We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional I defined on exact divergence-free vector fields of class C 1 on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that I is invariant under arbitrary volumepreserving diffeomorphisms if and only if it is a function of the helicity.helicity | integral invariant | volume-preserving transformation I ncompressible inviscid fluids are modeled by the 3D Euler equations, which assert that the velocity field uðx, tÞ of the fluid flow must satisfy the system of differential equationsHere the scalar function pðx, tÞ is another unknown of the problem, which physically corresponds to the pressure of the fluid.It is customary to introduce the vorticity ω := curl u to simplify the analysis of these equations, as it enables us to get rid of the pressure function. In terms of the vorticity, the Euler equations read aswhere ½ω, u := ðω · ∇Þu − ðu · ∇Þw is the commutator of vector fields and u can be written in terms of ω, using the Biot-Savart lawat least when the space variable is assumed to take values in the whole space R 3 . The transport Eq. 1 was first derived by Helmholtz, who showed that the meaning of this equation is that the vorticity at time t is related to the vorticity at initial time t 0 via the flow of the velocity field, provided that the equation does not develop any singularities in the time interval ½t 0 , t. More precisely, if ϕ t,t0 denotes the (timedependent) flow of the divergence-free field u, then the vorticity at time t is given by the action of the push forward of the volumepreserving diffeomorphism ϕ t,t0 on the initial vorticity:The phenomenon of the transport of vorticity gives rise to a new conservation law of the 3D Euler equations. Moffatt coined the term "helicity" for this conservation law in his influential paper (1) and exhibited its topological nature. Indeed, defining the helicity of a divergence-free vector field w in R 3 asit turns out that the helicity of the vorticity Hðωð ·, tÞÞ is a conserved quantity for the Euler equations. In fact, helicity is also conserved for the compressible Euler equations provided the fluid is barotropic (i.e., the pressure is a function of the density). It is well known that the relevance of the helicity goes well beyond that of being a new (nonpositive) conserved quantity for the Euler equations. On the one hand, the helicity appears in other natural phenomena that are also described by a divergence-free field whose evolution is given by a time-dependent family of volume-preserving diffeomorphisms (2). For instance, the case of magnetohydrodynamics (MHD), where one is interested in the helicity of the magnetic field of a conducting plasma, has attracted considerable attention. On the other hand, it turns out that the helicity not only corresponds to a conserved quantity for evolution equations such as Euler or MHD, but also in fact defines an integral invariant for vector fiel...

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a 3-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we show that the steady Euler flows cannot be constructed using plugs (as in Wilson's or Kuperberg's constructions). Analogous results in higher dimensions are also proved.

Let S be a finite union of (pairwise disjoint but possibly knotted and linked) closed curves and tubes in the round sphere S 3 or in the flat torus T 3 . In the case of the torus, S is further assumed to be contained in a contractible subset of T 3 . In this paper we show that for any sufficiently large odd integer λ there exists a Beltrami field on S 3 or T 3 satisfying curl u = λu and with a collection of vortex lines and vortex tubes given by S, up to an ambient diffeomorphism.

We construct finite dimensional families of non-steady solutions to the Euler equations, existing for all time, and exhibiting all kinds of qualitative dynamics in the phase space, for example: strange attractors and chaos, invariant manifolds of arbitrary topology, and quasiperiodic invariant tori of any dimension.The main theorem of the paper, from which these families of solutions are obtained, states that for any given vector field X on a closed manifold N , there is a Riemannian manifold M on which the following holds: N is diffeomorphic to a finite dimensional manifold in the space of divergence-free vector fields on M that is invariant under the Euler evolution, and on which the Euler equation reduces to a finite dimensional ODE that is given by an arbitrarily small perturbation of the vector field X on N .