Determine whether there exists a sharp Hölder-regularity threshold for anomalous dissipation of passively advected scalars when the drift is a random, autonomous (time-independent), divergence-free vector field on the torus and the anomaly is required to hold almost surely. For a passive scalar solving ∂tθε+(v⋅∇)θε=εΔθε\partial_t\theta_\varepsilon+(v\cdot\nabla)\theta_\varepsilon=\varepsilon\Delta\theta_\varepsilon, a dissipation anomaly means lim sup⁡ε↓0ε∫0T∫∣∇θε∣2>0\limsup_{\varepsilon\downarrow 0}\varepsilon\int_0^T\int|\nabla\theta_\varepsilon|^2>0 for some smooth initial datum. Rigidity is currently known at three levels: in two dimensions almost every nondegenerate random continuous divergence-free field carries no anomaly (Bagnara–Boutros–De Lellis–Mayboroda); in d≥3d\ge 3 the same holds for random Clebsch-structured fields of Hölder exponent above 1/81/8 (same authors) and, in any dimension, for general multiscale finite-range random fields — including Gaussian ensembles with power-law spectra — of exponent above 1/21/2 (Colombo–Hess-Childs–Rowan). Flexibility is known only near regularity −1-1: self-similar random incompressible flows exhibit almost-sure anomalous regularization and superdiffusion there (Armstrong–Bou-Rabee–Kuusi), while deterministic fields exhibit anomalous dissipation at every exponent below 11. The open problem is to close this gap: whether the exponents 1/81/8 and 1/21/2 (and the probabilistic Morse–Sard exponent j/(2+j)j/(2+j) governing the level-set geometry behind the 1/81/8 result) are sharp; what happens for unstructured ensembles in d=3d=3 below the sweeping-driven well-posedness threshold 1/21/2; and whether random autonomous divergence-free ensembles can be constructed that almost surely produce anomalous dissipation at regularity strictly above −1-1.

Background

A passively advected scalar transported by a velocity field vv on the torus Td\mathbb{T}^d and damped by molecular diffusivity ε>0\varepsilon>0 solves ∂tθε+(v⋅∇)θε=εΔθε\partial_t\theta_\varepsilon+(v\cdot\nabla)\theta_\varepsilon=\varepsilon\Delta\theta_\varepsilon. In turbulent mixing, the cascade of scalar variance toward small scales is expected to sustain a nonzero dissipation rate even as the diffusivity vanishes. A dissipation anomaly is the statement that lim sup⁡ε↓0ε∫0T∫Td∣∇θε∣2 dx dt>0\limsup_{\varepsilon\downarrow 0}\varepsilon\int_0^T\int_{\mathbb{T}^d}|\nabla\theta_\varepsilon|^2\,dx\,dt>0 for some smooth initial datum. This is the passive-scalar counterpart of the energy-dissipation anomaly predicted by Onsager's conjecture for incompressible Euler flows, which posits a sharp threshold at Hölder regularity 1/31/3 (spatial regularity measured by the exponent α\alpha in the Hölder class CαC^\alpha, meaning ∣v(x)−v(y)∣≲∣x−y∣α|v(x)-v(y)|\lesssim|x-y|^\alpha) separating energy-conserving weak solutions above from dissipative ones below; both directions are now theorems, with the dissipative direction due to Isett. Whether the passive-scalar problem admits an analogous regularity threshold — and at which exponent — is the subject of this problem, posed in the probabilistic setting of random autonomous (time-independent) divergence-free drifts.

For deterministic drifts no threshold exists in (0,1)(0,1): anomalous dissipation has been constructed for every Hölder exponent α<1\alpha<1, for time-dependent divergence-free velocities by Drivas–Elgindi–Iyer–Jeong, by Colombo–Crippa–Sorella, by Elgindi–Liss, and by Armstrong–Vicol (fractal homogenization, α<1/3\alpha<1/3), and for autonomous divergence-free velocities in two dimensions with arbitrary α∈(0,1)\alpha\in(0,1) by Johansson–Sorella. The latter construction exploits the Lagrangian fluctuation–dissipation relation of Drivas–Eyink, which identifies anomalous dissipation with spontaneous stochasticity — the failure of the variance of stochastically perturbed Lagrangian trajectories to vanish as the perturbation is removed. Anomalous dissipation also implies the constraint α+2β≤1\alpha+2\beta\le 1 between the spatial regularity α\alpha of the drift and any uniform Hölder bound β\beta of the scalar (so that anomalous regularization, i.e. Hölder bounds on θε\theta_\varepsilon uniform in the diffusivity, fails), and it implies failure of the DiPerna–Lions renormalization property described next. The vanishing-viscosity anomalous-dissipation problem for the unforced three-dimensional Navier–Stokes equations remains open and is regarded as extremely challenging.

The rigidity side rests on the DiPerna–Lions theory: a bounded divergence-free field vv has the renormalization property if β(θ)\beta(\theta) is again a weak solution of the transport equation ∂tθ+(v⋅∇)θ=0\partial_t\theta+(v\cdot\nabla)\theta=0 whenever θ\theta is, for every smooth β\beta; this yields uniqueness and strong stability of the Cauchy problem and, via a fluctuation–dissipation argument, precludes anomalous dissipation for bounded data. Ambrosio extended the theory to BVBV fields. In two dimensions every smooth divergence-free field is Hamiltonian, v=∇⊥ψv=\nabla^\perp\psi with stream function ψ\psi, and Alberti–Bianchini–Crippa proved for Lipschitz fields that renormalization is equivalent to a weak Sard property of ψ\psi (a surrogate of the classical Morse–Sard property on the level sets of ψ\psi) and to uniqueness of the regular Lagrangian flow (the canonical measure-preserving selection of ODE trajectories). Bagnara–Boutros–De Lellis–Mayboroda proved that for a random continuous divergence-free field on T2\mathbb{T}^2 whose pointwise law is nondegenerate, in the sense that P(v(x)=0)=0\mathbb{P}(v(x)=0)=0 for a.e. xx, almost surely the weak Sard property holds, hence the renormalization property; consequently almost surely there is no anomalous dissipation, no anomalous regularization, no Richardson dispersion (mean-square two-particle separation growing like t1/(1−α)t^{1/(1-\alpha)}), and Yaglom's law (the third-order structure-function flux relation for passive scalars, the analogue of Kolmogorov's four-fifths law) fails. Thus in two dimensions the random question is settled on the rigidity side at every positive regularity — in sharp contrast with the deterministic autonomous two-dimensional anomalies cited above.

In three dimensions a broad supply of divergence-free fields is given by the Clebsch representation v=∇φ1×∇φ2v=\nabla\varphi_1\times\nabla\varphi_2, which is automatically divergence-free. Bagnara–Boutros–De Lellis–Mayboroda prove that if the random pair φ=(φ1,φ2)\varphi=(\varphi_1,\varphi_2) is almost surely of class C1,αC^{1,\alpha} with α>1/8\alpha>1/8 and satisfies a small-ball nondegeneracy condition on the Jacobian, P(∣Dφ(x0)−M0∣≤r)≤C0 r6\mathbb{P}(|D\varphi(x_0)-M_0|\le r)\le C_0\,r^{6} for all x0x_0 and matrices M0M_0, then almost surely the associated field has the renormalization property, and hence no dissipation anomaly. The proof passes through a probabilistic Morse–Sard theorem of independent interest: a random φ∈C1,α(Td,Rd−j)\varphi\in C^{1,\alpha}(\mathbb{T}^d,\mathbb{R}^{d-j}) with small-ball control P(∣Dφ(x0)−M0∣≤r)≤C0 rd(d−j)\mathbb{P}(|D\varphi(x_0)-M_0|\le r)\le C_0\,r^{d(d-j)} almost surely satisfies the classical Morse–Sard property once α>j/(2+j)\alpha>j/(2+j); the authors state that they do not know whether this exponent is sharp for any choice of dd and jj. The 1/81/8 threshold emerges from level-set geometry of the two potentials (an absence-of-triods argument, excluding the Y-shaped triple junctions that obstruct the deterministic theory). Their Conjecture 1.3 — that a deterministic planar divergence-free field allows dissipation anomalies if and only if its stream function violates the weak Sard property — was disproved by Pappalettera, who constructed for every α∈(0,1)\alpha\in(0,1) compactly supported planar divergence-free CαC^\alpha fields whose stream function violates the weak Sard property while no dissipation anomaly occurs; weak-Sard failure is thus strictly weaker than anomalous dissipation.

For unstructured ensembles the state of the art is due to Colombo–Hess-Childs–Rowan, who study multiscale finite-range random fields: superpositions of independent, compactly supported pieces living on geometrically decaying length scales, a class that contains Gaussian fields with power-law spectra ∣k∣−d/2−α|k|^{-d/2-\alpha} and is thus a standard model of rough turbulent velocities. Their central mechanism is the sweeping effect: in an autonomous field the large scales transport particles rapidly across the small scales, and the resulting stochastic averaging regularizes the effective Lagrangian dynamics, raising the effective Lagrangian regularity (the regularity of the particle trajectories, as opposed to that of the field itself) above the bare Eulerian one (at least to α/(1−α)\alpha/(1-\alpha) for a CαC^\alpha field). They prove that for such fields of Hölder exponent α>1/2\alpha>1/2 the zero level set is almost surely small, the regular Lagrangian flow is almost surely unique, and renormalization — hence absence of anomalous dissipation — holds, a dimension-independent statement requiring no geometric structure. Conversely, in dimensions d≥4d\ge 4 and for every α<1/2\alpha<1/2 they construct random divergence-free multiscale fields for which almost surely the ODE flow and the transport equation are nonunique, refuting the hope that every nondegenerate random continuous divergence-free field in d≥3d\ge 3 is almost surely renormalized. What remains open for their framework is whether the 1/21/2 threshold is sharp for typical (e.g. Gaussian) fields, whether the nonuniqueness examples extend to d=3d=3, and whether nonuniqueness can be upgraded to genuine anomalous dissipation.

On the flexibility side, the only almost-sure results for random autonomous incompressible velocities are those of Armstrong–Bou-Rabee–Kuusi, whose self-similar ensembles v=∇ ⁣⋅kv=\nabla\!\cdot k, with kk a random field of small positive Hurst exponent (the exponent controlling the roughness of kk), have velocity regularity close to −1-1 and exhibit anomalous regularization and superdiffusion almost surely; the regime of positive Hölder exponent is untouched, and they leave the associated spontaneous stochasticity and anomalous dissipation open. In summary, between the rigidity exponents (00 in d=2d=2; 1/81/8 for structured and 1/21/2 for unstructured ensembles in d≥3d\ge 3) and the flexibility frontier near −1-1, the almost-sure regularity threshold for anomalous dissipation under random autonomous divergence-free advection is undetermined.

Problem Statement

Fix d≥2d\ge 2 and consider probability measures on autonomous (time-independent), mean-free (of spatial average zero), divergence-free vector fields on the torus Td\mathbb{T}^d ('random autonomous drifts'). Say that a measure P\mathbb{P} allows almost-sure anomalous dissipation if, for P\mathbb{P}-almost every sampled field vv of Hölder class CαC^\alpha, there exist T>0T>0 and a smooth initial datum whose advection-diffusion solution θε\theta_\varepsilon satisfies lim sup⁡ε↓0ε∫0T∫Td∣∇θε∣2 dx dt>0\limsup_{\varepsilon\downarrow 0}\varepsilon\int_0^T\int_{\mathbb{T}^d}|\nabla\theta_\varepsilon|^2\,dx\,dt>0 (for sampled fields of nonpositive regularity, θε\theta_\varepsilon is understood in the established renormalized/singular solution sense for such drifts). The problem is to determine whether there is a sharp regularity threshold for this property, and its value, along three linked facets.

(1) Rigidity. Establish, or disprove, almost-sure renormalization (which in particular precludes anomalous dissipation, anomalous regularization, Richardson dispersion, and spontaneous stochasticity) for the widest classes of random autonomous divergence-free fields in each dimension. Currently known: all nondegenerate random continuous fields in d=2d=2; random Clebsch-structured fields in d≥3d\ge 3 with Hölder exponent above 1/81/8; multiscale finite-range fields, including Gaussian power-law ensembles, with exponent above 1/21/2 in any dd. Determine: whether 1/81/8 is sharp for the Clebsch class, or can be lowered; what the threshold is for natural unstructured ensembles (e.g. Gaussian) in d=3d=3 below 1/21/2; and whether the d≥4d\ge 4 nonuniqueness examples of Colombo-Hess-Childs-Rowan below 1/21/2 extend to d=3d=3 and to typical fields, and can be upgraded from transport nonuniqueness to genuine anomalous dissipation.

(2) Flexibility. Construct, for the largest possible regularity exponent α\alpha, a nondegenerate probability measure on autonomous divergence-free CαC^\alpha fields on Td\mathbb{T}^d (nondegenerate: satisfying a natural nondegeneracy hypothesis analogous to the pointwise-law condition P(v(x)=0)=0\mathbb{P}(v(x)=0)=0 for a.e. xx used in two dimensions or the small-ball condition on the law of DφD\varphi used in the Clebsch theorem, and not a point mass on a deterministic example) that allows almost-sure anomalous dissipation; the current frontier is regularity exponent close to −1-1. Decide, in particular, whether almost-sure anomalous dissipation is possible for some nondegenerate ensemble at some α<1/2\alpha<1/2 in d≥3d\ge 3 (in d=2d=2 the rigidity theorem already confines any construction to degenerate pointwise laws or genuinely negative regularity), or whether randomness forces a genuine threshold strictly above the deterministic frontier.

(3) Mechanism. Determine whether the exponent j/(2+j)j/(2+j) in the probabilistic Morse-Sard theorem is sharp for any choice of dd and jj — i.e. whether there exist random maps φ∈C1,α(Td,Rd−j)\varphi\in C^{1,\alpha}(\mathbb{T}^d,\mathbb{R}^{d-j}) satisfying the small-ball nondegeneracy condition that almost surely fail the classical Morse-Sard property at the borderline or below — since this exponent governs the level-set geometry behind the 1/81/8 rigidity result; and more broadly identify which mechanism (level-set geometry of potentials versus sweeping-driven stochastic cancellation) controls the threshold in each dimension.

The common objective is the existence, location, and mechanism of the almost-sure regularity threshold for anomalous dissipation under random autonomous divergence-free advection. The problem is solved for a given class when matching rigidity and flexibility results pin the threshold, or when it is proved that no threshold exists (almost-sure anomalous dissipation for a natural ensemble at regularity arbitrarily close to 11, matching the deterministic landscape). Each facet’s standing relative to this criterion is explicit: facets (1) and (2) resolve it only jointly, since a threshold for a class is pinned only when a result on one side meets its counterpart on the other, whereas the facet (3) determination — Morse-Sard sharpness and identification of the controlling mechanism — does not by itself pin the threshold for any class. Substantial partial progress — improved exponents in either direction, removal of structural hypotheses, sharpness counterexamples, mechanism determinations, or almost-sure anomalous-dissipation constructions above regularity −1+ε-1+\varepsilon — counts toward this objective without solving it.

References

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