The Coadjoint Operator, Conjugate Points, and the Stability of Ideal Fluids

For a closed Riemannian manifold $M$ with metric $g$ and volume form $\mu$, let $\mathcal{D}_{\mu}^{s}(M)$ denote those diffeomorphisms of Sobolev class $H^{s}$ preserving $\mu$. When $s>\frac{\dim M}{2}+1$, $\mathcal{D}_{\mu}^{s}(M)$ becomes an infinite dimensional manifold whose tangent space at the identity $T_{e}\mathcal{D}_{\mu}^{s}(M)$ is given by $H^{s}$ divergence free vector fields on $M$. Using right translations, the $L^{2}$ inner-product

defines a weak, right-invariant Riemannian metric with smooth exponential map $\exp_{e}^{\mu}$, right-invariant Levi-Civita connection $\nabla^{\mu}$ and right invariant curvature tensor $R^{\mu}$, cf. Ebin and Marsden ([Ebin and Marsden1970]).

A curve $\eta(t)$ in the group $\mathcal{D}_{\mu}^{s}$ is a geodesic of the $L^{2}$ metric (1) if and only if the time dependent vector field $v\in T_{e}\mathcal{D}_{\mu}^{s}$, defined by $v=\dot{\eta}\circ\eta^{-1}$, solves the Euler equations of hydrodynamics

which describe the motion of a perfect incompressible fluid filling a compact, boundary-less domain $M$ with Levi-Civita connection $\nabla$ and $P^{\mu}$ the orthogonal projection onto the space of divergence free vector fields on $M$. We refer the reader to Arnold ([Arnold1966]), who first made this observation for the group of smooth volume-preserving diffeomorphisms, and [Ebin and Marsden1970] where this was made precise in the context of Sobolev $H^{s}$ diffeomorphisms. When $M$ is two dimensional it is well known that solutions of the Euler equations (1) exist for all time, see Wolibner ([Wolibner1933]). Consequently the manifold $\mathcal{D}_{\mu}^{s}$ is $L^{2}$ geodesically complete. However, when $M$ is three dimensional solutions of (1) are defined only on short time intervals, cf. ([Ebin and Marsden1970]), and the global existence problem is wide open. In this case, $L^{2}$ geodesics in $\mathcal{D}_{\mu}^{s}$ exist only locally in time. Geodesics in the group $\mathcal{D}_{\mu}^{s}$ describe the paths of fluid particles and this known as the Lagrangian formulation of hydrodynamics. The Euler equations (1) describe particle velocities and this is known as the Eulerian formulation of hydrodynamics. The battleground for the Euler equations (1) has not been the space of vector fields but rather its corresponding dual space; not the adjoint representation, but the coadjoint representation. The coadjoint representation is the representation that associates to each element $\eta$ of $\mathcal{D}_{\mu}^{s}(M)$ the linear operator $\mathrm{Ad}_{\eta}^{*}.$ The coadjoint orbit of a point $\xi$ in the dual is the set of all points $\mathrm{Ad}_{\eta}^{*}\xi$ ($\eta\in\mathcal{D}_{\mu}^{s}(M)$) in the dual. When $M$ is a closed surface the natural dual to the space of divergence free vector fields is the space of functions on $M$ with zero mean. The coadjoint representation is given by the natural action of diffeomorphisms, preserving the area form of $M$, on functions on $M$. When $M$ is three dimensional, the dual to the space of divergence free vector fields is the space of 1-forms on $M$. The adjoint and coadjoint representations are the standard actions of diffeomorphisms on the corresponding vector fields and 1-forms. We refer the reader to Arnold and Khesin ([Arnold and Khesin1998]) for a discussion and proof of these statements.

In this paper we give a new description of the coadjoint operator $\mathrm{Ad}_{\eta^{-1}(t)}^{*}$ along an $L^{2}$ geodesic $\eta(t)$ in $\mathcal{D}_{\mu}^{s}$. We first identify the space of divergence free vector fields with its weak $L^{2}$ dual through the metric (1). The adjoint representation is given by the usual pushforward of vector fields and the formal coadjoint representation is defined as the $L^{2}$ operator adjoint of the pushforward operation. In particular, we study the coadjoint operator defined on the space of divergence free vector fields; one can always transform the coadjoint representation in this space of vector fields back to the coadjoint representation defined on the dual. When $M$ is two dimensional the coadjoint operator $\mathrm{Ad}_{\eta^{-1}(t)}^{*}$ is given by the solution operator to the linearized Euler equations modulo a compact operator. When $M$ is three dimensional the coadjoint operator $\mathrm{Ad}_{\eta^{-1}(t)}^{*}$ is given by the solution operator to the linearized Euler equations plus a bounded linear operator. This is the content of Theorem 5 and sheds some light on the subtle difference between coadjoint orbits in two and three dimensional hydrodynamics.

We give two applications of this result when $M$ is a closed surface. The first solves the Jacobi equation explicitly and gives a new characterization of conjugate points. These results are contained in Proposition 6 and Theorem 7. Conjugate points are characterized completely in terms of the coadjoint operator and finding the conjugate locus is a purely algebraic question. The proofs make use of a result by Ebin, Misiołek, and Preston which states that the exponential map of the $L^{2}$ metric on $\mathcal{D}_{\mu}^{s}(M)$ is a non-linear Fredholm map of index zero, cf. ([Ebin et al.2006]). The second application concerns the stability of a stationary solution $v_{o}$ to the Euler equations (1) with corresponding stationary geodesic $\eta(t)$ in $\mathcal{D}_{\mu}^{s}$. Preston ([Preston2004]) demonstrated that if the $L^{2}$ norm of a solution $z(t)$, $z(0)=z_{o}$, to the linearized Euler equations (at $v_{o}$) remains bounded in time then the $L^{2}$ norm of the Jacobi field $J(t)$ along $\eta(t)$, with $J(0)=0$ and $J^{\prime}(0)=z_{o}$, grows at most quadratically in time. Also, if $v_{o}$ is analytic with isolated non-degenerate fixed points then $v_{o}$ is at most Lagrangian polynomial unstable if it is at most Eulerian polynomial unstable. Here we prove that the notions of Eulerian and Lagrangian stabilities of the Euler equations (1) are equivalent in that the $L^{2}$ norm of a Jacobi field remains bounded in time if and only if the $L^{2}$ norm of the corresponding linearized Euler solution remains bounded in time. We make no assumptions on the topology of $M^{2}$, nor on the qualitative behavior of the stationary solutions to (1). This completes the results of ([Preston2004]). Furthermore, we find that the instabilities present in the 2D Euler equations are contained and small whereas the instabilities in the 3D Euler equations seem to be much more wide-spread, see Remark 14.

In the next section we define the objects relevant to our discussion. Section 3 contains the main results, Proposition 4 and Theorem 5, while Sects. 4 and 5 contain the applications. We close with some concluding remarks in Sect. 6.

In what follows $M^{n}$ denotes a closed Riemannian manifold of dimension $n=2,3$. Define a map $g^{\flat}:TM^{n}\rightarrow T^{*}M^{n}$ which assigns to each vector field $v$ the one form $i_{v}g$, where $i_{v}$ denotes the interior product with $v$. This is an isomorphism from the tangent bundle to the cotangent bundle of $M^{n}$ with inverse defined as $g^{\sharp}:T^{*}M^{n}\rightarrow TM^{n}$ and corresponds to contracting with the inverse components of the metric tensor.

The space of all Sobolev $H^{s}$ vector fields on $M^{n}$ decomposes, according to the Hodge decomposition, as

where $d$ denotes the usual differential operator, $\delta$ its formal $L^{2}$ adjoint and $\mathcal{H}$ denotes the finite dimensional space of harmonic vector fields, cf. Morrey ([Morrey1966]). The set of divergence free vector fields is given by

and since the action of $\mathcal{D}_{\mu}^{s}$ on $\mathcal{D}^{s}$ by composition on the right is an isometry of (1) we obtain an $L^{2}$-orthogonal splitting in each tangent space

Denote by $P_{\eta}^{\mu}$ the orthogonal projection onto the first summand and the corresponding projection $Q_{\eta}^{\mu}=I-P_{\eta}^{\mu}$ onto the second. We refer the reader to [E-M] where it is shown that the projections depend smoothly on the base-point $\eta$.

For $\sigma\geq 0$, let $T_{e}\mathcal{D}_{\mu}^{\sigma}(M^{n})$ denote the closure of the space of exact divergence free vector fields in the $H^{\sigma}$ norm. By the Hodge decomposition this is a closed subspace in the space of all $H^{\sigma}$ vector fields, cf. ([Morrey1966]). For $\sigma>\frac{n}{2}+1$ this coincides with the actual tangent space to $\mathcal{D}_{\mu}^{\sigma}$, however, for smaller $\sigma$ the group $\mathcal{D}_{\mu}^{\sigma}$ is not necessarily a smooth manifold. In what follows, $s>\frac{n}{2}+1$ and $\sigma\leq s-1$.

In standard Lie group notation, the group adjoint representation is given by $\mathrm{Ad}_{\eta}=dL_{\eta}dR_{\eta^{-1}}$ for any $\eta\in\mathcal{D}_{\mu}^{s}$, where $dR_{\eta^{-1}}v=v\circ\eta^{-1}$ and $dL_{\eta}v=D\eta\cdot v$. Thus the adjoint representation is

which is the usual pushfoward operation on vector fields. Observe that right multiplication $dR_{\eta}$ is smooth whilst left multiplication $dL_{\eta}$ is only continuous. Therefore, if $\eta\in\mathcal{D}_{\mu}^{s}$ the adjoint representation is only defined on $T_{e}\mathcal{D}_{\mu}^{\sigma}$.

For a curve $\eta(s)\in\mathcal{D}_{\mu}^{s}$, with $\eta(0)=e$, $\dot{\eta}(0)=v\in T_{e}\mathcal{D}^{s}$, the differential of $\mathrm{Ad}_{\eta(s)}$ at the group identity defines the algebra adjoint representation $\mathrm{ad}_{v}$. The vector field $v$ is of class $C^{1}$, by the Sobolev embedding theorem, and $\mathrm{ad}_{v}$ is the unbounded operator from $T_{e}\mathcal{D}_{\mu}^{\sigma}$ to itself, which is given by the negative of the usual Lie bracket of vector fields:

cf. ([Arnold and Khesin1998]).

The formal $L^{2}$ adjoint of the operator (5) in $T_{e}\mathcal{D}_{\mu}^{0}$ is called the group coadjoint operator and is defined by

The algebra coadjoint operator in $T_{e}\mathcal{D}_{\mu}^{0}$ is defined by

Explicitly, we can compute that the group coadjoint operator is given by

where $\eta^{*}$ is the pullback operator on differential forms. Although $\mathrm{Ad}_{\eta}$ maps $T_{e}\mathcal{D}_{\mu}^{0}$ to itself, the $L^{2}$ adjoint does not, which is why we must compose with the projection $P_{e}^{\mu}$. Using the metric (1), the definition of $\mathrm{ad}_{v}$ and the definition of $\mathrm{ad}^{*}_{v}$ we compute that

Recall that the $L^{2}$ metric admits a smooth exponential mapping $\exp_{e}^{\mu}$ which is a local diffeomorphorphism in a nighbourhood of the identity, cf. ([Ebin and Marsden1970]). Consider a geodesic $\eta(t)=\exp_{e}^{\mu}(tv_{o})$ starting from the identity in the direction $v_{o}\in T_{e}\mathcal{D}_{\mu}^{s}$. The Jacobi field $J(t)$ along $\eta(t)$, with initial conditions $J(0)=0$ and $\dot{J}(0)=z_{o}\in T_{e}\mathcal{D}_{\mu}^{s}$, is defined by

and satisfies the Jacobi equation

The fact that $R^{\mu}$ is a bounded multi-linear operator implies that Jacobi fields exist, are unique and are defined on $T_{e}\mathcal{D}_{\mu}^{s}$ for as long as $\eta$ is, cf. ([Misiołek1993]). Denote the solution operator to the Jacobi equation (11) by $\Phi(t)$,

which is a bounded operator from $T_{e}\mathcal{D}_{\mu}^{s}$ to $T_{\eta(t)}\mathcal{D}_{\mu}^{s}$.

Although $v(t)$ is defined on $T_{e}\mathcal{D}_{\mu}^{s}$, Eqs. (13) and (13) are only defined on $T_{e}\mathcal{D}_{\mu}^{\sigma}$ due to the loss of derivatives in $v$. Equation (13) are the linearized Euler equations and we denote the solution operator to (13) by

which is a bounded operator from $T_{e}\mathcal{D}_{\mu}^{\sigma}$ to $T_{e}\mathcal{D}_{\mu}^{\sigma}$, for each $t$, and is defined for as long as $v(t)$ is defined.

For any $v\in T_{e}\mathcal{D}_{\mu}^{s}$ define an operator

which, using the metric (1), may be written explicitly as

The operator $K_{v}$ is bounded on $T_{e}\mathcal{D}_{\mu}^{\sigma}$ since $dg^{\flat}v$ is of class $H^{s-1}$ and $P_{e}^{\mu}$ is continuous. It is skew self adjoint since $dg^{\flat}v$ is antisymmetric.

The next Proposition holds only for two dimensional manifolds $M^{2}$.

The adjoint representation is given by the usual pushforward of vector fields and the formal coadjoint representation on $T_{e}\mathcal{D}_{\mu}^{0}(M)$ is defined as the $L^{2}$ operator adjoint of the pushforward operation, as in Sect. 2 above. One can always transform the coadjoint representation on $T_{e}\mathcal{D}_{\mu}^{0}(M)$ to the coadjoint representation on the dual spaces mentioned in the introduction. A volume form is enough to define the space of divergence free vector fields and its dual so that the coadjoint operator is an algebraic object and does not rely on the metric; here we are identifying the space of divergence free vector fields with its weak $L^{2}$ dual and considering the coadjoint operator defined on the space of divergence free vector fields. In the next Proposition we relate the coadjoint operator in the space of vector fields, which is algebraic, to solutions of the linearized Euler equations and Jacobi fields, which are Riemannian.

Here $M^{2}$ is a closed surface. Using Propositions 3 and 4 we will derive an explicit expression for Jacobi fields along $L^{2}$ geodesics in $\mathcal{D}_{\mu}^{s}$. A new characterization of conjugate points is given in terms of the coadjoint operator; thus, finding the conjugate locus is a purely algebraic question.

Let $\eta(t)$ be a geodesic of the $L^{2}$ metric (1) in $\mathcal{D}_{\mu}^{s}(M^{2})$. The point $\eta(t^{*})$ is said to be conjugate to $\eta(0)$, $t^{*}\in(0,t]$, if the linear operator

fails to be an isomorphism. If $\dim\ker D\exp_{e}^{\mu}(t^{*}v_{o})=k$ , $k$ is called the multiplicity of the conjugate point.

A linear operator between Hilbert spaces with empty kernel need not be an isomorphism. Therefore, $\eta(t^{*})$ may be a conjugate point even if $D\exp_{e}^{\mu}(t^{*}v_{o})$ has empty kernel. A point $\eta(t^{*})$ is monoconjugate to $e$ if $D\exp_{e}^{\mu}\left(t^{*}v_{o}\right)$ fails to be injective and a point $\eta(t^{*})$ is epiconjugate to $e$ if $D\exp_{e}^{\mu}\left(t^{*}v_{o}\right)$ fails to be surjective. If $M^{2}$ has positive curvature and enough symmetry then conjugate points can be located along those geodesics which are contained in the isometry subgroup of $\mathcal{D}_{\mu}^{s}(M^{2})$. It is also known that conjugate points exist in $\mathcal{D}_{\mu}^{s}(M^{2})$ even if $M^{2}$ is flat; see Misiołek ([Misiołek1996]) for the case where $M^{2}$ is the flat torus and Shnirelman ([Shnirelman1994]) for the ball in $\mathbb{R}^{3}$. It was shown in ([Ebin et al.2006]) that the exponential mapping $\exp_{e}^{\mu}$ is a non-linear Fredholm map of index zero. In particular, conjugate points in $\mathcal{D}_{\mu}^{s}(M^{2})$ are isolated, of finite multiplicity, and the two types of conjugacies coincide.

Let $\eta(t)$ be a geodesic with velocity field $v(t)\in T_{e}\mathcal{D}_{\mu}^{s}$, $v(0)=v_{o}$. For each $t$, the operator $K_{v(t)}$ is compact and skew self-adjoint on $T_{e}\mathcal{D}_{\mu}^{0}$, by Proposition 3. According to the spectral theorem for compact (skew) self-adjoint operators there exists an orthonormal basis of $T_{e}\mathcal{D}_{\mu}^{0}$ consisting of eigenvectors $\left\{\psi_{i}(t)\right\}$ of $K_{v(t)}$, and

Before proving Proposition 6 we state and prove the main result of this section.

The proof of Proposition 6 will be broken up into a series of Lemmas, all of which amount to keeping track of how the operators $\mathrm{Ad}_{\eta(t)}$ and $\mathrm{Ad}_{\eta(t)}^{*}$ act on the subspace $\ker K_{v_{o}}$. Let $\{\psi_{i}(t)\}$ be the time-dependent orthonormal basis of eigenvectors corresponding to $K_{v(t)}$ and spanning $T_{e}\mathcal{D}_{\mu,ex}^{0}$. Given a curve $x(t)$ in $T_{e}\mathcal{D}_{\mu,ex}^{0}$ we shall write $x^{\ker(t)}(t)$ for the part of $x(t)$ which lies in $\ker K_{v(t)}$, and $x^{\bot(t)}=x(t)-x^{\ker(t)}(t)$.

The solution operator $\Phi(t)$ to the Jacobi equation (11) is a non-linear Fredholm operator of index zero on $T_{e}\mathcal{D}_{\mu,ex}^{s}$, cf. ([Ebin et al.2006]). Any Fredholm operator decomposes as the sum of an invertible operator and a compact operator. The operator

is the invertible part of $\Phi(t)$ on $T_{e}\mathcal{D}_{\mu}^{s}$ and we will make use of this in the following.

As in Sect. 4 we will consider a closed surface $M^{2}$. We turn our attention to the stability of a stationary solution $v$ (and its corresponding stationary geodesic $\eta(t)$ in $\mathcal{D}_{\mu}^{s}(M^{2})$) to the Euler equations (1).

Let $k\geq 0$. A geodesic $\eta\in\mathcal{D}_{\mu}^{s}(M^{2})$ is said to be Lagrangian $H^{k}$ (linearly) stable if every solution of the Jacobi equation (11) along $\eta$, remains bounded in the Sobolev $H^{k}$ norm. Intuitively, a geodesic is Lagrangian stable if configurations of particles which are initially nearby remain nearby.

We also have the notion of Eulerian stability, in which the initial velocity of a solution to the Euler equations (1) is perturbed. A stationary solution $v$ to the Euler equations is said to be Eulerian $H^{k}$ (linearly) stable if every solution of the linearized Euler equations (13) remains bounded in the Sobolev $H^{k}$ norm. Intuitively, a solution is Eulerian stable if nearby velocity fields remain nearby.

In [Preston2004] it was shown that if a solution to the linearized Euler equations (13) (at $v$) remains bounded in time then the corresponding Jacobi field along $\eta(t)$ grows at most quadratically in time. Also, if $v$ is analytic with isolated non-degenerate fixed points, then $v$ is at most Lagrangian polynomial unstable if it is at most Eulerian polynomial unstable. We show more; that is, a solution to the linearized Euler equation remains bounded in time if and only if the corresponding Jacobi field remains bounded in time, and if a Jacobi field is unbounded in time then the corresponding solution to the linearized Euler equation is also unbounded and grows at the same rate. This holds for any stationary $H^{s}$ solution to the Euler equations and proves that the notions of Eulerian and Lagrangian stabilities of the Euler equations (1) are equivalent, which completes the results of [Preston2004].

Recall that the set of divergence free vector fields is given by

where the second summand is the finite dimensional space of harmonic vector fields and the first the set of rotated gradients; that is, vector fields of the form $v_{F}=\nabla^{\bot}F$ for some $H^{s+1}$ function $F$ with zero mean on $M^{2}$. $T_{e}\mathcal{D}_{\mu,ex}^{s}:=g^{\sharp}\left(d\delta H^{s+2}(T^{*}M^{2})\right)$ is known as the set of exact divergence free vector fields and is a closed subspace of finite codimension in $T_{e}\mathcal{D}_{\mu}^{s}$. The Lie bracket of any two exact divergence free vector fields is again an exact divergence free vector field and therefore we can restrict the adjoint and coadjoint operators to this space.

Following Smolentsev ([Smolentsev1986]), define another inner product on $T_{e}\mathcal{D}_{\mu,ex}^{s}$ by

The weak $L^{2}$ metric (1) on vector fields is related to the inner product (26) by

where $-\triangle$ is the Hodge Laplacian.

We now present two simple examples illustrating Theorem 13.

which differ in growth by a linear factor of $t$. To see that this agrees with the formula given in 2. of the Theorem, observe that $z(t)=v_{G}$ with $G(x,y)=-\cos(x)$, and $G$ is an eigenfunction of the laplacian with the property $G\circ\eta(t)=G$. Therefore $v_{G}$ is invariant under the adjoint and coadjoint action and

Let $M$ be a closed symplectic manifold of dimension $2n$. It has been shown by Ebin in ([Ebin2012]) that the group of Sobolev symplectic diffeomorphisms $\mathcal{D}_{\omega}^{s}(M)$, equipped with the $L^{2}$ metric (1), is geodesically complete and the corresponding symplectic Euler equations on $M$ are globally well-posed for $s>\frac{\dim M}{2}+1$. The subgroup of Hamiltonian diffeomorphisms plays a role in plasma dynamics analogous to the role played by the volume preserving diffeomorphism group in incompressible hydrodynamics, cf. [Arnold and Khesin1998]. It has also been shown in [Benn2016] that  3 remains true for any closed Symplectic manifold of dimension $2n$ and the exponential map of the $L^{2}$ metric on $\mathcal{D}_{\omega}^{s}(M)$ is a non-linear Fredholm map of index zero. Consequenty, all results derived in this paper hold true for the symplectomorphism group $\mathcal{D}_{\omega}^{s}(M)$. Theorem 7 also provides an interesting contrast to the Hofer metric which is a bi-invariant metric on the group of symplectomorphisms whose conjugate points do not have such a nice description, see Ustilovsky ([Ustilovsky1996]).