Elgindi and Liss constructed a divergence-free velocity field on the two-dimensional torus, smooth in time and with spatial modulus of continuity ω(s)=s(1+∣log⁡s∣4)\omega(s)=s(1+|\log s|^4), for which the advection–diffusion equation exhibits anomalous dissipation for every smooth mean-zero initial datum, and asked whether the Osgood condition — the exact integrability threshold on the modulus of continuity that guarantees uniqueness for the transport equation — is the sharp phase boundary for non-uniqueness and anomalous dissipation. The non-uniqueness half was resolved affirmatively by Colombo and Kumar (2026), who constructed, for every non-Osgood modulus, a divergence-free field of that modulus with non-unique flow maps and non-unique solutions of the continuity equation. The surviving open core, and the subject of this problem, is the dissipative half: can anomalous dissipation be produced by a divergence-free velocity field whose modulus of continuity is at least as strong as the s∣log⁡s∣2s|\log s|^2 level at which both the Elgindi–Liss alternating-shear technique and the non-parallelized Colombo–Kumar scheme saturate (that level itself is not reached: Elgindi–Liss achieve only powers p>2p>2, and even those only for suitable initial data) — in particular ω(s)=s∣log⁡s∣p\omega(s)=s|\log s|^p with 1<p≤21<p\le 2 — or, in full generality, by a field with any prescribed non-Osgood modulus? An answer would fix the exact minimal spatial regularity of an incompressible flow compatible with a scalar dissipative anomaly, the passive-scalar analogue of Onsager's critical regularity resolved to logarithmic precision.

Background

A passive scalar is a quantity transported by a fluid without reacting back on the flow: on the torus Td\mathbb{T}^d it solves the advection–diffusion equation ∂tθκ+u⋅∇θκ=κΔθκ\partial_t\theta^\kappa+u\cdot\nabla\theta^\kappa=\kappa\Delta\theta^\kappa with molecular diffusivity κ>0\kappa>0 and a prescribed divergence-free (incompressible) velocity field uu; setting κ=0\kappa=0 gives the transport equation ∂tθ+u⋅∇θ=0\partial_t\theta+u\cdot\nabla\theta=0. Obukhov–Corrsin theory of scalar turbulence predicts that a sufficiently rough flow transfers scalar variance to arbitrarily small scales fast enough that the dissipated amount κ∫0T∫∣∇θκ∣2 dx dt\kappa\int_0^T\int|\nabla\theta^\kappa|^2\,dx\,dt stays bounded away from zero as κ→0\kappa\to0 even for smooth initial data; this dissipative anomaly is the passive-scalar analogue of Onsager's conjecture on energy dissipation in ideal turbulence. Rigorous deterministic constructions were given by Drivas–Elgindi–Iyer–Jeong (alternating 'sawtooth' shear flows, for a large class of data), Colombo–Crippa–Sorella (finite-time mixers, with lack of selection in the vanishing-diffusivity limit), and Armstrong–Vicol (fractal homogenization, dissipating all H1H^1 data with energy continuous in time). All of these velocity fields are only Hölder- or Sobolev-type regular in space.

The present question lives on a much finer, near-Lipschitz scale. A modulus of continuity is a concave, increasing function ω\omega with ω(0)=0\omega(0)=0; a velocity field has modulus ω\omega if ∣u(t,x)−u(t,y)∣≤C ω(∣x−y∣)|u(t,x)-u(t,y)|\le C\,\omega(|x-y|). Hölder regularity CαC^\alpha corresponds to ω(s)=sα\omega(s)=s^\alpha, while ω(s)=s ∣log⁡s∣p\omega(s)=s\,|\log s|^p is strictly more regular than every Hölder class yet weaker than Lipschitz. The modulus is called Osgood if ∫01 ds/ω(s)=∞\int_0^1\,ds/\omega(s)=\infty; for the family ω(s)=s∣log⁡s∣p\omega(s)=s|\log s|^p this holds exactly when p≤1p\le 1. Osgood's classical criterion guarantees uniqueness of the ordinary differential equation X˙=u(t,X)\dot X=u(t,X) whose solutions define the flow map (the Lagrangian trajectories), and its PDE counterpart — in the framework of the DiPerna–Lions theory of transport equations, extended to Osgood moduli (see Fjordholm–Mæhlen) — guarantees uniqueness of the transport equation. Under Osgood uniqueness the vanishing-diffusivity limit converges to the unique transport solution, which merely rearranges the scalar and conserves its L2L^2 norm; anomalous dissipation is therefore impossible for fields with an Osgood modulus, and anomalous dissipation is known to imply non-uniqueness of the underlying transport equation (Drivas–Elgindi–Iyer–Jeong).

Elgindi and Liss constructed a divergence-free field u∈C∞([0,T];Cα(T2))u\in C^\infty([0,T];C^\alpha(\mathbb{T}^2)) for every α<1\alpha<1 with modulus ω(s)=s(1+∣log⁡s∣4)\omega(s)=s(1+|\log s|^4), smooth in time, such that the advection–diffusion equation exhibits anomalous dissipation for every mean-zero smooth initial datum and the transport equation is non-unique. Their construction accumulates alternating shears with exploding frequencies at a single 'singular time'. They note that the power 4 can be lowered to any p>2p>2 (with modifications, for suitable data) but that the technique appears to cap at ω(s)=s∣log⁡s∣2\omega(s)=s|\log s|^2, and they pose the sharpness question: is the Osgood condition the exact threshold, i.e. for any non-Osgood modulus can one construct a field of that modulus exhibiting anomalous dissipation, or transport non-uniqueness? Colombo and Kumar resolved the non-uniqueness half: for every non-Osgood modulus ω\omega and every d≥2d\ge 2 there is a divergence-free field v∈C([0,T];Cω(Rd))v\in C([0,T];C^\omega(\mathbb{R}^d)) whose flow has at least two distinct flow maps (non-uniqueness of trajectories on a set of full measure inside the supporting cube) and whose continuity equation admits two distinct L1L^1 solutions from the same absolutely continuous initial datum. Their key new mechanism, 'parallelization' (simultaneous motion across nested spatial scales, as opposed to one scale at a time), was needed precisely to pass the z∣log⁡z∣2z|\log z|^2 barrier; without it they recover exactly the Elgindi–Liss threshold. Their construction, however, yields no anomalous dissipation: the solutions are L1L^1 densities and no vanishing-diffusivity analysis is performed. The dissipative half of the threshold question is what remains open.

Problem Statement

Let ω\omega be a modulus of continuity that fails Osgood's condition, ∫01 ds/ω(s)<∞\int_0^1\,ds/\omega(s)<\infty (in the family ω(s)=s∣log⁡s∣p\omega(s)=s|\log s|^p this means p>1p>1). Determine whether the Osgood condition is sharp for anomalous dissipation, not only for non-uniqueness:

(i) For every non-Osgood modulus of continuity ω\omega, does there exist a bounded divergence-free velocity field uu on [0,T]×Td[0,T]\times\mathbb{T}^d (d≥2d\ge 2) with ∣u(t,x)−u(t,y)∣≤C ω(∣x−y∣)|u(t,x)-u(t,y)|\le C\,\omega(|x-y|) uniformly in time, such that the solutions θκ\theta^\kappa of the advection–diffusion equation ∂tθκ+u⋅∇θκ=κΔθκ\partial_t\theta^\kappa+u\cdot\nabla\theta^\kappa=\kappa\Delta\theta^\kappa exhibit anomalous dissipation, i.e.

lim sup⁡κ→0 κ∫0T∫Td∣∇θκ∣2 dx dt>0,\limsup_{\kappa\to 0}\,\kappa\int_0^T\int_{\mathbb{T}^d}|\nabla\theta^\kappa|^2\,dx\,dt>0,

for at least one explicitly given nontrivial smooth initial datum (a stronger answer would cover a full class of smooth data, as in the all-data theorem of Elgindi–Liss)?

(ii) The concrete landmark case: can this be achieved with ω(s)=s∣log⁡s∣p\omega(s)=s|\log s|^p for some 1<p≤21<p\le 2, i.e. at or strictly below the s∣log⁡s∣2s|\log s|^2 level at which the alternating-shear construction of Elgindi–Liss and the non-parallelized Colombo–Kumar scheme both saturate (Elgindi–Liss reach only p>2p>2, and even that only for suitable initial data), yet still above the Osgood threshold p=1p=1 below which anomalous dissipation is impossible? Sub-question (i) is the mandatory general statement and (ii) is a strictly weaker landmark sub-case of (i): a positive resolution of (i) settles (ii) as well, whereas an affirmative answer to (ii) alone — a single construction for some 1<p≤21<p\le 2 — is only a partial result.

The variant of this question with 'the transport equation has non-unique solutions' in place of anomalous dissipation has been answered affirmatively by Colombo–Kumar and is not part of this problem. An admissible answer to (i) is either an explicit construction with a complete rigorous proof, or a negative answer: a specific non-Osgood modulus for which every bounded divergence-free field of that modulus conserves the scalar's L2L^2 norm in the vanishing-diffusivity limit for all smooth data.

References

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