d}\tau\end{align*}$$

is a curve in |$H^{s-1}({\mathbb {T}}^n, \mathbb {R}^n)$|⁠. Evaluating at |$t=1$| and denoting |$G(1)$| by |$G$| we arrive at (7).

Suppose now that |$\gamma (t)$| is a geodesic of the metric |$\langle \cdot , \cdot \rangle $| in |$\mathscr {G}^s$| with |$\gamma (0)=e$| and |$\dot {\gamma }(0)=u_0 \in T_e\mathscr {G}^s$|⁠. Suppose further that at time |$t_0=1$|⁠, |$\gamma $| passes through |$\eta $|⁠. Since |${\operatorname {Ad}}^{-1}_{\gamma (t)} = D\gamma (t)^{-1} R_{\gamma (t)}$|⁠, using the conservation law (6) and Lemma 4.1, we may rewrite (7) as

$$ \begin{align}& \Delta \eta = D\eta \int_{0}^{1} {\operatorname{Ad}}_{\gamma(t)}^{-1} P_\lambda (t) \big({\operatorname{Ad}}^{-1}_{\gamma(t)}\big)^{*} u_0 \ \text{d}t + G, \end{align}$$

(11)

where * denotes the metric adjoint as before.

We can see from (11) that |$P_\lambda $| will play a central role in establishing a relationship between the regularity of |$\eta $| and our initial data |$u_0$|⁠. To this end, we establish that it defines a norm equivalent to the |$H^1$| norm for sufficiently large |$\lambda>0$|⁠.

 

Lemma 4.2.

There exists a |$\lambda> 0$| such that, for any |$t \in [0,1]$| and |$v \in H^{1}$|⁠, the operator |$P_\lambda $| satisfies the estimate

$$ \begin{align*}& \langle P_\lambda(t) v, \ v \rangle _{L^2} \ \simeq \left\lVert v\right\rVert _{H^1}^2 \end{align*}$$

with the constants depending on the curve |$\gamma : [0,1] \rightarrow \mathscr {D}^s_\mu ({\mathbb {T}}^n)$|⁠.

 

Proof.

We first derive an estimate for the coefficients |$p_{ij}$|⁠. For any |$w \in {\mathbb {R}}^n$|⁠, |$t \in [0, 1]$|⁠, and |$1 \leq i,j \leq n$|⁠, we have

$$ \begin{align*} \sum_{i,j = 1}^n p_{ij}(t) w_i w_j &= w^{\top} [D\gamma(t)][D\gamma(t)^{\top}] \circ \gamma^{-1}(t) w \\ &= \lvert D\gamma(t)^{\top} \circ \gamma^{-1} w\rvert^2. \end{align*}$$

As |$s>\frac {n}{2} + 3$|⁠, it follows from a compactness argument and the fact that |$D\gamma \circ \gamma ^{-1}$| is a linear isomorphism of |${\mathbb {R}}^n$| for all |$t\in [0, 1]$| that

$$ \begin{align}& \sum_{i,j = 1}^n p_{ij}(t) w_i w_j \ \simeq \vert w \vert^2. \end{align}$$

(12)

Integrating by parts and using estimate (12), we have

$$ \begin{align*} \langle P_\lambda(t) v, \ v \rangle _{L^2} &= \lambda \langle v, v \rangle_{L^2} - \sum_{i,j}\langle p_{ij}(t)\partial_i \partial_j v, v \rangle_{L^2} \\ &= \lambda \left\lVert v\right\rVert_{L^2}^2 + \frac{1}{2} \int_{{\mathbb{T}}^n}\sum_{i,j} \partial_i p_{ij}(t) \partial_j \vert v \vert^2 \ \text{d}x + \int_{{\mathbb{T}}^n} \sum_{i,j}p_{ij}(t) \partial_jv \partial_iv \ \text{d}x \\ &= \lambda \left\lVert v\right\rVert_{L^2}^2 - \frac{1}{2} \sum_{i,j}\langle \partial_j\partial_i p_{ij}(t) v, v \rangle_{L^2} + \int_{{\mathbb{T}}^n} \sum_{i,j}p_{ij}(t) \partial_jv \partial_iv \ \text{d}x \\ &\gtrsim \lambda \left\lVert v\right\rVert_{L^2}^2 - \frac{1}{2} \sum_{i,j}\left\lVert \partial_j\partial_i p_{ij}(t) v\right\rVert_{L^2}\left\lVert v\right\rVert_{L^2} + \int_{{\mathbb{T}}^n} \sum_{i} \vert \partial_iv_1 \vert^2 + \vert \partial_iv_2 \vert^2 \ \text{d}x \\ &\gtrsim \left\lVert v\right\rVert_{H^1}^2, \end{align*}$$

where the penultimate and last inequalities follow from the uniform boundedness in |$t \in [0, 1]$| of the coefficients |$\partial _i\partial _j p_{ij}$| and taking |$\lambda $| sufficiently large.

On the other hand, using (12) again, we have

$$ \begin{align*} \langle P_\lambda(t) v, \ v \rangle _{L^2} &\lesssim \lambda \left\lVert v\right\rVert_{L^2}^2 + \frac{1}{2} \sum_{i,j}\left\lVert \partial_j\partial_i p_{ij}(t) v\right\rVert_{L^2}\left\lVert v\right\rVert_{L^2} + \int_{{\mathbb{T}}^n} \sum_{i} \vert \partial_iv_1 \vert^2 + \vert \partial_iv_2 \vert^2 \ \text{d}x \\ & \lesssim \left\lVert v\right\rVert_{H^1}^2. \end{align*}$$

4.1 Diffeomorphisms of the flat |$n$|-torus |${\mathbb {T}}^n$|

We are now ready to present our 1st theorem, which generalises to |$n$| dimensions the corresponding results proved in [9, 10, 18]. Our method follows along the lines of the aforementioned. The main difference is the explicit use of the coadjoint operators and the conservation law (6), which shortens the argument.

 

Theorem 4.3.

Let |$r \geq 1$| be an integer, |$s>\frac {n}{2}+2r+5$|⁠, and consider the space of orientation preserving diffeomorphisms of |${\mathbb {T}}^n$| equipped with a right-invariant |$H^r$| metric (4). Given |$u_0 \in T_e\mathscr {D}^s$|⁠, let |$\gamma (t)$| denote the corresponding geodesic of the weak |$H^r$| metric. If at time |$t=1$|⁠, |$\gamma $| passes through a point |$\eta \in \mathscr {D}^{s+1}$|⁠, then we have |$u_0 \in T_e\mathscr {D}^{s+1}$| and consequently |$\gamma (t)$| evolves entirely in |$\mathscr {D}^{s+1}$|⁠.

Recall from [23, Section 3] that the group coadjoint on |$\mathscr {D}^s({\mathbb {T}}^n)$| equipped with a |$H^r$| metric has the form

$$ \begin{align*} &{\operatorname{Ad}}^{*_r} _\eta v = A^{-r}\big( \det(D\eta) D\eta^{\top} \big(A^r v \circ \eta \big)\big).\end{align*}$$

Hence, we have the following version of (6):

$$ \begin{align}& u(t) = \big({\operatorname{Ad}}_{\gamma(t)}^{-1}\big)^{*_r} u_0 = A^{-r} R_{\gamma(t)}^{-1} \big(\det\big(D\gamma(t)\big) D\gamma(t)^{\top}\big)^{-1} A^r u_0. \end{align}$$

(13)

 

Remark 4.4.

Later in our argument, we will require that, for each |$t \in [0,1]$|⁠, |$\big ({\operatorname {Ad}}_{\gamma (t)}^{-1}\big )^{*_r}: T_e\mathscr {D}^{s+1} \rightarrow T_e\mathscr {D}^{s+1}$| be a bounded invertible linear operator, so it is at this point that we can explicitly see the necessity of |$r\geq 1$|⁠. A simple derivative count shows that |$A^r$| must be at least of order |$2$| to prevent a loss of derivatives coming from the multiplication by the coefficients of the matrix |$D\gamma (t)$| that are a priori only of class |$H^{s-1}$| (recall that only the endpoints of |$\gamma (t)$| will be assumed to be of class |$H^{s+1}$|⁠).

The argument for Theorem 4.3 consists of three stages. We begin with a lemma that defines an operator |$M$| and establishes that it is a linear isomorphism from |$H^{s+1}$| to |$H^{s-1}$|⁠. We then combine (11) with some commutator estimates to acquire an expression of the form |$\Delta \eta = D \eta M u_0 + \widetilde {G}$|⁠, where |$\widetilde {G}$| is of class |$H^{s-1}$|⁠. We finish by concluding that, as |$Mu_0 = D \eta ^{-1}\big (\Delta \eta - \widetilde {G}\big ) \in H^{s-1}$|⁠, we have that |$u_0 \in H^{s+1}$|⁠. Hence, by the results for the initial value problem, we have that |$\gamma (t)$| evolves entirely in |$H^{s+1}$| and the proof of Theorem 4.3 will be complete. We proceed to the lemma.

 

Lemma 4.5.

The operator defined by

$$ \begin{align}& Mv = \int_{0}^{1} M_t v \ \text{d}t:= \int_{0}^{1} {\operatorname{Ad}}_{\gamma(t)}^{-1} A^{-\frac{r}{2}} P_\lambda (t) A^{\frac{r}{2}} \big({\operatorname{Ad}}^{-1}_{\gamma(t)}\big)^{{*_r}} v \ \text{d}t \end{align}$$

(14)

is a linear isomorphism from |$H^{s+1}$| to |$H^{s-1}$|⁠.

 

Proof.

We can see that, for |$r+1 \leq k \leq s+1$|⁠, |$M: H^{k} \rightarrow H^{k-2}$| is a bounded linear operator, as the integrand is a composition of bounded linear operators (with uniform bounds in |$t$|⁠). Consider the case |$k = r+1$|⁠, and define the bilinear form

$$ \begin{align*} & \Lambda: H^{r+1} \times H^{r+1} \rightarrow {\mathbb{R}} \quad ; \quad \Lambda(v,w) = \langle Mv, w \rangle_{H^r}. \end{align*}$$

It follows from the boundedness of |$M$| that |$\Lambda $| is bounded. Furthermore, by Lemma 4.2, we have

$$ \begin{align*} \Lambda(v,v) &= \int_0^{1} \Big\langle A^{-\frac{r}{2}} P_\lambda(t) A^{\frac{r}{2}} \big({\operatorname{Ad}}^{-1}_{\gamma(t)}\big)^{{*_r}} v \, \ \big({\operatorname{Ad}}^{-1}_{\gamma(t)}\big)^{{*_r}} v \Big\rangle_{H^r} \ \text{d}t \\ &= \int_0^{1} \Big\langle P_\lambda(t) A^{\frac{r}{2}} \big({\operatorname{Ad}}^{-1}_{\gamma(t)}\big)^{{*_r}} v \, \ A^{\frac{r}{2}} \big({\operatorname{Ad}}^{-1}_{\gamma(t)}\big)^{{*_r}} v \Big\rangle_{L^2} \ \text{d}t \\ &\gtrsim \int_0^{1} \left\lVert A^{\frac{r}{2}} \big({\operatorname{Ad}}^{-1}_{\gamma(t)}\big)^{{*_r}} v\right\rVert_{H^1}^2 \ \text{d}t \\ &= \int_0^1 \left\lVert \big({\operatorname{Ad}}^{-1}_{\gamma(t)}\big)^{{*_r}} v\right\rVert_{H^{r+1}}^2 \ \text{d}t \\ &\gtrsim \left\lVert v\right\rVert_{H^{r+1}}^2. \end{align*}$$

Hence, by the Lax–Milgram theorem, we have that |$M: H^{r+1} \rightarrow H^{r-1}$| is a linear isomorphism. We now proceed by induction. Assume |$M: H^{k} \rightarrow H^{k-2}$| is a linear isomorphism for some |$r+1 \leq k \leq s$|⁠, and let |$g \in H^{k-1}$|⁠. By the induction hypothesis, there exists |$f \in H^{k}$| such that |$Mf = g$|⁠. We claim that |$f$| is in fact of class |$H^{k+1}$|⁠. For any |$j = 1,...,n$|⁠, consider

$$ \begin{align*} \partial_j f &= \partial_j M^{-1}g \\ &= M^{-1} \partial_j g \ + \ [\partial_j, M^{-1}]g \\ &= M^{-1} \partial_j g \ + \ M^{-1}[M, \partial_j]M^{-1}g \\ &= M^{-1} \partial_j g \ + \ M^{-1}[M, \partial_j]f. \end{align*}$$

By assumption, the 1st term on the RHS is in |$H^{k}$|⁠. As for the commutator term, from (14), it suffices to show that for |$j = 1,...,n$| and |$r+1 \leq k \leq s$|⁠, |$[M_t, \partial _j]: H^{k} \rightarrow H^{k-2}$| is a bounded linear operator with the constant depending continuously on |$t$|⁠. This will give us that |$[M, \partial _j]: H^{k} \rightarrow H^{k-2}$| and hence |$\partial _j f \in H^{k}$|⁠.

First, for an operator |$B$|⁠, define |$C_{\gamma }( B ) = R_{\gamma } B R_{\gamma }^{-1}$|⁠, and its inverse |$\widetilde {C}_{\gamma }( B ) = R_{\gamma }^{-1} B R_{\gamma }$|⁠. Then, using (13), the fact that |$[A^{r}, \partial _j]=0$| and some standard commutator algebra, we have

$$ \begin{align*} \big[M_t, \partial_j\big] &= \big[D\gamma R_{\gamma} A^{-\frac{r}{2}} P_{\lambda} A^{-\frac{r}{2}} R_{\gamma}^{-1} \big(\det\big(D\gamma\big) D\gamma^{\top}\big)^{-1} A^r, \partial_j\big] \\ &= \big[D\gamma C_{\gamma} \big( A^{-\frac{r}{2}} P_{\lambda} A^{-\frac{r}{2}} \big) \big(\det\big(D\gamma\big) D\gamma^{\top}\big)^{-1}, \partial_j\big] A^r \\ &= \big[D\gamma, \partial_j\big] C_{\gamma} \big( A^{-\frac{r}{2}} P_{\lambda} A^{-\frac{r}{2}} \big) \big(\det\big(D\gamma\big) D\gamma^{\top}\big)^{-1} A^r \\ & \qquad + D\gamma C_\gamma(A^{-\frac{r}{2}} P_{\lambda} A^{-\frac{r}{2}}) \big[\big(\det\big(D\gamma\big) D\gamma^{\top}\big)^{-1}, \partial_j\big] A^r \\ & \qquad + D\gamma \big[C_\gamma(A^{-\frac{r}{2}} P_{\lambda} A^{-\frac{r}{2}}), \partial_j\big] \big(\det\big(D\gamma\big) D\gamma^{\top}\big)^{-1} A^r. \end{align*}$$

The 1st and 2nd term in the sum mapping to the correct space follows from the fact that, for |$\phi \in H^{s-1}$|⁠, we have |$\big [\phi ,\partial _j\big ]: H^{k-2} \rightarrow H^{k-2}$| and |$H^{k-2r} \rightarrow H^{k-2r}$| is a bounded linear operator, which is clear. All that remains is to show that |$\big [C_\gamma (A^{-\frac {r}{2}} P_{\lambda } A^{-\frac {r}{2}}), \partial _j\big ]: H^{k-2r} \rightarrow H^{k-2}$| continuously, which is equivalent to showing |$\big [A^{-\frac {r}{2}} P_{\lambda } A^{-\frac {r}{2}}, \widetilde {C}_\gamma (\partial _j)\big ] = \big [A^{-\frac {r}{2}} P_{\lambda } A^{-\frac {r}{2}}, \left (\partial _j\gamma ^i \circ \gamma ^{-1}\right ) \partial _i\big ]: H^{k-2r} \rightarrow H^{k-2}$| continuously. Observe we have

$$ \begin{align*} \big[A^{-\frac{r}{2}} P_{\lambda} A^{-\frac{r}{2}}, \left(\partial_j\gamma^i \circ \gamma^{-1}\right) \partial_i\big] &= A^{-\frac{r}{2}} \big[\left(\partial_j\gamma^i \circ \gamma^{-1}\right), A^{\frac{r}{2}}\big] \partial_i A^{-\frac{r}{2}} P_\lambda A^{-\frac{r}{2}} \\&\qquad\qquad+ A^{-\frac{r}{2}} \big[P_\lambda, \left(\partial_j\gamma^i \circ \gamma^{-1}\right) \partial_i\big] A^{-\frac{r}{2}} \\ & \qquad \qquad + A^{-\frac{r}{2}} P_\lambda A^{-\frac{r}{2}} \big[\left(\partial_j\gamma^i \circ \gamma^{-1}\right), A^{\frac{r}{2}}\big] \partial_i A^{-\frac{r}{2}} \end{align*}$$

and the result follows by observing that

• |$\big [P_\lambda , \left (\partial _j\gamma ^i \circ \gamma ^{-1}\right ) \partial _i\big ]$| is a 2nd-order differential operator with coefficients in |$H^{s-3}$|⁠;

• |$\big [\left (\partial _j\gamma ^i \circ \gamma ^{-1}\right ),A^{\frac {r}{2}}\big ]: H^{k-3} \rightarrow H^{k-r-2}$|⁠;

• |$\big [\left (\partial _j\gamma ^i \circ \gamma ^{-1}\right ),A^{\frac {r}{2}}\big ]: H^{k-r-1} \rightarrow H^{k-2r}$|⁠,

where the latter two follow from the Kato–Ponce [19] type commutator estimate, cf. [29],

$$ \begin{align}& \left\lVert A^{\frac{r}{2}}(fg) - fA^{\frac{r}{2}}g\right\rVert_{H^{s-r-1}} \lesssim \left\lVert \nabla f\right\rVert_{\infty} \left\lVert g\right\rVert_{H^{s-2}} + \left\lVert f\right\rVert_{H^{s-1}} \left\lVert g\right\rVert_{\infty}. \end{align}$$

(15)

Note now that all of the bounds established here depend on the operator norms of |$R_{\gamma (t)}$|⁠, |$R_{\gamma (t)}^{-1}$| and the multiplication operator for each |$\partial _j\gamma ^i(t)$|⁠, all of which are continuous in |$t$| in each of the topologies considered. Hence, |$[M, \partial _j]: H^{k} \rightarrow H^{k-2}$| and we have |$\partial _j f \in H^{k}$| for any |$j=1,...,n$|⁠, which implies that |$f \in H^{k+1}$|⁠. Hence, by the induction argument, |$M: H^{s+1} \rightarrow H^{s-1}$| is a linear isomorphism.

With this lemma in hand, we are ready to prove our main theorem for this section.

 

Proof of Theorem 4.3.

Recall (11):

$$ \begin{align*} \Delta \eta &= D\eta \int_{0}^{1} {\operatorname{Ad}}_{\gamma(t)}^{-1} P_\lambda (t) \big({\operatorname{Ad}}^{-1}_{\gamma(t)}\big)^{{*_r}} u_0 \ \text{d}t + G, \end{align*}$$

where again |$G$| is of class |$H^{s-1}$|⁠. Introducing a commutator term |$P_\lambda = A^{-\frac {r}{2}} P_\lambda (t) A^{\frac {r}{2}} + A^{-\frac {r}{2}} \big [ A^{\frac {r}{2}}, P_\lambda (t)\big ]$|⁠, we have

$$ \begin{align*} \Delta \eta &\!=\! D\eta\! \int_{0}^{1} \!{\operatorname{Ad}}_{\gamma(t)}^{-1} A^{-\frac{r}{2}} P_\lambda (t) A^{\frac{r}{2}} \big({\operatorname{Ad}}^{-1}_{\gamma(t)}\big)^{{*_r}} u_0 \ \text{d}t \!+\! D\eta\! \int_{0}^{1}\! {\operatorname{Ad}}_{\gamma(t)}^{-1} A^{-\frac{r}{2}} \big[ A^{\frac{r}{2}}, P_\lambda (t)\big] \big({\operatorname{Ad}}^{-1}_{\gamma(t)}\big)^{{*_r}} u_0 \ \text{d}t \!+\! G. \end{align*}$$

Notice now that |$\big [ A^{\frac {r}{2}}, P_\lambda (t)\big ] = \big [ p^{ij}, A^{\frac {r}{2}}\big ] \partial _i \partial _j$| and, as |$p^{ij}$| are of class |$H^{s-1}$|⁠, we have that |$\big [ A^{\frac {r}{2}}, P_\lambda (t)\big ]: H^{s} \rightarrow H^{s-r-1}$| by (15). Hence, the term |$D\eta \int _{0}^{1} {\operatorname {Ad}}_{\gamma (t)}^{-1} A^{-\frac {r}{2}} \big [ A^{\frac {r}{2}}, P_\lambda (t)\big ] \big ({\operatorname {Ad}}^{-1}_{\gamma (t)}\big )^{{*_r}} u_0 \ \text {d}t$| belongs to |$H^{s-1}$| and we rewrite

$$ \begin{align*} \Delta \eta &= D\eta \int_{0}^{1} {\operatorname{Ad}}_{\gamma(t)}^{-1} A^{-\frac{r}{2}} P_\lambda (t) A^{\frac{r}{2}} \big({\operatorname{Ad}}^{-1}_{\gamma(t)}\big)^{{*_r}} u_0 \ \text{d}t + \widetilde{G}, \end{align*}$$

where |$\widetilde {G}:= D\eta \int _{0}^{1} {\operatorname {Ad}}_{\gamma (t)}^{-1} A^{-\frac {r}{2}} \big [ A^{\frac {r}{2}}, P_\lambda (t)\big ] \big ({\operatorname {Ad}}^{-1}_{\gamma (t)}\big )^{{*_r}} u_0 \ \text {d}t + G$| is of class |$H^{s-1}$|⁠.

Finally, we have

$$ \begin{align*} \Delta \eta &= D\eta M (u_0) + \widetilde{G}, \end{align*}$$

which implies

$$ \begin{align*} u_0 = M^{-1} \big( D\eta^{-1} \big( \Delta \eta - \widetilde{G} \big) \big) \in H^{s+1}. \end{align*}$$

Hence, by Lemma 4.5, we have |$u_0 \in H^{s+1}$| and the proof of Theorem 4.3 is complete.

4.2 Volume-preserving diffeomorphisms of the flat 2-torus |${\mathbb {T}}^2$|

The main result presented in this section is as follows.

 

Theorem 1.1.

Let |$s> 6$|⁠, and consider the group of volume-preserving diffeomorphisms of |${\mathbb {T}}^2$| equipped with a right-invariant |$L^2$| metric. Given |$u_0 \in T_e\mathscr {D}^s_\mu $|⁠, let |$\gamma (t)$| denote the corresponding geodesic of the weak |$L^2$| metric. If at time |$t=1$|⁠, |$\gamma $| passes through a point |$\eta \in \mathscr {D}^{s+1}_\mu $|⁠, then we have |$u_0 \in T_e\mathscr {D}^{s+1}_\mu $| and consequently |$\gamma (t)$| evolves entirely in |$\mathscr {D}^{s+1}_\mu $|⁠.

For the proof, we make use of the conservation of vorticity in 2D, which takes the form

$$ \begin{align}& \text{rot} (u)\circ \gamma = \text{rot} (u_0), \end{align}$$

(16)

where |$u$| is the Eulerian velocity of the flow |$\gamma $|⁠; cf. (6). Applying the symplectic gradient |$\nabla ^{\perp }:= (-\partial _2, \partial _1)$| to both sides of (16), we obtain the following conservation law:

$$ \begin{align}& \Delta u = {\operatorname{Ad}}_{\gamma} \left( \Delta u_0 \right). \end{align}$$

(17)

Now, recall by the Hodge decomposition theorem (2) that we may decompose the tangent space |$T_e\mathscr {D}^s_\mu $| into an |$L^2$|-orthogonal sum:

$$ \begin{align*}& T_e\mathscr{D}^s_\mu = T_e\mathscr{D}^s_{\mu,ex} \oplus \mathcal{H}, \end{align*}$$

where |$T_e\mathscr {D}^s_{\mu ,ex} = g^\sharp \big (d \delta H^{s+2}(T^{*}{\mathbb {T}}^2)\big )$| is known as the space of exact volume-preserving vector fields and again |$\mathcal {H}$| denotes the space of harmonic vector fields. Hence, we can rewrite |$u_0 = \nabla ^{\perp } f_0 + h_0$|⁠, where |$f_0 \in H^{s+1}_0({\mathbb {T}}^2, {\mathbb {R}}):= \big \{ g \in H^{s+1}({\mathbb {T}}^2, {\mathbb {R}}) \ \vert \ \widehat {g}(0) = 0 \big \}$| and |$h_0 \in \mathcal {H}$|⁠. So we may reformulate (17) as follows.

 

Lemma 4.6.

For |$u_0 = \nabla ^{\perp } f_0 + h$|⁠, |$\gamma $| and |$u$| as above, we have

$$ \begin{align}& u(t) = \nabla^{\perp} \big( \Delta^{-1} R_{\gamma(t)}^{-1} \Delta f_0 \big) + h(t), \end{align}$$

(18)

where |$h(t)$| evolves in |$\mathcal {H}$| and |$\Delta ^{-1}: H^{s-1}_0({\mathbb {T}}^2, {\mathbb {R}}) \rightarrow H^{s+1}_0({\mathbb {T}}^2, {\mathbb {R}})$|⁠, defined in frequency space by

$$ \begin{align*} & \widehat{\Delta^{-1}f}(\xi):= \begin{cases} 0 & \textrm{if } \xi = (0,0) \\ \dfrac{\widehat{f}(\xi)}{\vert \xi \vert^2} & \textrm{if } \xi \in {\mathbb{Z}}^2 \setminus {(0,0)} \end{cases} \end{align*}$$

is a linear isomorphism.

 

Proof.

Using (17), we have

$$ \begin{align*} \Delta u &= {\operatorname{Ad}}_\gamma (\Delta u_0) \\ &= {\operatorname{Ad}}_\gamma \big(\Delta (\nabla^{\perp} f_0 + h)\big) \\ &= {\operatorname{Ad}}_\gamma(\nabla^{\perp} \Delta f_0) \\ & = \nabla^{\perp} (R_\gamma^{-1} \Delta f_0), \end{align*}$$

which, as |$\Delta $| acts component-wise on vector fields, immediately yields (18).

The argument for Theorem 1.1 consists of three stages. We begin with a lemma that defines an operator |$M$| and establishes that, for |$v \in H_0^{s-2}$|⁠, |$Mv$| being of class |$H^{s-1}$| implies that |$v$| is of class |$H_0^{s-1}$|⁠. We then combine (11) and (18), along with some commutator estimates to acquire an expression of the form |$\Delta \eta = D \eta M \nabla ^{\perp } \Delta f_0 + \widetilde {G}$|⁠, where |$\widetilde {G}$| is of class |$H^{s-1}$|⁠. We finish by concluding that, as |$M \nabla ^{\perp } \Delta f_0 = D \eta ^{-1}\big (\Delta \eta - \widetilde {G}\big ) \in H^{s-1}$|⁠, we have that |$f_0 \in H^{s+2}$|⁠, and hence |$u_0 \in H^{s+1}$|⁠. Then, by the results for the initial value problem, we have that |$\gamma (t)$| evolves entirely in |$H^{s+1}$| and the proof of Theorem 1.1 will be complete. We proceed with the lemma.

 

Lemma 4.7.

For |$0\leq k \leq s-1$|⁠,

$$ \begin{align}& Mv = \int_0^{1} M_t v \ \text{d}t:= \int_{0}^{1} R_{\gamma(t)} \Delta^{-1/2} \pi_0 P_\lambda(t) \Delta^{-1/2} R_{\gamma(t)}^{-1} v \ \text{d}t \end{align}$$

(19)

is a bounded linear operator from |$H^k_0$| to |$H^k$|⁠. Furthermore, if |$M v \in H^{k}$|⁠, then |$v \in H^k_0$|⁠.

 

Proof.

The boundedness of |$M$| on |$H^{k}_0$| for |$0\leq k \leq s-1$| follows from the uniform boundedness in |$t\in [0,1]$| of the operators comprising the integrand |$M_t$|⁠. Our goal is now to show that, for |$0 \leq k \leq s-1$|⁠, if |$M v \in H^{k}$|⁠, then |$v \in H^k_0$|⁠. We proceed by induction. For |$k=0$|⁠, we have from Lemma 4.2 and the |$L^2$|-orthogonality in the Hodge decomposition (2) that

$$ \begin{align*} \left\lVert M v\right\rVert_{L^2}\left\lVert v\right\rVert_{L^2} &\geq \big\vert \langle Mv, v \rangle_{L^2} \big\vert \\ &= \bigg\vert \int_0^1 \langle \pi_0 P_\lambda(t) \Delta^{-1/2} R_{\gamma(t)}^{-1}v \, \ \Delta^{-1/2} R_{\gamma(t)}^{-1}v \rangle_{L^2} \ \text{d}t \bigg\vert \\ &\gtrsim \bigg\vert \int_0^1 \left\lVert \Delta^{-1/2} R_{\gamma(t)}^{-1}v\right\rVert^2_{H^1} \ \text{d}t \bigg\vert \\ &\simeq \left\lVert v\right\rVert^2_{L^2}. \end{align*}$$

From which it follows that if |$Mv \in L^2 $|⁠, then |$v \in L^2$|⁠. Now assume the statement holds for some |$k$| with |$0 \leq k \leq s-2$| and |$Mv \in H^{k+1}$|⁠. By the inductive hypothesis, |$v \in H^{k}_0$|⁠. Furthermore, for |$j = 1, 2$|⁠, we have

$$ \begin{align*}& M\partial_j v = \partial_j M v + [M, \partial_j] v. \end{align*}$$

By assumption, |$\partial _j M v \in H^{k}$|⁠. As for the latter term, we show that, for |$0 \leq k \leq s-2$|⁠, |$[M_t, \partial _j]: H^k_0 \rightarrow H^k_0$| is a bounded linear operator with the constant depending continuously on |$t$|⁠. This will give us that |$[M, \partial _j]: H^k_0 \rightarrow H^k_0$| from which it will follow that |$M \partial _j v \in H^k$|⁠. Expanding yields

$$ \begin{align*} [M_t, \partial_j] &= C_\gamma \big( \big[ \Delta^{-1/2} \pi_0 P_\lambda \Delta^{-1/2}, \widetilde{C}_\gamma(\partial_j) \big] \big), \end{align*}$$

where again |$C_{\gamma }( B ):= R_{\gamma } B R_{\gamma }^{-1}$| and |$\widetilde {C}_{\gamma }( B ):= R_{\gamma }^{-1} B R_{\gamma }$|⁠. Examining the term inside the conjugation above, we have

$$ \begin{align*} \big[ \Delta^{-1/2} \pi_0 P_\lambda \Delta^{-1/2}, \widetilde{C}_\gamma(\partial_j)\big] &= \Delta^{-1/2} \pi_0 P_\lambda \Delta^{-1/2} \widetilde{C}_\gamma(\partial_j) - \widetilde{C}_\gamma(\partial_j) \Delta^{-1/2} \pi_0 P_\lambda \Delta^{-1/2} \\ &= \Delta^{-1/2} \pi_0 P_\lambda \Delta^{-1/2} \widetilde{C}_\gamma(\partial_j) - \Delta^{-1/2} \pi_0 P_\lambda \widetilde{C}_\gamma(\partial_j) \Delta^{-1/2} \\ & \qquad + \Delta^{-1/2} \pi_0 P_\lambda \widetilde{C}_\gamma(\partial_j) \Delta^{-1/2} - \widetilde{C}_\gamma(\partial_j) \Delta^{-1/2} \pi_0 P_\lambda \Delta^{-1/2} \\ &= \Delta^{-1/2} \pi_0 P_\lambda \big[ \Delta^{-1/2}, \widetilde{C}_\gamma(\partial_j)\big] + \big[ \Delta^{-1/2} \pi_0 P_\lambda, \widetilde{C}_\gamma(\partial_j) \big] \Delta^{-1/2} \\ &= \Delta^{-1/2} \pi_0 P_\lambda \Delta^{-1/2} \big[\widetilde{C}_\gamma(\partial_j), \Delta^{1/2}\big] \Delta^{-1/2} + \big[ \Delta^{-1/2} \pi_0 P_\lambda, \widetilde{C}_\gamma(\partial_j) \big] \Delta^{-1/2}. \end{align*}$$

The commutator estimate, cf. [29] or [18, Lemma B.3] (this lemma establishes a slightly different estimate, but the method of proof is the same),

$$ \begin{align}& \left\lVert \Delta^{1/2}(fg) - f\Delta^{1/2}g\right\rVert_{H^{k}} \lesssim \left\lVert \nabla f\right\rVert_{\infty}\left\lVert g\right\rVert_{H^{k}} + \left\lVert f\right\rVert_{H^{k+1}}\left\lVert g\right\rVert_{\infty} \end{align}$$

(20)

tells us that

$$ \begin{align*}& \big[\widetilde{C}_\gamma(\partial_j), \Delta^{1/2}\big] = \big[\left( \partial_j \gamma^i \circ \gamma^{-1} \right) \partial_i, \Delta^{1/2}\big] = \big[\partial_j \gamma^i \circ \gamma^{-1}, \Delta^{1/2}\big]\partial_i \end{align*}$$

is a bounded linear operator from |$H^{k+1}_0 \rightarrow H^{k}_0$| for any |$0\leq k \leq s-2$|⁠. Furthermore, we have

$$ \begin{align*} \big[ \Delta^{-1/2} \pi_0 P_\lambda, \widetilde{C}_\gamma(\partial_j) \big] &= \Delta^{-1/2} \pi_0 P_\lambda \widetilde{C}_\gamma(\partial_j) - \widetilde{C}_\gamma(\partial_j) \Delta^{-1/2} \pi_0 P_\lambda \\ &= \Delta^{-1/2} \pi_0 P_\lambda \widetilde{C}_\gamma(\partial_j) \!-\! \Delta^{-1/2} \pi_0 \widetilde{C}_\gamma(\partial_j) P_\lambda \!+\! \Delta^{-1/2} \pi_0 \widetilde{C}_\gamma(\partial_j) P_\lambda \!-\! \widetilde{C}_\gamma(\partial_j) \Delta^{-1/2} \pi_0 P_\lambda \\ &= \Delta^{-1/2} \pi_0 \big[P_\lambda, \widetilde{C}_\gamma(\partial_j)\big] + \big[\Delta^{-1/2}\pi_0, \widetilde{C}_\gamma(\partial_j)\big]P_\lambda. \end{align*}$$

The commutator term |$\big [P_\lambda , \widetilde {C}_\gamma (\partial _j)\big ]$| can be explicitly calculated to be a 2nd-order operator with |$H^{s-3}$| coefficients. As for |$\big [\Delta ^{-1/2}\pi _0, \widetilde {C}_\gamma (\partial _j)\big ]P_\lambda $|⁠, we compute

$$ \begin{align*} [\Delta^{-1/2}\pi_0, \widetilde{C}_\gamma(\partial_j)\big] &= \Delta^{-1/2} \pi_0 \widetilde{C}_\gamma(\partial_j) - \widetilde{C}_\gamma(\partial_j) \Delta^{-1/2} \pi_0 \\ &= \Delta^{-1/2} \pi_0 \widetilde{C}_\gamma(\partial_j)(\pi_0 + \pi_{\mathcal{H}}) - \widetilde{C}_\gamma(\partial_j) \Delta^{-1/2} \pi_0. \end{align*}$$

To proceed, we introduce the identity on |$H^{k}_0$| as |$\Delta ^{1/2} \Delta ^{-1/2}$| into the term involving the projection |$\pi _0$|⁠. So we have

$$ \begin{align*} [\Delta^{-1/2}\pi_0, \widetilde{C}_\gamma(\partial_j)\big] &= \Delta^{-1/2} \pi_0 \widetilde{C}_\gamma(\partial_j)\Delta^{1/2} \Delta^{-1/2}\pi_0 - \widetilde{C}_\gamma(\partial_j) \Delta^{-1/2} \pi_0 \\ &\qquad + \Delta^{-1/2} \pi_0 \widetilde{C}_\gamma(\partial_j) \pi_{\mathcal{H}} \\ &= \Delta^{-1/2} \big[ \widetilde{C}_\gamma(\partial_j), \Delta^{1/2}\big]\Delta^{-1/2}\pi_0 + \Delta^{-1/2} \pi_0 \widetilde{C}_\gamma(\partial_j) \pi_{\mathcal{H}}. \end{align*}$$

As before, |$\big [ \widetilde {C}_\gamma (\partial _j), \Delta ^{1/2}\big ]: H^{k+1}_0 \rightarrow H^{k}_0$| is a bounded linear operator. Note now that all of the bounds established here depend on the operator norms of |$R_{\gamma (t)}$|⁠, |$R_{\gamma (t)}^{-1}$| and the multiplication operator for each |$\partial _j\gamma ^i(t)$|⁠, all of which are continuous in |$t$| in each of the topologies considered. Hence, we have |$[M, \partial _j]: H^k_0 \rightarrow H^k_0$| for |$0 \leq k \leq s-2$| and we have shown that |$M\partial _j v = \partial _j M v + [M, \partial _j] v \in H^{k}$|⁠. So, by the inductive hypothesis, we have |$\partial _j v \in H^k_0$| for any |$j=1,2$| that gives us |$v \in H^{k+1}_0$|⁠. Therefore, we have that, for |$0 \leq k \leq s-1$|⁠, if |$M v \in H^{k}$|⁠, then |$v \in H^k_0$|⁠.

With this lemma in hand, we are now ready to prove our main theorem for this section.

 

Proof of Theorem 1.1.

Combining (11) and (18),

$$ \begin{align*} \Delta \eta &= D\eta \int_{0}^{1} D\gamma(t)^{-1} R_{\gamma(t)} P_\lambda(t) \nabla^{\perp} \big( \Delta^{-1} R_{\gamma(t)}^{-1} \Delta f_0 \big)\ \text{d}t + D\eta \int_{0}^{1} D\gamma(t)^{-1} R_{\gamma(t)} P_\lambda(t) h(t) \ \text{d}t + G, \end{align*}$$

where again |$G$| is of class |$H^{s-1}$|⁠. Some straightforward calculus now yields

$$ \begin{align*} \Delta \eta &= D\eta\! \int_{0}^{1} \!D\gamma(t)^{-1} R_{\gamma(t)} P_\lambda(t) \Delta^{-1} R_{\gamma(t)}^{-1} \big( D\gamma(t) \nabla^{\perp} \Delta f_0 \big) \ \text{d}t \!+\! D\eta\! \int_{0}^{1} \!D\gamma(t)^{-1} R_{\gamma(t)} P_\lambda(t) h(t) \ \text{d}t + G \\ &= D\eta \int_{0}^{1} R_{\gamma(t)} \Gamma^{-1}(t) P_\lambda(t) \Delta^{-1} \Gamma(t) R_{\gamma(t)}^{-1} \nabla^{\perp} \Delta f_0 \ \text{d}t + G, \end{align*}$$

where |$G$| has absorbed the term |$D\eta \int _{0}^{1} D\gamma (t)^{-1} R_{\gamma (t)} P_\lambda (t) h(t) \ \text {d}t$| and is still of class |$H^{s-1}$| and we denote |$\Gamma :=D\gamma \circ \gamma ^{-1}$| for notational simplicity. Next, we introduce commutator terms in order to achieve a more advantageous symmetry. We recall the Hodge projections:

$$ \begin{align*}& \pi_{0}: T_e\mathscr{D}^{\sigma} \rightarrow T_e\mathscr{D}^{\sigma}_{\mu,ex} \oplus \nabla H^{\sigma+1} \ \text{ and } \ \pi_{\mathcal{H}}: T_e\mathscr{D}^{\sigma} \rightarrow \mathcal{H}, \end{align*}$$

cf. (3) and rewrite

$$ \small{\begin{align*} \Delta \eta &= D\eta\! \int_{0}^{1}\! R_{\gamma(t)} \big( \Gamma^{-1}(t) P_\lambda(t) \Delta^{-1} \Gamma(t) \!+\! \Delta^{-1/2} \pi_{0} P_\lambda(t) \Delta^{-1/2} \!\!-\! \Delta^{-1/2} \pi_{0} P_\lambda(t) \Delta^{-1/2} \big) R_{\gamma(t)}^{-1} \nabla^{\perp} \Delta f_0 \ \text{d}t +\! G \\ &= D\eta \int_{0}^{1} R_{\gamma(t)} \Delta^{-1/2} \pi_{0} P_\lambda(t) \Delta^{-1/2} R_{\gamma(t)}^{-1} \big( \nabla^{\perp} \Delta f_0 \big) \ \text{d}t + \ D\eta \int_{0}^{1} R_{\gamma(t)} \Omega(t) R_{\gamma(t)}^{-1} \big( \nabla^{\perp} \Delta f_0 \big) \ \text{d}t + G, \end{align*}}$$

where

$$ \begin{align*} \Omega(t) &= \Gamma^{-1}(t) P_\lambda(t) \Delta^{-1} \Gamma(t) - \Delta^{-1/2} \pi_{0} P_\lambda(t) \Delta^{-1/2}. \end{align*}$$

We claim the integral involving |$\Omega (t)$| is of class |$H^{s-1}$| and hence can be absorbed into |$G$|⁠. Observe that |$\Gamma (t)$| and its inverse have entries in |$H^{s-1}$|⁠. We suppress |$t$| and continue

$$ \begin{align*} \Omega &= \Gamma^{-1} P_\lambda \Delta^{-1} \Gamma - \Delta^{-1/2} \pi_0 P_\lambda \Delta^{-1/2} \\ &= \Gamma^{-1} P_\lambda \Delta^{-1} \Gamma - P_\lambda \Delta^{-1} + P_\lambda \Delta^{-1} - \Delta^{-1/2} \pi_0 P_\lambda \Delta^{-1/2} \\ &= \Gamma^{-1} \big[ P_\lambda \Delta^{-1}, \Gamma \big] + \pi_0 P_\lambda \Delta^{-1} + \pi_{\mathcal{H}} P_\lambda \Delta^{-1} - \Delta^{-1/2} \pi_0 P_\lambda \Delta^{-1/2}. \end{align*}$$

As |$\pi _0 = \Delta ^{-1/2} \pi _0 \Delta ^{1/2}$|⁠, we have

$$ \begin{align*} \Omega &= \Gamma^{-1} \big[ P_\lambda \Delta^{-1}, \Gamma \big] + \Delta^{-1/2} \pi_0 \Delta^{1/2} P_\lambda \Delta^{-1} - \Delta^{-1/2} \pi_0 P_\lambda \Delta^{-1/2} + \pi_{\mathcal{H}} P_\lambda \Delta^{-1} \\ &= \Gamma^{-1} \big[ P_\lambda \Delta^{-1}, \Gamma \big] + \Delta^{-\frac{1}{2}} \pi_0 \big[ \Delta^{\frac{1}{2}}, P_\lambda \big] \Delta^{-1} + \pi_{\mathcal{H}} P_\lambda \Delta^{-1}. \end{align*}$$

Examining the terms separately, we have

$$ \begin{align*} \big[ P_\lambda \Delta^{-1}, \Gamma \big] &= \lambda \big[ \Delta^{-1}, \Gamma \big] - [p^{ij}\partial_i\partial_j\Delta^{-1}, \Gamma] \\ &= \lambda \big[ \Delta^{-1}, \Gamma \big] - [p^{ij} \Delta^{-1} \partial_i\partial_j, \Gamma] \\ &= \lambda \big[ \Delta^{-1}, \Gamma \big] - p^{ij} \Delta^{-1} \partial_i\partial_j \Gamma + \Gamma p^{ij} \Delta^{-1} \partial_i\partial_j. \end{align*}$$

Commuting |$p^{ij}$| with |$\Gamma $| and introducing commutator terms, we have

$$ \begin{align*} \big[ P_\lambda \Delta^{-1}, \Gamma \big] &= \lambda \big[ \Delta^{-1}, \Gamma \big] - p^{ij} \Delta^{-1} \partial_i\partial_j \Gamma + p^{ij} \Gamma \Delta^{-1} \partial_i\partial_j \\ &= \lambda \big[ \Delta^{-1}, \Gamma \big] - p^{ij} \Delta^{-1} \partial_i\partial_j \Gamma + p^{ij} \Delta^{-1} \Gamma \partial_i\partial_j - p^{ij} \Delta^{-1} \Gamma \partial_i\partial_j + p^{ij} \Gamma \Delta^{-1} \partial_i\partial_j \\ &= \lambda \big[ \Delta^{-1}, \Gamma \big] - p^{ij} \Delta^{-1} [\partial_i\partial_j, \Gamma] - p^{ij} [\Delta^{-1}, \Gamma] \partial_i\partial_j \\ &= \lambda \Delta^{-1} \big[\Gamma, \Delta \big] \Delta^{-1} - p^{ij} \Delta^{-1} [\partial_i\partial_j, \Gamma] - p^{ij} \Delta^{-1} \big[\Gamma, \Delta \big] \Delta^{-1} \partial_i\partial_j. \end{align*}$$

We can see by direct calculation of |$\big [\Gamma , \Delta \big ]$| and |$\big [\partial _i \partial _j, \Gamma \big ]$| that these terms map |$H^{s-2}$| to |$H^{s-3}$|⁠. As for the term |$\Delta ^{-\frac {1}{2}} \pi _0 \big [ \Delta ^{\frac {1}{2}}, P_\lambda \big ] \Delta ^{-1}$|⁠, we have

$$ \begin{align*} [\Delta^{\frac{1}{2}}, P_\lambda] &= [\Delta^{\frac{1}{2}}, p^{ij}]\partial_i\partial_j \end{align*}$$

and, as |$p^{ij}$| are of class |$H^{s-1}$|⁠, the fact that this maps |$H^{s}$| to |$H^{s-2}$| is a consequence of setting |$k=s-2$| in (20). Lastly, observe that the term |$\pi _{\mathcal {H}} P_\lambda \Delta ^{-1}$| maps |$H^{s-2} \rightarrow C^\infty $|⁠. Hence, we have

$$ \begin{align} \Delta \eta &= D\eta \int_{0}^{1} R_{\gamma(t)} \Delta^{-1/2} \pi_0 P_\lambda(t) \Delta^{-1/2} R_{\gamma(t)}^{-1} \big( \nabla^{\perp} \Delta f_0 \big) \ \text{d}t + \widetilde{G}, \end{align}$$

(21)

where now |$\widetilde {G} = G + D\eta \int _{0}^{1} R_{\gamma (t)} \Omega (t) R_{\gamma (t)}^{-1} \big ( \nabla ^{\perp } \Delta f_0 \big ) \ \text {d}t$| evolves in |$H^{s-1}$|⁠.

Notice now that (21) can be rearranged as

$$ \begin{align}& M(\nabla^{\perp} \Delta f_0) = D\eta^{-1} \big( \Delta \eta - \widetilde{G} \big) \in H^{s-1}({\mathbb{T}}^2, {\mathbb{R}}^2). \end{align}$$

(22)

Hence, from Lemma 4.7, we have

$$ \begin{align*}& \nabla^{\perp} \Delta f_0 \in H^{s-1}, \end{align*}$$

which implies that

$$ \begin{align*}& \nabla^{\perp} f_0 \in H^{s+1}, \end{align*}$$

which finally gives us that |$u_0 \in H^{s+1}$| and the proof is complete.

4.2.1 Higher-order Euler–Arnold equations

In this section, we equip the group of volumorphisms with a right-invariant |$H^r$| metric (4) induced by the inertia operator |$A^r$|⁠, for integer |$r\geq 1$|⁠, then (5) becomes

$$ \begin{align}& \begin{cases} \partial_t A^ru + \nabla_u A^r u + \big( \nabla u \big)^{\top}A^r u = -\nabla p \\ \text{div}(u) = 0.\\ \end{cases} \end{align}$$

(23)

The main result in this section is as follows.

 

Theorem 4.8.

Let |$r \geq 1$| be an integer, |$s> 2r + 6$|⁠, and consider the group of volume-preserving diffeomorphisms of |${\mathbb {T}}^2$| equipped with a right-invariant |$H^r$| metric (4). Given |$u_0 \in T_e\mathscr {D}^s_\mu $|⁠, let |$\gamma (t)$| denote the corresponding geodesic of the weak |$H^r$| metric. If at time |$t=1$|⁠, |$\gamma $| passes through a point |$\eta \in \mathscr {D}^{s+1}_\mu $|⁠, then we have |$u_0 \in T_e\mathscr {D}^{s+1}_\mu $| and consequently |$\gamma (t)$| evolves entirely in |$\mathscr {D}^{s+1}_\mu $|⁠.

Taking the curl of both sides of (23), we acquire a “conservation of vorticity”-type equation in this context.

$$ \begin{align}& \text{rot}(A^r u) \circ \gamma = \text{rot} (A^r u_0). \end{align}$$

(24)

This allows us to derive an analogous conservation law to (18).

 

Lemma 4.9.

For |$u_0 = \nabla ^{\perp } f_0 + h$|⁠, |$\gamma $| and |$u$| as above, we have

$$ \begin{align}& u(t) = \nabla^{\perp} \big( \Delta^{-1} A^{-r} R_{\gamma(t)}^{-1} A^r \Delta f_0 \big) + h(t), \end{align}$$

(25)

where |$h(t)$| evolves in |$\mathcal {H}$|⁠.

 

Proof.

Again, as in the |$L^2$| setting, applying the symplectic gradient |$\nabla ^{\perp }:=(-\partial _2, \partial _1)$| to both sides of (24) gives us

$$ \begin{align*}& A^r \Delta u = {\operatorname{Ad}}_\gamma A^r \Delta u_0, \end{align*}$$

which immediately yields (25).

 

Proof of Theorem 4.8.

Using Lemma 4.1 and (25), this follows from a completely analogous argument as in Theorem 1.1.

 

Remark 4.10.

If we define |$A = 1 - \alpha ^2 \Delta $|⁠, the above theorem covers the case of the Euler-|$\alpha $| equations studied in [15] and [27].

4.3 Swirl-free axisymmetric diffeomorphisms of the flat 3-torus |${\mathbb {T}}^3$|

In this section, we consider the group of axisymmetric diffeomorphisms of the flat periodic box |${\mathbb {T}}^3$|⁠, equipped with any of the Killing fields |$K = \frac {\partial }{\partial {x_i}}$| and a certain subclass of swirl-free initial data.

Given an axisymmetric vector field |$v$|⁠, we define its swirl to be the function |$g(v, K)$| on |${\mathbb {T}}^3$|⁠. A vector field is called swirl-free if this function identically vanishes. In our argument for this section, the swirl-free condition will deliver a conservation law that will play the same role that conservation of vorticity did in the previous section. It is shown in [21] that the swirl of an axisymmetric velocity field is transported by its flow. More precisely, if |$u_0 \in T_e\mathscr {A}_\mu ^s$| and |$\gamma (t)$| is the corresponding geodesic in |$\mathscr {A}_\mu ^s$|⁠, then |$g(u,K) \circ \gamma (t) = g(u_0,K)$| as long as it is defined. We denote the space of swirl-free axisymmetric vector fields by |$T_e\mathscr {A}_{\mu ,0}^s$|⁠. The proof of the following lemma can be found in [21].

 

Lemma 4.11.

Let |$K$| be any of the Killing fields |$\partial _i$| on |${\mathbb {T}}^3$|⁠. Then,

1. if |$v \in T_e\mathscr {A}_{\mu ,0}^{s+1}$|⁠, then |${\operatorname {curl}} v = \phi K$|⁠, where |$\phi $| is a function of class |$H^s$|⁠;

2. if |$u_0 \in T_e \mathscr {A}_{\mu ,0}^s$| and |$u(t)$| is the corresponding solution of the Euler equations (1), then, by the above, we can write |${\operatorname {curl}} u_0 = \phi _0 K$| and |${\operatorname {curl}} u(t,x) = \phi (t,x)K(x)$|⁠. The function |$\phi $| is transported along the flow lines: |$\phi (t,\gamma (t)) = \phi _0(x)$|⁠;

3. if |$u_0 \in T_e \mathscr {A}_{\mu ,0}^s$|⁠, then the corresponding solution |$u(t)$| of the Euler equations (1) can be extended globally in time;

4. if |$u_0$| and |$u(t)$| are as above, and |$\gamma (t) \in \mathscr {A}^{s}_{\mu ,0}$| is the flow of |$u(t)$|⁠, then |$K$| is preserved by the adjoint: |${\operatorname {Ad}}_{\gamma (t)}K = K$|⁠;

5. for |$v \in T_p{\mathbb {T}}^3$| with |$g\big (v, K(p)\big ) = 0$|⁠, we have |$g\big (D\gamma (t) v, K(\gamma (t,p))\big ) = 0$|⁠. We now proceed to the main result for this section.

 

Theorem 1.2.

Let |$s>\frac {13}{2}$|⁠, and consider the group of axisymmetric diffeomorphisms of |${\mathbb {T}}^3$| with respect to any of the Killing fields |$K=\partial _i$|⁠, equipped with a right invariant |$L^2$| metric. Given |$u_0 \in T_e\mathscr {A}^s_{\mu ,0}$|⁠, let |$\gamma (t)$| denote the corresponding geodesic of the weak |$L^2$| metric. If at time |$t=1$|⁠, |$\gamma $| passes through a point |$\eta \in \mathscr {A}^{s+1}_{\mu ,0}$|⁠, then we have |$u_0 \in T_e\mathscr {A}^{s+1}_{\mu , 0}$| and consequently |$\gamma (t)$| evolves entirely in |$\mathscr {A}^{s+1}_{\mu ,0}$|⁠.

Without loss of generality, we will assume, for the duration of the section, that |$K=\partial _3$|⁠. So, if we assume a vector field is axisymmetric and swirl-free, this is now equivalent to saying |$v = v_1\partial _1 + v_2 \partial _2$|⁠, where |$\partial _3v_1 = \partial _3v_2 = 0$|⁠. Proceeding as before, we establish a conservation law.

 

Lemma 4.12.

Let |$u(t)$| be the Eulerian velocity of the flow |$\gamma (t)$|⁠. Then we have

$$ \begin{align}& \Delta u (t) = {\operatorname{Ad}}_{\gamma(t)} \Delta u_0. \end{align}$$

(26)

 

Proof.

As |$u(t)$| evolves in |$T_e\mathscr {A}^s_{\mu , 0}$|⁠, we may write |$u(t) =u_1(t)\partial _1 + u_2(t)\partial _2$|⁠, where |$\partial _3u_1(t) = \partial _3u_2(t) = 0$|⁠. Hence, by taking the curl, we have

$$ \begin{align*} {\operatorname{curl}} u (t) &= \big(-\partial_2u_1(t) + \partial_1u_2(t)\big) \partial_3. \end{align*}$$

Comparing this with Lemma 4.11, we see |$\phi = -\partial _2u_1(t) + \partial _1u_2(t)$|⁠, and thus, as |$\phi $| is preserved along the flow lines (⁠|$\phi \circ \gamma = \phi _0$|⁠), we have as in 2D:

$$ \begin{align}& \big(-\partial_2u_1(t) + \partial_1u_2(t)\big) \circ \gamma = -\partial_2u_{0,1}(t) + \partial_1u_{0,2}(t). \end{align}$$

(27)

Since

$$ \begin{align*} &\Delta u_0 = \Delta u_{0,1} \partial_1 + \Delta u_{0,2} \partial_2,\end{align*}$$

$$ \begin{align*} &\Delta u(t) = \Delta u_{1}(t) \partial_1 + \Delta u_{2}(t) \partial_2,\end{align*}$$

and

$$ \begin{align*} &D \gamma(t) = \begin{bmatrix} \partial_1\gamma_1 & \partial_2\gamma_1 & 0\\ \partial_1\gamma_2 & \partial_2\gamma_2 & 0\\ 0 & 0 & 1 \end{bmatrix},\end{align*}$$

the lemma follows in an identical fashion to the computation for (17).

We proceed with the proof of Theorem 1.2.

 

Proof Theorem 1.2.

Using (7) and (26), this follows from a completely analogous argument as in Theorem 1.1.

4.4 Symplectomorphisms of the flat torus |${\mathbb {T}}^{2k}$|

In this section, we consider the group of symplectomorphisms of the torus |${\mathbb {T}}^{2k}$|⁠, equipped with the standard symplectic form |$\omega $|⁠. The main result presented in this section is the following.

 

Theorem 4.13.

Let |$s>k+5$|⁠, and consider the group of symplectomorphisms of |${\mathbb {T}}^{2k}$| equipped with a right-invariant |$L^2$| metric. Given |$u_0 \in T_e\mathscr {D}^s_\omega $|⁠, let |$\gamma (t)$| denote the corresponding geodesic of the weak |$L^2$| metric. If at time |$t=1$|⁠, |$\gamma $| passes through a point |$\eta \in \mathscr {D}^{s+1}_\omega $|⁠, then we have |$u_0 \in T_e\mathscr {D}^{s+1}_\omega $| and consequently |$\gamma (t)$| evolves entirely in |$\mathscr {D}^{s+1}_\omega $|⁠.

As before, we begin with a consequence of (6) and establish a new conservation law from it. The following can be found in [11].

$$ \begin{align}& \gamma^*\delta\omega^\flat u = \delta \omega^\flat u_0. \end{align}$$

(28)

Using this, we obtain the following proposition.

 

Proposition 4.14.

$$ \begin{align}& \Delta u = {\operatorname{Ad}}_\eta \Delta u_0 \end{align}$$

(29)

 

Proof.

Applying |$g^\sharp d$| to both sides of (28),

$$ \begin{align*} g^\sharp \eta^* \big( \Delta \omega^\flat u \big) &= g^\sharp \Delta \omega^\flat u_0\\ \Rightarrow D\eta^\top g^\sharp \omega^\flat \big( \Delta u \circ \eta \big) &= g^\sharp \omega^\flat \Delta u_0 \\ \Rightarrow D\eta^\top J \big( \Delta u \circ \eta \big) &= J \Delta u_0 \\ \Rightarrow \Delta u &= R_\eta^{-1} J^{-1} \big(D\eta^\top\big)^{-1} J \Delta u_0 \\ &= R_\eta^{-1} D\eta \Delta u_0 \\ &= {\operatorname{Ad}}_\eta \Delta u_0, \end{align*}$$

where, in the penultimate line, we have used the fact that |$D_\eta $| is a symplectic matrix.

 

Proof of Theorem 4.13.

This follows in a completely analogous fashion to the proof of Theorem 1.1, using the above conservation law (29).

5 The Frechét Setting

As mentioned in the introduction, to the best of the author’s knowledge, the earliest investigations into the kind of regularity property considered in this paper are due to Constantin and Kolev [9, 10] and Kappeler et al. [17, 18]. In each instance, the authors used this property to construct exponential maps that were local “|$C^1_F$|-diffeomorphisms” in the Frechét category. We obtain analogous results here for the settings we have considered. As in the above, we make use of a Nash–Moser-type inverse function theorem for Frechét spaces admitting Hilbert approximations; cf. [14] and [18, Theorem A.5]. Throughout this section, we again use |$\mathscr {G}^s$| to refer to any of the manifolds of mappings of Sobolev class |$H^s$| that we have considered (orientation-preserving, volume-preserving, etc.) and |$\mathscr {G}$| to refer to the corresponding Frechét manifolds of smooth mappings. First, we recall some basic definitions.

Let |$X$| and |$Y$| be Frechét spaces with |$U \subseteq X$| and |$V \subseteq Y$| open subsets. We say |$f: U \rightarrow Y$| is differentiable at |$u \in U$| in the direction |$x \in X$| if the limit

$$ \begin{align*}& \delta_{u}f(x):= \lim_{\varepsilon \to 0} \frac{f(u + \varepsilon x) - f(u)}{\varepsilon} \end{align*}$$

converges in |$Y$| with respect to the Frechét topology. If |$\delta _uf(x)$| exists for all |$(u,x) \in U \times X$| and the map given by

$$ \begin{align*} \begin{split} \delta f: \ &U \times X \rightarrow Y \\: \ &(u,x) \mapsto \delta_uf(x) \end{split} \end{align*}$$

is continuous with respect to the Frechét topologies on |$U \times X$| and |$Y$|⁠, then |$f$| is said to be continuously differentiable or |$C^{1}_F$| (it is important to note that this notion of continuously differentiable is weaker than the standard definition; cf. [14, Page 73]). A map |$f:U \rightarrow V$| is called a |$C^1_F$|-diffeomorphism if it is a homeomorphism where both |$f$| and |$f^{-1}$| are |$C^1_F$|⁠. We now state our version of the inverse function theorem.

 

Lemma 5.1.

Let |$s_0>\frac {n}{2}+1$|⁠, and assume we have a well-defined exponential map on |$\big (\mathscr {G}^{s_0}, \langle \cdot , \cdot \rangle \big )$|⁠:

$$ \begin{align*} &\exp^{s_0}: U^0 \rightarrow V^0,\end{align*}$$

which is a |$C^1$|-diffeomorphism, where |$U^0$| is a neighbourhood of |$0 \in T_e\mathscr {G}^{s_0}$| and |$V^{0}$| is a neighbourhood of |$e \in \mathscr {G}^{s_0}$|⁠, lying in the image of one of our chart maps for |$\mathscr {G}^{s_0}$|⁠. Define, for any |$k \in {\mathbb {Z}}_{\geq 0}$|⁠, |$s_k:=s_0+k$|⁠, |$U^k:= U^0 \cap T_e\mathscr {G}^{s_0 + k}$|⁠, |$U:= U^{0} \cap T_e\mathscr {G}$|⁠, |$V^k:= V^0 \cap \mathscr {G}^{s_0 + k}$|⁠, |$V:= V^{0} \cap \mathscr {G}$|⁠, and |$\exp ^{s_k}:= \exp ^{s_0}\big \vert _{U^k}$|⁠. If, for all |$k$|⁠, the following properties hold:

1. |$\exp ^{s_k}: U^k \rightarrow V^k$| is a bijective |$C^1$|-map;

2. for any |$u \in U$|⁠, |$d_u \exp ^{s_0}: T_e\mathscr {G}^{s_0} \rightarrow T_{\exp ^{s_0}(W)}\mathscr {G}^{s_0}$| is a linear isomorphism with the property that

   $$ \begin{align*} &d_u \exp^{s_0} (T_e\mathscr{G}^{s_k} \setminus T_e\mathscr{G}^{s_{k+1}}) \subseteq T_{\exp^{s_0}(u)}\mathscr{G}^{s_k}\setminus T_{\exp^{s_0}(u)}\mathscr{G}^{s_{k+1}}.\end{align*}$$

Then |$\exp := \exp ^{s_0}\big \vert _{U}: U \rightarrow V$| is a |$C^1_F$|-diffeomorphism.

 

Proof.

By |$(1)$|⁠, |$\exp : U \rightarrow V$| is a well-defined bijection. Examining the derivatives, we note that, for any |$u \in U$|⁠, |$d_u \exp ^{s_k} = d_u\exp ^{s_0} \big \vert _{T_e\mathscr {G}^{s_k}}$|⁠. Hence, by our assumptions, the map |$d_u \exp ^{s_k}: T_e \mathscr {G}^{s_k} \rightarrow T_{\exp ^{s_0}(u)}\mathscr {G}^{s_k}$| is a bounded linear bijection in the |$H^{s_k}$| topologies, and hence, by the open mapping theorem, is a linear isomorphism. Now, as |$U$| is a dense subset of each |$U^k$| in the |$H^{s_k}$| topology, by applying the inverse function theorem at each point |$u \in U$|⁠, we have |$\left ({\exp ^{s_k}}\right )^{-1}: V^k \rightarrow U^k$| is a |$C^1$| map in the |$H^{s_k}$| topologies.

So, for any |$k \in {\mathbb {Z}}_{\geq 0}$|⁠, |$u\in U$| and |$\xi \in V$|⁠, we have

$$ \begin{align*} &\exp = \exp^{s_0}\big\vert_{U} = \exp^{s_k}\big\vert_{U} \,\end{align*}$$

$$ \begin{align*} &\delta_u\exp = d_u \exp^{s_k} \big\vert_{T_e\mathscr{G}},\end{align*}$$

and

$$ \begin{align*} & \delta_\xi \big(\exp^{-1}\big) = d_\xi \big(\left({\exp^{s_k}}\right)^{-1}\big)\big\vert_{T_{\xi}\mathscr{G}}.\end{align*}$$

Hence, |$\delta \exp : U \times T_e\mathscr {G} \rightarrow TV$| and |$\delta \big ( \exp ^{-1} \big ): TV \rightarrow U \times T_e\mathscr {G}$| are continuous in the Frechét topologies.

Hence, |$\exp : U \rightarrow V$| is a |$C^{1}_F$| diffeomorphism.

We now state the main theorem for this section, which encompasses Theorems 1.3 and 1.4 from the introduction.

 

Theorem 5.2.

For each of the settings, we have considered in Section 4, we can construct an exponential map on a neighbourhood of |$0 \in T_e\mathscr {G}$| that is a |$C^1_F$| diffeomorphism onto its image.

 

Remark 5.3.

We will prove Theorem 5.2 for the case of |$\mathscr {D}_\mu ({\mathbb {T}}^2)$| equipped with the |$L^2$| metric. We note that this case is not a new result, cf. [27]; however, we have used a different method of proof. The analogous results in the other settings considered in Section 4 follow from an analogous argument, making use of the literature pertaining to Fredholmness of exponential maps on groups of diffeomorphisms; cf. [5–7, 21, 23]. We will use the notation defined in Lemma 5.1.

 

Proof.

Let |$s_0> 6$|⁠. From [12], we have that |$\mathscr {D}^{s_0}_\mu ({\mathbb {T}}^2)$| equipped with the |$L^2$| metric admits a well-defined exponential map that is a local |$C^\infty $| diffeomorphism at the identity |$\exp ^{s_0}: U^{s_0} \rightarrow V^{s_0}$|⁠. We may shrink |$U^{s_0}$| if necessary so that |$V^{s_0}$| lies in the image of a chart map for |$\mathscr {D}^{s_0}_\mu ({\mathbb {T}}^2)$|⁠.

We know from [12, Theorem 12.1] that, for all |$k \in {\mathbb {Z}}_{\geq 0}$|⁠, |$\exp ^{s_0}(U^k) \subseteq V^k$|⁠. Furthermore, uniqueness and smooth dependence of Lagrangian solutions on initial data in each |$H^{s_k}$| topology gives us that |$\exp ^{s_k}:= \exp ^{s_0}\big \vert _{U^k}: U^k \rightarrow V^k$| is a well-defined |$C^\infty $| injection. Theorem 1.1 now guarantees that, for all |$k\in {\mathbb {Z}}_{\geq 0}$|⁠, |$\exp ^{s^k}: U^k \rightarrow V^k$| is in fact a |$C^{\infty }$| bijection.

Next, from [13], we know that |$\exp ^{s_0}$| is a nonlinear Fredholm map of index zero. Hence, we can further restrict |$U^{s_0}$| if necessary to guarantee that, for all |$u \in U$|⁠, |$d_u \exp ^{s_0}: T_e \mathscr {D}_\mu ^{s_0}({\mathbb {T}}^2) \rightarrow T_{\exp ^{s_0}(u)} \mathscr {D}_\mu ^{s_0}({\mathbb {T}}^2)$| is a linear isomorphism. Furthermore, defining |$\eta :=\exp ^{s_0}(u) \in \mathscr {D}_\mu ({\mathbb {T}}^2)$|⁠, we in fact have

$$ \begin{align*}& d_u \exp^{s_0} = D \eta \big( \Omega_u - \Gamma_u \big), \end{align*}$$

where, for any |$k \in {\mathbb {Z}}_{\geq 0}$|⁠, |$D\eta : T_e\mathscr {D}_\mu ^{s_k}({\mathbb {T}}^2) \rightarrow T_\eta \mathscr {D}_\mu ^{s^k}({\mathbb {T}}^2)$|⁠, and |$\Omega _u: T_e\mathscr {D}_\mu ^{s_k}({\mathbb {T}}^2) \rightarrow T_e\mathscr {D}_\mu ^{s_k}({\mathbb {T}}^2)$| are linear isomorphisms and |$\Gamma _u: T_e\mathscr {D}_\mu ^{s_k}({\mathbb {T}}^2) \rightarrow T_e\mathscr {D}_\mu ^{s_{k+1}}({\mathbb {T}}^2) \subset T_e\mathscr {D}_\mu ^{s_k}({\mathbb {T}}^2)$| is a compact operator. So, if for |$w \in T_e \mathscr {D}_\mu ^{s_k}({\mathbb {T}}^2)$|⁠, |$d_u\exp ^{s_0}(w) \in T_\eta \mathscr {D}_\mu ^{s_{k+1}}(T^2)$|⁠, we have that |$w = \Omega _u^{-1}\bigg ( D\eta ^{-1}\big (d_u\exp ^{s_0}(w)\big ) + \Gamma _u(w) \bigg ) \in T_e\mathscr {D}_\mu ^{s_{k+1}}({\mathbb {T}}^2)$|⁠.

Hence, |$d_u\exp ^{s_0}(T_e\mathscr {D}_\mu ^{s_k}({\mathbb {T}}^2) \setminus T_e\mathscr {D}_\mu ^{s_{k+1}}({\mathbb {T}}^2)) \subseteq T_\eta \mathscr {D}_\mu ^{s_k}({\mathbb {T}}^2) \setminus T_\eta \mathscr {D}_\mu ^{s_{k+1}}({\mathbb {T}}^2)$| and we may apply Lemma 5.1.

 

Remark 5.4.

It is important to note that Theorem 5.2 does not follow immediately from the work of Ebin and Marsden [12]. While they define an exponential map for each Sobolev index |$s>\frac {n}{2}+1$|⁠, |$\exp ^s: \widetilde {U}^s \rightarrow \widetilde {V}^s$| and, indeed, their Theorem 12.1 ensures that each |$\exp ^s$| will map smooth initial data to a geodesic in |$\mathscr {D}_\mu ({\mathbb {T}}^2)$|⁠, they do so by applying the inverse function theorem in separately for each index. Hence, there is no a priori relationship between |$\widetilde {U}^s$| that guarantees that their intersection is not a single point; cf. [24, page 87].

Funding

This work was supported in part by the National University of Ireland Dr. Éamon de Valera Travelling Studentship in Mathematics.

Acknowledgments

The bulk of this work was completed during the author’s PhD studies at the University of Notre Dame. The author wishes to thank their advisor, Professor Gerard Misiołek, for introducing them to the problem and for many inspiring conversations.

Communicated by Prof. Jonatan Lenells