The Exponential Map Near Conjugate Points In 2D Hydrodynamics

Let $M$ be an $n$-dimensional compact Riemannian manifold. Suppose that $M$ is filled with an incompressible inviscid fluid whose particles cannot fuse or split. The Lagrangian approach to hydrodynamics aims at describing trajectories of these particles as they trace out a geodesic of a right-invariant metric defined by fluid’s kinetic energy in the group of Sobolev class diffeomorphisms $\mathscr{D}_{\mu}^{s}=\mathscr{D}_{\mu}^{s}(M)$ preserving the volume form $\mu$ (the volumorphism group). The metric is given at the identity diffeomorphism $e$ by the $L^{2}$ inner product

where $v,w\in T_{e}\mathscr{D}_{\mu}^{s}$ are divergence free Sobolev $H^{s}$ vector fields on $M$. This metric is only weak-Riemannian¹¹In the (complete) Hilbert case the metric is called strong-Riemannian [Ebin and Marsden1970]. in that the tangent spaces to $\mathscr{D}_{\mu}^{s}$ with the induced inner products are (incomplete) pre-Hilbert spaces. Its geodesics are the critical points of the associated $L^{2}$ length functional

and satisfy the equations

which rewritten on the Lie algebra of divergence-free vector fields become the Euler equations of hydrodynamics

where $p$ is the pressure function of the fluid, $v=\dot{\eta}\circ\eta^{-1}$ and $v\cdot\nabla$ denotes the covariant derivative on $M$ in the direction of $v$.

As in finite-dimensional differential geometry the weak-Riemannian metric on $\mathscr{D}_{\mu}^{s}$ defined by (1.1) has an exponential map however its global properties are not yet fully understood. Recall that two points $p$ and $q=\exp_{p}v$ along a geodesic are conjugate if the derivative $d\exp_{p}(v)$ fails to be an isomorphism. In infinite dimensions this may happen if either the kernel is non-empty or the range is not the whole target space. Accordingly, one has to distinguish mono-conjugate points and epi-conjugate points, respectively. Roughly speaking, the former are related to minimizing properties of geodesics while the latter to covering properties of the exponential map. What further complicates matters is the fact that conjugate points (of either type) can accumulate along finite geodesic segments or that their multiplicities can be of infinite order, see e.g. [Grossman1965].

However, even in finite dimensions singularities of Riemannian exponential maps are of a rather special kind. [Morse and Littauer1932] proved that a singularity of $d\exp_{p}(v)$ necessarily implies non-injectivity of the exponential map near the vector $v$ for any analytic Riemannian or Finslerian manifold. [Savage1943] reproved this result in the smooth case and [Warner1965] extended it to a more general class of maps. More recently, [Biliotti et al.2006] revisited it in the tooltip of strong-Riemannian Hilbert manifolds.

It turns out that there is a hydrodynamical analogue of the Morse-Littauer result. Namely, our main goal is to prove the following

In particular, we have the following simple hydrodynamical interpretation: every neighbourhood of $t_{c}v_{0}$ in $T_{e}\mathscr{D}_{\mu}^{s}$ contains a vector whose image under the exponential map corresponds to a configuration of a 2D fluid which can be reached from the initial state by two distinct fluid flows in the same amount of time.

An important fact used in the proof below is that the weak-Riemannian exponential map in 2D hydrodynamics is a Fredholm map of index zero. Combined with a hydrodynamical Morse Index theorem and a Schauder type invariance of domain result this fact allows us to apply classical Riemannian geometric constructions in infinite dimensions. In the last section we describe several open problems concerning the exponential map and its singularities.

The differential geometric framework for hydrodynamics outlined above was formulated by [Arnold1966]. Its functional analytic foundations were developed subsequently by [Ebin and Marsden1970]. For any $s>n/2+1$ the group $\mathscr{D}_{\mu}^{s}$ has the structure of an infinite-dimensional Hilbert manifold modelled on the Sobolev space of $H^{s}$ divergence-free vector fields on $M$ and the geodesic equation (1.3) of the $L^{2}$ metric is an ordinary differential equation on $\mathscr{D}_{\mu}^{s}$.

This fact provides an important technical advantage of the Lagrangian approach which allows to solve the equations for the particle trajectories (at least on short time intervals) using standard Picard iterations. Consequently, as in finite dimensions, it yields a smooth exponential map, the main object of our study, along with a smooth Levi-Civita connection $\nabla$ and its curvature tensor $\mathcal{R}$. Furthermore, by the inverse function theorem the exponential map is a local diffeomorphism in an open neighbourhood $\mathcal{U}$ of the origin in the tangent space at the identity

Motivated by the problems of hydrodynamic stability [Arnold1966] calculated sectional curvatures of (1.1) when $M$ is the flat 2-torus and showed that they can take on both signs. This led him to conjecture, by analogy with finite dimensions, that positive curvatures are related to the existence of conjugate points in the volumorphism group, see also [Arnold2004], [Arnold and Khesin1998]. Because the curvature tensor $\mathcal{R}$ of (1.1) is known to be a continuous multilinear operator in the $H^{s}$ topology, conjugate points in $\mathscr{D}_{\mu}^{s}$ can be studied using the equation of geodesic deviation and the Jacobi fields.²²For these results we refer to [Misiołek1993]. First examples of conjugate points in $\mathscr{D}_{\mu}^{s}$ were constructed in [Misiołek1993], [Misiołek1996] and further examples were given by [Shnirelman1994] and [Preston2006]. Epi-conjugate points that are not mono-conjugate and other pathological situations were shown to occur in 3D hydrodynamics, cf. [Ebin et al.2006], [Preston2006]. On the other hand the 2D case turns out to be entirely different.

In particular, observe that mono-conjugate points and epi-conjugate points along any $L^{2}$ geodesic in $\mathscr{D}_{\mu}^{s}$ must coincide. Note also that the result of [Wolibner1933] on global existence and uniqueness of solutions to the Euler equations in 2D implies that the exponential map can be defined on the whole tangent space $T_{e}\mathscr{D}_{\mu}^{s}$. In order to prove our Theorem we will need three more lemmas.

Next, we have the following result.³³Lemma 3 can also be proved using a result of A. Shnirelman on analyticity of the $L^{2}$ exponential map and decomposing $d\exp_{e}$ into a sum of an invertible and a compact operator as in [Ebin et al.2006]. See also Problem 2 below.

The last lemma we will need is the following extension of Schauder’s invariance of domain result.

Suppose to the contrary that the $L^{2}$ exponential map is injective in some neighbourhood of its conjugate vector $t_{c}v_{0}$ in $T_{e}\mathscr{D}_{\mu}^{s}$. Since it is a Fredholm map of index zero it then follows from Lemma 4 that $\exp_{e}$ is bijective (in fact a homeomorphism) on some open ball $B(\delta)$ of radius $\delta>0$ centered at $t_{c}v_{0}$ onto the open set $\mathcal{V}=\exp_{e}{B(\delta)}$. We will show that together with Lemma 2 this leads to a contradiction with the fact that $\eta(t)$ can be embedded in a family of curves of strictly shorter $L^{2}$ lengths. As before we assume that $\|v_{0}\|_{L^{2}}=1$.

To this end observe that since by Lemma 3 conjugate points along $\eta(t)$ are isolated we can choose $t_{1}$ and $t_{2}$ such that

for some $T>0$. Furthermore, we can arrange so that $\eta(t_{c})$ is the only conjugate point to $e$ in the segment $\eta([t_{1},t_{2}])\subset\mathcal{V}$. Let $Y$ be a Jacobi field along $\eta(t)$ with $Y(0)=Y(t_{c})=0$ and extend it to a broken Jacobi field by setting $Y(t)=0$ for $t_{c}\leq t\leq T$. Let $\varphi:\mathbb{R}\to\mathbb{R}$ be any smooth function such that

Recall that the Levi-Civita connection of the $L^{2}$ metric induces a parallel translation operator along curves in $\mathscr{D}_{\mu}^{s}$ which is an isometry of the tangent spaces with their pre-Hilbert structure (1.1) and commutes with $\nabla$, cf. [Misiołek1993]. Let $Z$ denote the parallel translation along $\eta(t)$ of the vector $-\nabla_{\dot{\eta}}Y(t_{c})\neq 0$ and for any $\varepsilon>0$ set $X_{\varepsilon}=Y+\varepsilon\varphi Z$. Then, a standard calculation of the $L^{2}$ index form using integration by parts gives

provided that $\varepsilon$ is chosen sufficiently small. Next, define a piecewise smooth variation of $\eta(t)$ by

for any $-\delta<s<\delta$. Note that the variation field of $\alpha(s,t)$ is precisely $X_{\varepsilon}$. Thus, choosing $\varepsilon>0$ as in (3.1) we find that

Observe that the equality in the second formula above holds because $\eta(t)$ has $L^{2}$ unit speed and the Jacobi field $Y$ and its covariant derivative are $L^{2}$ orthogonal to the geodesic (note that $Y$ vanishes at two distinct points). Consequently, the variation $\alpha(s,t)$ is length decreasing, that is

for sufficiently small $s$.

On the other hand observe that, by construction, $\exp_{e}$ is a bijection from the ball $B(\delta)$ onto $\mathcal{V}$. Therefore, for any such $s$ we can also write

where

Approximating $c_{s}(t)$ in $T_{e}\mathscr{D}_{\mu}^{s}$ by a smooth curve, if necessary, we conclude from Lemma 2 that

which contradicts (3.3) and proves the Theorem.

Since the $L^{2}$ exponential map on $\mathscr{D}_{\mu}^{s}$ is the (Lagrangian) solution map for the Euler equations (1.4) further in-depth studies of its analytic and geometric properties are certainly a matter of considerable interest. Let us first point out a possible alternative strategy which was suggested to us by P. Piccione.

It seems plausible that such an approach could in fact produce a stronger result which we can formulate as

A result of this type in the finite-dimensional case can be found in [Piccione et al.2004] while a more Riemannian-oriented strategy was described by B. Schmidt. Another possible approach when $M$ is real analytic would be to apply Shnirelman’s result [Shnirelman2012] in order to make use of the powerful methods of [Gohberg et al.1978] developed for holomorphic Fredholm operator-valued functions.

As mentioned above, Fredholmness of the exponential map suggests strong similarities with Riemannian exponential maps on finite dimensional manifolds. Since by Wolibner’s result [Wolibner1933] the volumorphism group $\mathscr{D}_{\mu}^{s}$ equipped with the $L^{2}$ metric is geodesically complete it is a natural question whether there is a hydrodynamical analogue of the classical Hopf–Rinow theorem. The most complete results on this to date can be found in [Shnirelman1994] and [Misiołek and Preston2010]. A related problem of independent interest is the following

Combined with the fact that the exponential map is Fredholm of index zero the answer to this question could help in determining whether $\exp_{e}$ is a covering map, see [Misiołek and Preston2010]. Some recent progress in this direction has been made in [Benn2014] in the case of certain stationary geodesics.

The failure of Fredholmness in 3D and the possible consequences that this may entail need to be well understood. Explicit examples where this happens can be found in [Ebin et al.2006] and [Preston2006]. In fact, it is known that this phenomenon is borderline in that the exponential map of any right-invariant $H^{r}$ metric on $\mathscr{D}_{\mu}^{s}(M^{3})$ with $r>0$ is Fredholm of index zero, cf. [Misiołek and Preston2010]. The proofs of Fredholmness in the above papers rely essentially on expressing the derivative of $\exp_{e}$ as a sum of an invertible and a compact operator. In the 3D case it may be possible to measure the deviation from compactness using the tools of spectral theory. A closer examination of explicit examples would be very useful.

The existing Fredholmness results were obtained for closed manifolds. A different sort of difficulty arises when the underlying manifold $M$ has a boundary. Even the situation in 2D has not yet been fully understood although it is known that in this case the kernel of $d\exp_{e}$ must be finite-dimensional, see [Ebin et al.2006], [Misiołek and Preston2010].

The result of this paper above can be viewed as a starting point for an in-depth study of the conjugate locus. This should be possible in certain special cases using the tools of singularity theory and the Fredholm property of the exponential map in 2D. For finite dimensional Riemannian manifolds such a study was carried out by Warner in his thesis [Warner1965] and may serve as a useful guide.