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Anomalous regularization of passive scalars up to the Obukhov-Corrsin exponent

A mix of human-written and AI-generated textHuman understanding: some partsmath.AP — Analysis of PDEsmath-ph — Mathematical Physics

Contributed by Scott Armstrong ↗, Vlad Vicol ↗

Version 1 / Nov 06, 2026 / CC BY 4.0

Abstract

We consider the advection-diffusion equation ∂tθ+b⋅∇θ−κΔθ=0\partial_t\theta+\mathbf{b}\cdot\nabla\theta-\kappa\Delta\theta=0 on the two-dimensional torus T2\mathbb{T}^2, where b\mathbf{b} is a divergence-free velocity field constructed as in arxiv:2305.05048, H\"older continuous with an exponent $\alpha<\nf13$ and built from shear flows alternating in direction at infinitely many scales. We prove anomalous regularization: for every ss below the Obukhov--Corrsin exponent 12(1−α)\frac12(1-\alpha), with b\mathbf{b} depending on ss, the solutions are bounded in~C0,s(T2)C^{0,s}(\mathbb{T}^2) at positive times by a constant times the L2(T2)L^2(\mathbb{T}^2) norm of the initial data, uniformly in the diffusivity κ\kappa in a union of bands accumulating at 00. As a consequence, we obtain anomalous dissipation at a uniform rate: solutions dissipate a fixed fraction of their variance in unit time, so that the variance decays exponentially, with constants independent of κ\kappa, of the initial datum and of the initial time. The proof combines iterated quantitative homogenization with large-scale regularity: at each scale the solution is compared with a solution of the homogenized equation of that scale. Since the derivatives consumed by each comparison are regenerated at every scale, the H\"older norm is carried from the coarsest scale to the finest by induction. Each of our main results, together with its proof, is fully formalized in the Lean 4 proof assistant.

Provenance statement

This paper is based on a proof outlined by the authors as early as 2022. We have written about it in our papers arxiv:2305.05048 and in more detail in the proceedings article arxiv:2503.11744. In October 2026, this proof outline was fed to LLMs (Claude Opus 5.5 and GPT 6 Astra) who, after working closely with the authors, produced a complete but very long (612 page) natural language (slop) proof. We then autoformalized this slop, and worked with the agents to simplify the proof as much as we could. The paper is split into two parts: the first part (30 pages) outlines the proof without all the technical details. The second part (135 pages) presents those details in full. The authors spent some effort rewriting the first 30 pages; the last 135 pages is the best slop they could get the agents to write. The point of this paper and its lean formalization is to demonstrate that the original vision of the authors from 2022 is correct, even if the details are quite complicated. In particular, interior regularity is indeed obtained by a large-scale regularity cascade, which is dual to the homogenization cascade that gives anomalous dissipation.

Formalizations

References

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Version history

  1. v1 · 165 pagesSubmitted by Scott ArmstrongInitial depositCurrentNov 06, 2026