Group Of Volume-preserving Diffeomorphisms Research Articles

Arnold stability and Misiołek curvature

Let M be a compact 2-dimensional Riemannian manifold with smooth boundary and consider the incompressible Euler equation on M. In the case that M is the straight periodic channel, the annulus or the disc with the Euclidean metric, it was proved by T. D. Drivas, G. Misiołek, B. Shi, and the second author that all Arnold stable solutions have no conjugate point on the volume-preserving diffeomorphism group $${{\mathcal {D}}}_{\mu }^{s}(M)$$ . They also proposed a question which asks whether this is true or not for any M. In this article, we give a partial positive answer. More precisely, we show that the Misiołek curvature of any Arnold stable solution is nonpositive. The positivity of the Misiołek curvature is a sufficient condition for the existence of a conjugate point.

Нахождение симметрий для задачи о волнах на воде с поверхностным натяжением

T. Brooke Benjamin and P. J. Olver “Hamiltonian structure, symmet­ries and conservation laws for water waves” study the behavior of Hamil­to­nian systems with an infinite-dimensional phase space. The methods of va­riational problems and infinite-dimensional differential geometry are applicable to this problem. A special case of the problem is an abstract prob­lem of hydrodynamics for an ideal fluid. Its configuration space is the group of volume-preserving diffeomorphisms of some manifold in or filled with fluid. Even more special is the problem of waves on water. Its non-standard nature is due to the presence of boundary con­di­tions on the free surface. These boundary conditions can be interpreted in terms of the functional derivatives of the energy integral, which plays the role of the Hamiltonian. Here we consider in detail the case of this prob­lem in R2, taking into account surface tension, and find symmetries for it, which was not considered in detail in the article. Finding symmet­ries can be achieved without recourse to the Hamiltonian structure of the gi­ven problem.

Conjugate points in [formula omitted

Rossby-Haurwitz waves on the sphere S2 form a set of exact time-dependent solutions to the Euler equations of hydrodynamics and generate a family of non-stationary geodesics of the L2 metric in the volume preserving diffeomorphism group of S2. Restricting to a particular subset of Rossby-Haurwitz waves, this article shows that under certain conditions on the physical characteristics of the waves each corresponding geodesic contains conjugate points. In addition, a physical interpretation of conjugate points is given and links the result to the stability analysis of meteorological Rossby-Haurwitz waves.

Hofer-Like Geometry and Flux Theory

This paper meticulously revisit and study the flux geometry of any compact connected oriented manifold (M, Ω). We generalize several well- known factorization results, exhibit some orbital conditions under which flux geometry can be studied, give a proof of the discreteness of the flux group for volume-preserving diffeomorphisms, derive that any smooth isotopy in the group of all vanishing-flux volume-preserving diffeomorphisms is a vanishing- flux path, and show that the kernel of flux for volume-preserving diffeomorphisms is C 1−closed inside the group of all volume-preserving diffeomorphisms isotopic to the identity map: We recover several well-known results from symplectic geometry. We use the above studies to construct a right-invariant metric on the group of all volume-preserving diffeomorphisms isotopic to the identity map and study the induced geometry. In the case of a symplectic volume form, the restriction of our metric to the group Ham(N, ω), of all Hamiltonian diffeomorphisms of a closed symplectic manifold (N, ω), is controlled from above by the usual Hofer metric in general, while the Hofer-like metric control our metric in the case where the Riemannian structure is compatible with the symplectic structure (in particular, our construction implies the non-degeneracy of the Hofer and Hofer-like norms).

The helicity uniqueness conjecture in 3d hydrodynamics

We prove that the helicity is the only regular Casimir function for the coadjoint action of the volume-preserving diffeomorphism group $\text{SDiff}(M)$ on smooth exact divergence-free vector fields on a closed three-dimensional manifold $M$. More precisely, any regular $C^1$ functional defined on the space of $C^\infty$ (more generally, $C^k$, $k\ge 4$) exact divergence-free vector fields and invariant under arbitrary volume-preserving diffeomorphisms can be expressed as a $C^1$ function of the helicity. This gives a complete description of Casimirs for adjoint and coadjoint actions of $\text{SDiff}(M)$ in 3D and completes the proof of Arnold-Khesin's 1998 conjecture for a manifold $M$ with trivial first homology group. Our proofs make use of different tools from the theory of dynamical systems, including normal forms for divergence-free vector fields, the Poincar\'e-Birkhoff theorem, and a division lemma for vector fields with hyperbolic zeros.

Positivity for the curvature of the diffeomorphism group corresponding to the incompressible Euler equation with Coriolis force

Abstract We investigate the geometry of the central extension $\widehat{\mathcal D}_{\mu}(S^{2})$ of the group of volume-preserving diffeomorphisms of the 2-sphere equipped with an $L^{2}$-metric, for which geodesics correspond to solutions of the incompressible Euler equation with Coriolis force. In particular, we calculate the Misiołek curvature of this group. This value is related to the existence of a conjugate point and its positivity directly implies the positivity of the sectional curvature.

Existence and characterisation of magnetic energy minimisers on oriented, compact Riemannian 3-manifolds with boundary in arbitrary helicity classes

In this paper we deal with the existence, regularity and Beltrami field property of magnetic energy minimisers under a helicity constraint. We in particular tackle the problem of characterising local as well as global minimisers of the given minimisation problem. Further we generalise Arnold’s results concerning the problem of finding the minimum magnetic energy in an orbit of the group of volume-preserving diffeomorphisms to the setting of abstract manifolds with boundary.

Axisymmetric Diffeomorphisms and Ideal Fluids on Riemannian 3-Manifolds

Abstract We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove that the $L^2$ exponential map on the group of volume-preserving diffeomorphisms of a $3$-manifold is Fredholm along axisymmetric flows with sufficiently small swirl. Along the way, we define the notions of axisymmetric and swirl-free diffeomorphisms of any manifold with suitable symmetries and show that such diffeomorphisms form a totally geodesic submanifold of infinite $L^2$ diameter inside the space of volume-preserving diffeomorphisms whose diameter is known to be finite. As examples, we derive the axisymmetric Euler equations on $3$-manifolds equipped with each of Thurston’s eight model geometries.

A Note on the Stability of Geodesics on Diffeomorphism Groups with One-side Invariant Metric

In this note, we consider the stability of geodesics on volume-preserving diffeomorphism groups with one-side invariant metric. We showed that for non-Beltrami fields on a three-dimensional compact manifold, there does not exist Eulerian stable flow which is Lagrangian exponential unstable. We noticed that a stationary flow corresponding to the KdV equation can be Eulerian stable while the corresponding motion of the fluid is at most exponentially unstable.

A Basis of Casimirs in 3D Magnetohydrodynamics

Abstract We prove that any regular Casimir in 3D magnetohydrodynamics (MHD) is a function of the magnetic helicity and cross-helicity. In other words, these two helicities are the only independent regular integral invariants of the coadjoint action of the MHD group $\textrm{SDiff}(M)\ltimes \mathfrak X^*(M)$, which is the semidirect product of the group of volume-preserving diffeomorphisms and the dual space of its Lie algebra.

Least action principles for incompressible flows and geodesics between shapes

As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an Eulerian framework, we study incompressible flows of shapes as critical paths for action (kinetic energy) along transport paths constrained to have characteristic-function densities. The formal geodesic equations for this problem are Euler equations for incompressible, inviscid potential flow of fluid with zero pressure and surface tension on the free boundary. The problem of minimizing this action exhibits an instability associated with microdroplet formation, with the following outcomes: any two shapes of equal volume can be approximately connected by an Euler spray—a countable superposition of ellipsoidal geodesics. The infimum of the action is the Wasserstein distance squared, and is almost never attained except in dimension 1. Every Wasserstein geodesic between bounded densities of compact support provides a solution of the (compressible) pressureless Euler system that is a weak limit of (incompressible) Euler sprays.

Geodesic motion on groups of diffeomorphisms with H1 metric as geometric generalised Lagrangian mean theory

ABSTRACTGeneralized Lagrangian mean theories are used to analyse the interactions between mean flows and fluctuations, where the decomposition is based on a Lagrangian description of the flow. A systematic geometric framework was recently developed by Gilbert and Vanneste who cast the decomposition in terms of intrinsic operations on the group of volume preserving diffeomorphisms or on the full diffeomorphism group. In this setting, the mean of an ensemble of maps can be defined as the Riemannian centre of mass on either of these groups. We apply this decomposition in the context of Lagrangian averaging where equations of motion for the mean flow arise via a variational principle from a mean Lagrangian, obtained from the kinetic energy Lagrangian of ideal fluid flow via a small amplitude expansion for the fluctuations. We show that the Euler-α equations arise as Lagrangian averaged Euler equations when using the -geodesic mean on the volume preserving diffeomorphism group of a manifold without boundaries, imposing a “Taylor hypothesis”, which states that first order fluctuations are transported as a vector field by the mean flow, and assuming that fluctuations are statistically nearly isotropic. Similarly, the EPDiff equations arise as the Lagrangian averaged Burgers' equations using the same argument on the full diffeomorphism group. A serious drawback of this construction is that the assumptions of Lie transport of the fluctuation vector field and isotropy of fluctuations cannot persist except for an asymptotically vanishing interval of time. To remedy the problem of persistence of isotropy, we suggest adding strong mean-reverting stochastic term to the Taylor hypothesis and identify a scaling regime in which the inclusion of the stochastic term leads to the same averaged equations up to a constant.

Vortex sheets and diffeomorphism groupoids

In 1966 V. Arnold suggested a group-theoretic approach to ideal hydrodynamics in which the motion of an inviscid incompressible fluid is described as the geodesic flow of the right-invariant L2-metric on the group of volume-preserving diffeomorphisms of the flow domain. Here we propose geodesic, group-theoretic, and Hamiltonian frameworks to include fluid flows with vortex sheets. It turns out that the corresponding dynamics is related to a certain groupoid of pairs of volume-preserving diffeomorphisms with common interface. We also develop a general framework for Euler–Arnold equations for geodesics on groupoids equipped with one-sided invariant metrics.

The Exponential Map of the Group of Area-Preserving Diffeomorphisms of a Surface with Boundary

We prove that the Riemannian exponential map of the right-invariant L2 metric on the group of volume-preserving diffeomorphisms of a two-dimensional manifold with a nonempty boundary is a nonlinear Fredholm map of index zero.

Geometric generalised Lagrangian-mean theories

Many fluctuation-driven phenomena in fluids can be analysed effectively using the generalised Lagrangian-mean (GLM) theory of Andrews & McIntyre (J. Fluid Mech., vol. 89, 1978, pp. 609–646) This finite-amplitude theory relies on particle-following averaging to incorporate the constraints imposed by the material conservation of certain quantities in inviscid regimes. Its original formulation, in terms of Cartesian coordinates, relies implicitly on an assumed Euclidean structure; as a result, it does not have a geometrically intrinsic, coordinate-free interpretation on curved manifolds, and suffers from undesirable features. Motivated by this, we develop a geometric generalisation of GLM that we formulate intrinsically using coordinate-free notation. One benefit is that the theory applies to arbitrary Riemannian manifolds; another is that it establishes a clear distinction between results that stem directly from geometric consistency and those that depend on particular choices. Starting from a decomposition of an ensemble of flow maps into mean and perturbation, we define the Lagrangian-mean momentum as the average of the pull-back of the momentum one-form by the perturbation flow maps. We show that it obeys a simple equation which guarantees the conservation of Kelvin’s circulation, irrespective of the specific definition of the mean flow map. The Lagrangian-mean momentum is the integrand in Kelvin’s circulation and distinct from the mean velocity (the time derivative of the mean flow map) which advects the contour of integration. A pseudomomentum consistent with that in GLM can then be defined by subtracting the Lagrangian-mean momentum from the one-form obtained from the mean velocity using the manifold’s metric. The definition of the mean flow map is based on choices made for reasons of convenience or aesthetics. We discuss four possible definitions: a direct extension of standard GLM, a definition based on optimal transportation, a definition based on a geodesic distance in the group of volume-preserving diffeomorphisms, and the ‘glm’ definition proposed by Soward & Roberts (J. Fluid Mech., vol. 661, 2010, pp. 45–72). Assuming small-amplitude perturbations, we carry out order-by-order calculations to obtain explicit expressions for the mean velocity and Lagrangian-mean momentum at leading order. We also show how the wave-action conservation of GLM extends to the geometric setting. To make the paper self-contained, we introduce in some detail the tools of differential geometry and main ideas of geometric fluid dynamics on which we rely. These include variational formulations which we use for alternative derivations of some key results. We mostly focus on the Euler equations for incompressible inviscid fluids but sketch out extensions to the rotating–stratified Boussinesq, compressible Euler, and magnetohydrodynamic equations. We illustrate our results with an application to the interaction of inertia-gravity waves with balanced mean flows in rotating–stratified fluids.

Lagrangian averaging with geodesic mean

This paper revisits the derivation of the Lagrangian averaged Euler (LAE), or Euler-α equations in the light of an intrinsic definition of the averaged flow map as the geodesic mean on the volume-preserving diffeomorphism group. Under the additional assumption that first-order fluctuations are statistically isotropic and transported by the mean flow as a vector field, averaging of the kinetic energy Lagrangian of an ideal fluid yields the LAE Lagrangian. The derivation presented here assumes a Euclidean spatial domain without boundaries.

Differential-geometrical approach to the dynamics of dissipationless incompressible Hall magnetohydrodynamics: II. Geodesic formulation and Riemannian curvature analysis of hydrodynamic and magnetohydrodynamic stabilities

In this study, the dynamics of a dissipationless incompressible Hall magnetohydrodynamic (HMHD) medium are formulated as geodesics on a direct product of two volume-preserving diffeomorphism groups. Formulations are given for the geodesic and Jacobi equations based on a linear connection with physically desirable properties, which agrees with the Levi-Civita connection. Derivations of the explicit normal-mode expressions for the Riemannian metric, Levi-Civita connection, and related formulae and equations are also provided using the generalized Elsässer variables (GEVs). Examinations of the stabilities of the hydrodynamic (HD, ) and magnetohydrodynamic (MHD, ) motions and the Hall-term effect in terms of the Jacobi equation and the Riemannian sectional curvature tensor are presented, where α represents the Hall-term strength parameter. It is very interesting that the sectional curvatures of the MHD and HMHD systems between two GEV modes were found to take both the positive (stable) and negative (unstable) values, while that of the HD system between two complex helical waves was observed to be negative definite. Moreover, for the MHD case, negative sectional curvatures were found to occur only when mode interaction was ‘local’, i.e. the wavenumber moduli of the main flow (say p) and perturbation (say k) were relatively close to each other. However, in the nonlocal limit ( or ), the sectional curvatures were always positive. This result leads to the conjecture that the MHD interactions mainly excite wavy or non-growing motions; however, some local interactions cause dynamical instability that leads to chaotic or turbulent plasma motions. Additionally, it was found that the tendencies of the effects are opposite between the ion cyclotron and whistler modes. Comparison with the energy-Casimir method is also discussed using a remarkable constant of motion which relates the Riemannian curvature to the second variation of the Hamiltonian.

On stable transitivity of finitely generated groups of volume-preserving diffeomorphisms

In this paper, we provide a new criterion for the stable transitivity of volume-preserving finite generated groups on any compact Riemannian manifold. As one of our applications, we generalize a result of Dolgopyat and Krikorian [On simultaneous linearization of diffeomorphisms of the sphere.Duke Math. J. 136(2007), 475–505] and obtain stable transitivity for random rotations on the sphere in any dimension. As another application, we show that for$\infty \geq r\geq 2$, for any$C^{r}$volume-preserving partially hyperbolic diffeomorphism$g$on any compact Riemannian manifold$M$having sufficiently Hölder stable or unstable distribution, for any sufficiently large integer$K$and for any$(f_{i})_{i=1}^{K}$in a$C^{1}$open$C^{r}$dense subset of$\text{Diff}^{r}(M,m)^{K}$, the group generated by$g,f_{1},\ldots ,f_{K}$acts transitively.

The Geometry of Axisymmetric Ideal Fluid Flows with Swirl

The sectional curvature of the volume preserving diffeomorphism group of a Riemannian manifold M can give information about the stability of inviscid, incompressible fluid flows on M. We demonstrate that the submanifold of the volumorphism group of the solid flat torus generated by axisymmetric fluid flows with swirl, denoted by $$\mathcal {D}_{\mu ,E}(M)$$ , has positive sectional curvature in every section containing the field $$X = u(r)\partial _\theta $$ iff $$\partial _r(ru^2)>0$$ . This is in sharp contrast to the situation on $$\mathcal {D}_{\mu }(M)$$ , where only Killing fields X have nonnegative sectional curvature in all sections containing it. We also show that this criterion guarantees the existence of conjugate points on $$\mathcal {D}_{\mu ,E}(M)$$ along the geodesic defined by X.

Riemannian geometry of the space of volume preserving immersions

Given a compact manifold M and a Riemannian manifold N of bounded geometry, we consider the manifold Imm(M,N) of immersions from M to N and its subset Immμ(M,N) of those immersions with the property that the volume-form of the pull-back metric equals μ. We first show that the non-minimal elements of Immμ(M,N) form a splitting submanifold. On this submanifold we consider the Levi-Civita connection for various natural Sobolev metrics, we write down the geodesic equation for which we show local well-posedness in many cases. The question is a natural generalization of the corresponding well-posedness question for the group of volume-preserving diffeomorphisms, which is of importance in fluid mechanics.