For smooth solutions of the incompressible Euler equations, Kelvin's theorem states that the circulation around any closed material loop is a Lagrangian invariant. The singular velocity fields expected in high-Reynolds-number turbulence violate every premise of this theorem: material loops are expected to become non-rectifiable fractal curves, Lagrangian trajectories are expected to lose uniqueness ('spontaneous stochasticity'), and circulations can be dissipated anomalously. Eyink (2006) proved that the coarse-grained circulation flux around fixed smooth loops vanishes under a Besov-regularity condition, and conjectured a statistical replacement of Kelvin's theorem for turbulent solutions: circulations around advected loops should be martingales of a suitable generalized Euler flow, i.e. a probability measure on time-histories of measure-preserving maps. Formulating and proving (or refuting) this statistical Kelvin theorem remains open. No construction of such a generalized Euler flow with the required almost-sure spatial regularity exists, circulations on fractal advected loops lack a rigorous definition, the required exchange of coarse-graining and conditioning limits is unjustified, and the proposed derivation via a stochastic least-action principle with a Noether theorem for particle relabelling is rigorous only for the viscous Navier-Stokes case. The problem asks for a rigorous foundation (or a rigorous refutation) of this conjectured statistical conservation law for dissipative Euler solutions arising in the inviscid limit.

Background

The incompressible Euler equations, ∂tu+(u⋅∇)u=−∇p\partial_t u+(u\cdot\nabla)u=-\nabla p with ∇⋅u=0\nabla\cdot u=0, describe ideal (inviscid) fluid flow. For classical smooth solutions they possess an infinite family of Lagrangian invariants: by Kelvin's circulation theorem, the line integral Γ(C,t)=∮C(t)u⋅dℓ\Gamma(C,t)=\oint_{C(t)}u\cdot d\ell around any closed material loop C(t)C(t) advected by the flow is constant in time. This conservation is a consequence of the particle-relabelling symmetry of Hamilton's principle for ideal fluids (a Noether-type relation), and it underlies the structure of vortex dynamics.

Onsager's conjecture asserts that high-Reynolds-number turbulent flows are described, in the inviscid limit, by weak (distributional) solutions of Euler with velocity roughly 1/31/3-Hölder singularities, for which kinetic energy need not be conserved (the dissipative anomaly). Convex integration constructions have confirmed the flexibility of such dissipative weak solutions, so the physically relevant inviscid limit is expected to be highly non-unique. For such singular velocities the premises of Kelvin's theorem fail: the ordinary differential equation for Lagrangian particle paths is non-unique for non-Lipschitz velocity fields, and in turbulent regimes the Lagrangian trajectories are expected to remain random even in the joint limit of zero viscosity and zero artificial noise, a phenomenon called spontaneous stochasticity (established rigorously in the Kraichnan model of turbulent advection and for inviscid Burgers flows). Moreover, numerical simulations indicate that material loops become fractal, non-rectifiable curves.

Brenier (1989) introduced generalized flows for ideal incompressible fluids: probability measures on space-time particle paths that replace deterministic flow maps when classical solutions break down; however, his notion does not define material lines. Eyink (2006) therefore proposed to work with probability measures PP on time-histories a↦x(a,t)a\mapsto x(a,t) of measure-preserving maps of the flow domain, carrying an action S[P]S[P] generalizing Hamilton's principle, and conjectured that for dissipative Euler solutions relevant to turbulence the circulations Γ(C,t)\Gamma(C,t) of every initially smooth loop CC are martingales of the resulting generalized (spontaneously stochastic) flow, i.e. E[Γ(C,t)∣Γ(C,τ), τ<t′]=Γ(C,t′)for t>t′.E[\Gamma(C,t)\mid \Gamma(C,\tau),\ \tau<t']=\Gamma(C,t')\quad\text{for } t>t'. A martingale is a stochastic process whose conditional expectation given its past equals its present value; the statement thus asserts that the statistical circulation is conserved on average, even though it is not conserved in individual realizations.

The viscous analogue of this statement is a theorem: Constantin and Iyer (2008) proved a stochastic Lagrangian representation of the Navier-Stokes equations in which the circulation around loops advected by Brownian-driven stochastic trajectories is a backward-in-time martingale, and Eyink (2010) showed that this stochastic Kelvin theorem follows from a stochastic least-action principle invariant under particle relabelling. Analogous martingale properties have been argued at physics level of rigor for the Kraichnan ensemble of Gaussian white-in-time velocities and proved rigorously for inviscid Burgers flows, where spontaneous stochasticity was established by Eyink and Drivas; spontaneous stochasticity itself has also been rigorously established for passive scalars in engineered non-smooth velocity fields. What has not been achieved, in any setting governed by the Euler equations, is the construction of a generalized stochastic flow with well-defined random advected loops and circulations for which the martingale property holds. The problem sits at the intersection of the Onsager program (dissipative anomaly for weak Euler solutions) and the theory of spontaneous stochasticity: it asks whether Kelvin's theorem survives turbulence in statistical form, and whether that property can select the physically admissible dissipative Euler solutions among the infinitely many produced by convex integration.

Problem Statement

Let uu be a dissipative distributional solution of the incompressible Euler equations on a bounded periodic domain Λ⊂Rd\Lambda\subset\mathbb{R}^d, with the Besov/Hölder spatial regularity (exponent below the Onsager threshold 1/31/3) expected for high-Reynolds-number turbulent flow, arising as a vanishing-viscosity or otherwise physical inviscid limit. Construct a generalized Euler flow associated with such solutions — a probability measure PP on time-histories a↦x(a,t)a\mapsto x(a,t) of measure-preserving maps of Λ\Lambda that is almost surely sufficiently regular in space (e.g. Hölder continuous) for the images C(t)=x(C,t)C(t)=x(C,t) of smooth initial loops CC to be well-defined (random, generally fractal) curves — together with a rigorous definition of the circulations Γ(C,t)=∮C(t)u⋅dℓ\Gamma(C,t)=\oint_{C(t)}u\cdot d\ell on those loops. Then prove or disprove the statistical Kelvin theorem conjectured by Eyink (2006): for every smooth initial loop CC, the process Γ(C,t)\Gamma(C,t) is a martingale of the generalized flow, E[Γ(C,t)∣Γ(C,τ), τ<t′]=Γ(C,t′)for t>t′,E[\Gamma(C,t)\mid\Gamma(C,\tau),\ \tau<t']=\Gamma(C,t')\quad\text{for }t>t', with the associated flux formulation lim⁡ℓ→0E[Kℓ(C,t)∣Γ(C,τ),τ<t′]=0\lim_{\ell\to0}E[K_\ell(C,t)\mid\Gamma(C,\tau),\tau<t']=0 holding with the coarse-graining limit ℓ→0\ell\to0 and the conditioning rigorously exchanged. The problem comprises, as jointly necessary components of this single objective: (i) the construction of the generalized (spontaneously stochastic) Euler flow with the stated regularity and its relation to dissipative Euler solutions; (ii) an integration theory for circulations on non-rectifiable random advected loops; and (iii) a rigorous justification of the small-scale homogeneity and cancellation arguments underlying the vanishing of the conditional circulation flux — or, alternatively, a rigorous construction of natural inviscid-limit Euler flows for which the circulation process provably fails the martingale property, refuting the conjecture.

References

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