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Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold. In this paper we present a collection of open problems along with several new constructions in fluid dynamics and a concise survey of recent developments and achievements in this area. The topics discussed include variational settings for different types of fluids, models for invariant metrics, the Cauchy and boundary value problems, partial analyticity of solutions to the Euler equations, their steady and singular vorticity solutions, differential and Hamiltonian geometry of diffeomorphism groups, long-time behaviour of fluids, as well as mechanical models of direct and inverse cascades., Comment: 41 pages, 5 figures, a new section and more details and references are added

Two fluid configurations along a flow are conjugate if there is a one parameter family of geodesics (fluid flows) joining them to infinitesimal order. Geometrically, they can be seen as a consequence of the (infinite dimensional) group of volume preserving diffeomorphisms having sufficiently strong positive curvatures which ‘pull’ nearby flows together. Physically, they indicate a form of (transient) stability in the configuration space of particle positions: a family of flows starting with the same configuration deviate initially and subsequently re-converge (resonate) with each other at some later moment in time. Here, we first establish existence of conjugate points in an infinite family of Kolmogorov flows—a class of stationary solutions of the Euler equations—on the rectangular flat torus of any aspect ratio. The analysis is facilitated by a general criterion for identifying conjugate points in the group of volume preserving diffeomorphisms. Next, we show non-existence of conjugate points along Arnold stable steady states on the annulus, disk and channel. Finally, we discuss cut points, their relation to non-injectivity of the exponential map (impossibility of determining a flow from a particle configuration at a given instant) and show that the closest cut point to the identity is either a conjugate point or the midpoint of a time periodic Lagrangian fluid flow.

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.

We introduce a geometric framework to study Newton's equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities. It turns out that several important PDEs of hydrodynamical origin can be described in this framework in a natural way. In particular, the Madelung transform between the Schr\"odinger equation and Newton's equations is a symplectomorphism of the corresponding phase spaces. Furthermore, the Madelung transform turns out to be a K\"ahler map when the space of densities is equipped with the Fisher-Rao information metric. We describe several dynamical applications of these results., Comment: 17 pages, 2 figures

We revisit the geodesic approach to ideal hydrodynamics and present a related geometric framework for Newton's equations on groups of diffeomorphisms and spaces of probability densities. The latter setting is sufficiently general to include equations of compressible and incompressible fluid dynamics, magnetohydrodynamics, shallow water systems and equations of relativistic fluids. We illustrate this with a survey of selected examples, as well as with new results, using the tools of infinite-dimensional information geometry, optimal transport, the Madelung transform, and the formalism of symplectic and Poisson reduction., Comment: 62 pages. Revised version, accepted in Bull. AMS

The Madelung transform is known to relate Schr\"odinger-type equations in quantum mechanics and the Euler equations for barotropic-type fluids. We prove that, more generally, the Madelung transform is a K\"ahler map (i.e. a symplectomorphism and an isometry) between the space of wave functions and the cotangent bundle to the density space equipped with the Fubini-Study metric and the Fisher-Rao information metric, respectively. We also show that Fusca's momentum map property of the Madelung transform is a manifestation of the general approach via reduction for semi-direct product groups. Furthermore, the Hasimoto transform for the binormal equation turns out to be the 1D case of the Madelung transform, while its higher-dimensional version is related to the problem of conservation of the Willmore energy in binormal flows., Comment: 27 pages, 2 figures

We prove that the weak-Riemannian exponential map of the $$L^2$$ metric on the group of volume-preserving diffeomorphisms of a compact two-dimensional manifold is not injective in any neighbourhood of its conjugate vectors. This can be viewed as a hydrodynamical analogue of the classical result of Morse and Littauer.

We show that the 2D Euler equations are not locally well-posed in the sense of Hadamard in the \(C^1\) space and in the Besov space \(B^1_{\infty ,1}\). Our approach relies on the technique of Lagrangian deformations of Bourgain and Li (Strong ill-posedness of the incompressible Euler equations in borderline Sobolev spaces. arXiv:1307.7090). We show that the assumption that the data-to-solution map is continuous in either \(C^1\) or \(B^1_{\infty ,1}\) leads to a contradiction with a result in \(W^{1,p}\) of Kato and Ponce (Rev Mat Iberoam 2:73–88, 1986).

We study the family of Amari–Chentsov α-connections on the homogeneous space $ {{{\mathcal{D}(M)}} \left/ {{{{\mathcal{D}}_{\mu }}(M)}} \right.} $ of diffeomorphisms modulo volume-preserving diffeomorphisms of a compact manifold M. We show that in some cases their geodesic equations yield completely integrable Hamiltonian systems. Bibliography: 10 titles.

We develop a geometric framework for Newton-type equations on the infinite-dimensional configuration space of probability densities. It can be viewed as a second order analogue of the "Otto calculus" framework for gradient flow equations. Namely, for an n-dimensional manifold M we derive Newton's equations on the group of diffeomorphisms Diff(M) and the space of smooth probability densities Dens(M), as well as describe the Hamiltonian reduction relating them. For example, the compressible Euler equations are obtained by a Poisson reduction of Newton's equation on Diff(M) with the symmetry group of volume-preserving diffeomorphisms, while the Hamilton-Jacobi equation of fluid mechanics corresponds to potential solutions. We also prove that the Madelung transform between Schrodinger-type and Newton's equations is a symplectomorphism between the corresponding phase spaces T* Dens(M) and PL2 (M, C). This improves on the previous symplectic submersion result of von Renesse [1]. Furthermore, we prove that the Madelung transform is a Kahler map provided that the space of densities is equipped with the (prolonged) Fisher-Rao information metric and describe its dynamical applications. This geometric setting for the Madelung transform sheds light on the relation between the classical Fisher-Rao metric and its quantum counterpart, the Bures metric. In addition to compressible Euler, Hamilton-Jacobi, and linear and nonlinear Schrodinger equations, the framework for Newton equations encapsulates Burgers' inviscid equation, shallow water equations, two-component and mu-Hunter-Saxton equations, the Klein-Gordon equation, and infinite-dimensional Neumann problems.

We study the geometry of the space of densities Dens(M), which is the quotient space Diff(M)/Diff μ (M) of the diffeomorphism group of a compact manifold M by the subgroup of volume-preserving diffeomorphisms, endowed with a right-invariant homogeneous Sobolev $${\dot{H}^1}$$ -metric. We construct an explicit isometry from this space to (a subset of) an infinite-dimensional sphere and show that the associated Euler–Arnold equation is a completely integrable system in any space dimension whose smooth solutions break down in finite time. We also show that the $${\dot{H}^1}$$ -metric induces the Fisher–Rao metric on the space of probability distributions and its Riemannian distance is the spherical version of the Hellinger distance.

We show that the solution map of the periodic CH equation is not uniformly continuous in Sobolev spaces with exponent greater than 3/2. This extends earlier results to the whole range of Sobolev exponents for which local well-posedness of CH is known. The crucial technical tools used in the proof of this result are a sharp commutator estimate and a multiplier estimate in Sobolev spaces of negative index.

We show that continuous dependence on initial data of solutions to the Euler equations of incompressible hydrodynamics is optimal. More precisely, we prove that the data-to-solution map is not uniformly continuous in Sobolev H s (Ω) topology for any $${s \in \mathbb{R}}$$ if the domain Ω is the (flat) torus $${\mathbb{T}^n=\mathbb{R}^n/2\pi\mathbb{Z}^n}$$ and for any s > 0 if the domain is the whole space $${\mathbb{R}^n}$$ .

We prove that exponential maps of right-invariant Sobolev Hr metrics on a variety of diffeomorphism groups of compact manifolds are nonlinear Fredholm maps of index zero as long as r is sufficiently large. This generalizes the result of Ebin et al. (Geom. Funct. Anal. 16, 2006) for the L2 metric on the group of volume-preserving diffeomorphisms important in hydrodynamics. In particular, our results apply to many other equations of interest in mathematical physics. We also prove an infinite-dimensional Morse Index Theorem, settling a question raised by Arnold and Khesin (Topological methods in hydrodynamics. Springer, New York, 1998) on stable perturbations of flows in hydrodynamics. Finally, we include some applications to the global geometry of diffeomorphism groups.

We study an equation lying ‘mid-way’ between the periodic Hunter–Saxton and Camassa–Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and admits both cusped and smooth traveling-wave solutions which are natural candidates for solitons. We also prove that it is locally well-posed and establish results on the lifespan of its solutions. Throughout the paper we argue that despite similarities to the KdV, CH and HS equations, the new equation manifests several distinctive features that set it apart from the other three.

We construct an example showing that the solution map of the Euler equations is not continuous in the H\"older space from $C^{1,\alpha}$ to $L^\infty_tC^{1,\alpha}_x$ for any $0

It is shown that if a classical solution $(u, n)$ of the modified Euler-Poisson equation (mEP) in one space dimension is such that $u$, $u_x$ and $n$ are initially decaying exponentially and for some later time the first component $u$ is also decaying exponentially, then $n$ must be identically equal to zero and $u$ must be a solution to the Burgers equation. In particular, if $n$ and $u$ are initially compactly supported then $n$ can not be compactly supported at any later time, unless $n$ is identically equal to zero and $u$ is a solution to the Burgers equation. It is also shown that the mEP equations are locally well-posed in $H^s \times H^{s-1}$ for $s>5/2$.

In this note we obtain the characterization for asymptotic directions on various subgroups of the diffeomorphism group. We give a simple proof of non-existence of such directions for area-preserving diffeomorphisms of closed surfaces of non-zero curvature. Finally, we exhibit the common origin of the Monge–Ampere equations in 2D fluid dynamics and mass transport.

In this paper we consider the periodic Cauchy problem for the Camassa-Holm equation with analytic initial data and prove that its solutions are analytic in both variables, globally in space and locally in time.

We study the periodic Cauchy problem for the Camassa—Holm equation and prove that it is locally well-posed in the space of continuously differentiable functions on the circle. The approach we use consists in rewriting the equation and deriving suitable estimates which permit application of o.d.e. techniques in Banach spaces. We also describe results in fractional Sobolev H s spaces and in Appendices provide a direct well-posedness proof for arbitrary real s > 3/2 based on commutator estimates of Kato and Ponce as well as include a derivation of the equation on the diffeomorphism group of the circle together with related curvature computations.

We present an elementary proof of an a priori estimate of Bourgain for a general class of multipliers on a circle using an extension of methods developed in our previous work. The main tool is a suitable version of a counting argument of Zygmund for unbounded regions.

It is known that solutions of the Euler equations of hydrodynamics correspond to geodesics on the group of volume–preserving diffeomorphisms of a compact manifold. We conjecture that, regardless of the dimension of the manifold, the associated Riemannian exponential map on the group is nonlinear Fredholm of index zero. Such a result has been established for the Riemannian exponential maps of natural Sobolev metrics on loop spaces and loop groups.

We find conditions for the exponential map on a weak riemannian Hilbert manifold to be a nonlinear Fredholm map of index zero and apply the result to left-invariant Sobolev metrics on loop groups.

The online version of the original article can be found under doi: 10.1007/s00208-015-1213-0 .

We study the exponential map on Hilbert riemannian manifolds and show that on the free loop space $ \Omega M^n = H^1 (S^1, M^n) $ it is a nonlinear Fredholm map of index zero. Among corollaries we obtain that only one type of conjugate points exists in $ \Omega M^n $ . This answers a question of W. Klingenberg.

We study the geometry of a right invariant metric on a central extension D ^ ( S 1 ) \widehat {\mathcal {D}}(S^1) of the diffeomorphism group of a circle (the Bott-Virasoro group) introduced by Ovsienko and Khesin. We obtain an expression for the curvature tensor of this metric and apply it to find conjugate points in D ^ ( S 1 ) \widehat {\mathcal {D}}(S^1) .

We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on the hyperbolic spaces of higher dimension. We also describe the corresponding general Hamiltonian setting of hydrodynamics on complete Riemannian manifolds, which includes the hyperbolic setting., Comment: 6 pages

Many conservative partial differential equations correspond to geodesic equations on groups of diffeomorphisms. Stability of their solutions can be studied by examining sectional curvature of these groups: negative curvature in all sections implies exponential growth of perturbations and hence suggests instability, while positive curvature suggests stability. In the first part of the paper we survey what we currently know about the curvature-stability relation in this context and provide detailed calculations for several equations of continuum mechanics associated to Sobolev $H^0$ and $H^1$ energies. In the second part we prove that in most cases (with some notable exceptions) the sectional curvature assumes both signs., Written for "Geometric and Algebraic Structures in Mathematics," a conference in celebration of Dennis Sullivan's 70th birthday

We establish a simple relation between curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group and, consequently, to a conjugate point along a geodesic in the Wasserstein space of densities. This establishes an intrinsic connection between ideal Euler hydrodynamics (via Arnold's approach), shock formation in the multidimensional Burgers equation and the Wasserstein geometry of the space of densities., 11 pages, 2 figures

We construct high-frequency smooth periodic solutions to the Camassa-Holm equation and use them to prove that the related solution map is not uniformly continuous on bounded sets in any Sobolev space of index greater than or equal to two.

In this paper we consider the periodic Cauchy problem for a fifth order modification of the Camassa–Holm equation. We prove local well-posedness in appropriate Bourgain spaces for initial data in a Sobolev space Hs(T), s>1/2. We also prove global well-posedness for data in H1(T) and of arbitrary size. The proofs are based on a priori estimates using Fourier analysis techniques, microlocalization in phase space, an interpolation argument and a fixed point theorem.

We show that the following three systems related to various hydrodynamical approximations: the Korteweg--de Vries equation, the Camassa--Holm equation, and the Hunter--Saxton equation, have the same symmetry group and similar bihamiltonian structures. It turns out that their configuration space is the Virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right-invariant metrics on this group or on appropriate homogeneous spaces. In particular, we describe how Arnold's approach to the Euler equations as geodesic flows of one-sided invariant metrics extends from Lie groups to homogeneous spaces. We also show that the above three cases describe all generic bihamiltonian systems which are related to the Virasoro group and can be integrated by the translation argument principle: they correspond precisely to the three different types of generic Virasoro orbits., 26 pages, 4 figures, LaTeX. Advances in Mathematics (to appear)

We study a family of equations defined on the space of tensor densities of weight $\lambda$ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct "peakon" and "multi-peakon" solutions for all $\lambda \neq 0,1$, and "shock-peakons" for $\lambda = 3$. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold's approach to Euler equations on Lie groups., Comment: 30 pages