Anomalous dissipation — the survival of scalar dissipation in the vanishing-diffusivity limit — is the scalar version of Kolmogorov's zeroth law of turbulence. Johansson and Sorella (Duke Mathematical Journal, 2026) constructed the first autonomous (time-independent) divergence-free Hölder velocity field on the two-dimensional torus that produces anomalous dissipation of a passive scalar, using the fluctuation–dissipation link between anomalous dissipation and spontaneous stochasticity of Lagrangian trajectories. Their work leaves open the structure theory for this most rigid class of advecting flows. First, no verifiable sufficient criterion of the balanced-growth type is known that any autonomous planar field can satisfy. Second, failure of the weak Sard property of the stream function is necessary for anomalous dissipation, and the 2026 conjecture of Bagnara, Boutros, De Lellis and Mayboroda that Sard-type failure is also sufficient (in their formulation, imposed on the full critical set) was disproved by Pappalettera, who built an autonomous planar field violating even the weak Sard property yet causing no anomalous dissipation for any initial datum — so the exact geometric boundary is unknown, and it is still undecided whether the classical weak-Sard-violating examples of Alberti, Bianchini and Crippa dissipate. Third, anomalous dissipation cannot hold for all initial data (steady states never dissipate), and it is open whether it can hold for every non-steady datum, or which data dissipate for a given field. The problem is to settle these three linked questions for autonomous planar velocity fields.

Background

A passive scalar is a transported quantity (for example dye concentration or temperature) that does not react back on the fluid motion. For an incompressible velocity field uu on the two-dimensional periodic torus T2\mathbb{T}^2, it obeys the advection–diffusion equation

∂tθκ+u⋅∇θκ=κΔθκ,θκ(0,⋅)=θin,\partial_t \theta_\kappa + u \cdot \nabla \theta_\kappa = \kappa \Delta \theta_\kappa, \qquad \theta_\kappa(0,\cdot) = \theta_{\mathrm{in}},

where κ>0\kappa>0 is the molecular diffusivity. Obukhov–Corrsin scalar-turbulence theory predicts that a turbulent flow stirs the scalar into arbitrarily fine structures so quickly that the total dissipation does not vanish even as κ↓0\kappa\downarrow 0:

lim sup⁡κ↓0 κ∫01 ⁣ ⁣∫T2∣∇θκ∣2 dx dt>0,\limsup_{\kappa\downarrow 0}\,\kappa\int_0^1\!\!\int_{\mathbb{T}^2} |\nabla\theta_\kappa|^2\,dx\,dt > 0,

a property called anomalous dissipation; it is the scalar counterpart of Kolmogorov's zeroth law of turbulence. Rigorous examples require spatially rough (Hölder-continuous, exponent α∈(0,1)\alpha\in(0,1)) velocities, since smooth flows mix too gently.

The autonomous (time-independent) planar case is the most rigid setting. Every autonomous average-free (zero-mean) divergence-free field on the torus has a single-valued periodic stream function, or Hamiltonian, HH with u=∇⊥H:=(−∂2H,∂1H)u = \nabla^\perp H := (-\partial_2 H, \partial_1 H) (a field with nonzero mean, even a constant one, admits no such HH), and trajectories of the inviscid dynamics are confined to level sets of HH; sublevel sets such as {H<c}\{H<c\} are invariant even for non-unique flows. Consequently mixing is at most linear in time (Bonicatto–Marconi), and Bruè–Coti Zelati–Marconi proved that the best possible enhanced dissipation rate (the acceleration of homogenization by the flow) is the algebraic κ−1/3\kappa^{-1/3}, in contrast with the logarithmic rates achievable by time-dependent planar flows. Whether anomalous dissipation can occur at all in this class was posed as an open problem by Johansson–Sorella (2023) and by Elgindi–Liss (2024), the latter calling it of great mathematical, and possibly physical, interest.

The key measure-theoretic obstruction comes from Alberti–Bianchini–Crippa. Write S={x:∇H(x)=0}S=\{x:\nabla H(x)=0\} for the critical set of the Hamiltonian and let E∗E^* be the union, over levels hh, of those connected components of the level set H−1({h})H^{-1}(\{h\}) that have positive length (one-dimensional Hausdorff measure). HH has the weak Sard property when the push-forward of Lebesgue measure on S∩E∗S\cap E^* under HH is singular with respect to one-dimensional Lebesgue measure (in symbols, H♯(1S∩E∗L2)⊥L1H_\sharp(\mathbf{1}_{S\cap E^*}\mathcal{L}^2)\perp\mathcal{L}^1; this is the sense of Alberti–Bianchini–Crippa used by Pappalettera). Alberti–Bianchini–Crippa proved that for autonomous planar bounded divergence-free fields this property is equivalent to uniqueness of bounded solutions of the advection equation and to the DiPerna–Lions renormalization property (compositions β(θ)\beta(\theta) of a solution with a smooth function again solve the equation), which rules out anomalous dissipation. Rowan's analysis of anomalous diffusion in the Kraichnan model (a random, white-in-time model of turbulent advection) shows that anomalous dissipation forces non-uniqueness of the underlying flow; consequently a continuous autonomous planar field that vanishes nowhere cannot produce anomalous dissipation, and any dissipative example must fail the weak Sard property (Johansson–Sorella, Remark 1.3, whose own wording imposes the condition on the full critical set, L1(H(S))=0\mathcal{L}^1(H(S))=0; the two readings are reconciled below).

All previously known constructions of anomalous dissipation exploit explicit time dependence. Drivas–Elgindi–Iyer–Jeong introduced the balanced growth of norms mechanism: time-dependent fields whose inviscid solution develops a self-similar cascade in which Sobolev norms grow in a coordinated way so that the time-integrated gradient blow-up survives the diffusivity tending to zero; Elgindi–Liss achieved anomalous dissipation for every non-constant smooth datum this way, and Armstrong–Vicol, by quantitative (fractal) homogenization — comparing the small-diffusivity equation to a homogenized effective problem — reached every non-constant H1H^1 datum, though only for fields rougher than C1/3C^{1/3} and necessarily time-dependent. The anchor paper details why each of these mechanisms (balanced growth, mixing, homogenization) breaks down in the autonomous planar class: alternating shears need time, homogenization correctors must be cancelled by time derivatives, and the Hamiltonian structure forbids the required rearrangements of the flow.

Johansson–Sorella settled the existence question (arXiv 2024, published in Duke Mathematical Journal 2026): for every α∈[0,1)\alpha\in[0,1) there is an autonomous divergence-free field u∈Cα(T2)u\in C^\alpha(\mathbb{T}^2) and a smooth datum whose unique diffusive solutions exhibit anomalous dissipation, with dissipation continuous in time. The proof uses the fluctuation–dissipation formula of Drivas–Eyink, which identifies anomalous dissipation with spontaneous stochasticity: the variance among stochastic Lagrangian trajectories dXt=u(Xt) dt+2κ dWtdX_t = u(X_t)\,dt + \sqrt{2\kappa}\,dW_t issuing from the same point stays positive as the noise vanishes. Their field funnels a portion of the torus into a fat Cantor set (a closed, nowhere-dense set of positive Lebesgue measure), along which backward stochastic trajectories separate. Via the (2+12)(2+\tfrac{1}{2})-dimensional ansatz (embedding a planar advection problem as one component of a three-dimensional flow), this also answered Questions 2.2 and 2.3 of Bruè–De Lellis on anomalous dissipation for the forced three-dimensional Navier–Stokes equations.

Two 2026 developments sharpen the landscape around the remaining questions. Bagnara–Boutros–De Lellis–Mayboroda proved that a random autonomous planar field (under a mild nondegeneracy condition on the randomness) satisfies the weak Sard property almost surely and hence yields no dissipation anomalies generically, and conjectured a deterministic dichotomy: an autonomous average-free divergence-free field allows dissipation anomalies if and only if its stream function violates the weak Sard property (in their formulation, imposed on the full critical set). Pappalettera disproved the sufficiency direction: for every α∈(0,1)\alpha\in(0,1) he constructed a compactly supported autonomous divergence-free field in Cα(R2)C^\alpha(\mathbb{R}^2) whose Hamiltonian fails even the weak Sard property, yet for which no initial datum dissipates — the limiting flow of his approximating fields prevents the Lagrangian spontaneous stochasticity. He notes that rescaling and periodic extension transfer the example to a field in Cα(T2)C^\alpha(\mathbb{T}^2), so it also disproves the torus-formulated conjecture above. Dribas–Golovnev–Gusev showed that the weak and relaxed Sard properties (the latter imposing singularity of the push-forward of the full critical set) are genuinely distinct for C1,αC^{1,\alpha} Hamiltonians; Pappalettera's example violates the weak Sard property in the S∩E∗S\cap E^* sense and hence also the full-critical-set relaxed version, so its failure of weak Sard is compatible with both readings of the necessary condition quoted above. Thus weak Sard failure is known to be necessary but not sufficient for anomalous dissipation by autonomous planar fields, and the exact geometric boundary, the availability of verifiable criteria, and the data-dependence of dissipation all remain open.

Problem Statement

Fix the class: autonomous (time-independent), average-free (zero-mean), divergence-free velocity fields u∈Cα(T2;R2)u\in C^\alpha(\mathbb{T}^2;\mathbb{R}^2) with u=∇⊥Hu=\nabla^\perp H, for arbitrary but fixed Hölder exponent α∈[0,1)\alpha\in[0,1), advecting a passive scalar through

∂tθκ+u⋅∇θκ=κΔθκ,θκ(0,⋅)=θin,\partial_t\theta_\kappa + u\cdot\nabla\theta_\kappa = \kappa\Delta\theta_\kappa, \qquad \theta_\kappa(0,\cdot)=\theta_{\mathrm{in}},

and say anomalous dissipation holds for a datum θin\theta_{\mathrm{in}} if lim sup⁡κ↓0 κ∫01 ⁣ ⁣∫T2∣∇θκ∣2 dx dt>0\limsup_{\kappa\downarrow 0}\,\kappa\int_0^1\!\!\int_{\mathbb{T}^2}|\nabla\theta_\kappa|^2\,dx\,dt>0. Initial data are taken smooth, θin∈C∞(T2)\theta_{\mathrm{in}}\in C^\infty(\mathbb{T}^2), as in the anchor's construction, unless a question explicitly concerns a wider class of data; for every κ>0\kappa>0 the diffusive problem then has a unique solution. The common objective is a structure theory of anomalous dissipation in this class: deciding, by checkable conditions, which fields dissipate and which initial data dissipate for them. Concretely, answer the following three linked but distinct open questions, which the anchor paper explicitly leaves open.

  1. Verifiable criteria (balanced-growth analogue). Exhibit a set of verifiable growth/explosion hypotheses — in the spirit of the balanced-growth conditions satisfied by time-dependent constructions, formulated for solutions of the inviscid advection equation with uu or with its regularizations — that (i) provably imply anomalous dissipation and (ii) are satisfied by some autonomous planar divergence-free field.

  2. The geometric boundary between dissipation and no dissipation. Failure of the weak Sard property of HH (in the S∩E∗S\cap E^* sense defined in the Background) is necessary for anomalous dissipation (Johansson–Sorella, Remark 1.3) and not sufficient (Pappalettera's weak-Sard-violating, non-dissipating example, available on the torus by rescaling and periodic extension). Determine which geometric/measure-theoretic properties of the Hamiltonian decide anomalous dissipation. At minimum — the source-anchored core — decide whether the known weak-Sard-violating constructions of Alberti–Bianchini–Crippa do or do not produce anomalous dissipation for some datum. Separating the dissipative example of Johansson–Sorella from the non-dissipative example of Pappalettera is a necessary test for any proposed determining property, but the mere separation of these two named examples does not by itself answer the question.

  3. Initial-data dependence. Decide whether one can construct an example in the class in which anomalous dissipation occurs for every initial datum that is not a steady state (data of the form F(H)F(H), for instance θin=H\theta_{\mathrm{in}}=H; Johansson–Sorella's Remark 1.2 proves it cannot occur for all data, since steady states never dissipate). More generally, for an autonomous planar field that dissipates some datum, characterize the initial data θin\theta_{\mathrm{in}} whose solutions do and do not exhibit anomalous dissipation.

The three questions are distinct open problems: resolving any one of them is progress on that item, not a completion of the others. The structure-theory objective is settled only when all three are resolved for the class above.

References

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