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Uniqueness of DiPerna-Lions renormalized solutions of the hard-sphere Boltzmann equation

DiPerna and Lions (1989) proved global existence and weak stability of renormalized solutions of the inhomogeneous Boltzmann equation (∂t+v⋅∇x)F=Q(F,F)(\partial_t+v\cdot\nabla_x)F=Q(F,F) for any nonnegative initial density with finite mass, energy and entropy, ∫F0(1+∣v∣2+∣log⁡F0∣)<∞\int F_0(1+|v|^2+|\log F_0|)<\infty; these bounds alone do not make the collision term locally integrable. Uniqueness is known only under much stronger control (Maxwellian upper bounds, perturbations of a Maxwellian in weighted Sobolev spaces), and for the spatially homogeneous hard-sphere equation. The uniqueness problem for general renormalized solutions is listed as open, for example in Silvestre's 2022 survey of open problems in kinetic equations, and Gismondi, Golding and Novack (2026) conjectured nonuniqueness. For the hard-sphere kernel on T3×R3\mathbb T^3\times\mathbb R^3, are renormalized (entropy) solutions with finite mass, energy and entropy determined uniquely by their initial datum?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Kinetic theory; Boltzmann equation, weak solutions
Posed by
Open since the DiPerna-Lions existence theory (1989); cited by the manuscript as open in L. Silvestre, Regularity estimates and open problems in kinetic equations (2022, Section 2)
Year posed
1989
Years open
37y
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
30 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1 (principal): there are R<∞R<\infty and a nonnegative F0F_0 with suppvF0⊂BR(0)\mathrm{supp}_vF_0\subset B_R(0) and ∫F0(1+∣v∣2+∣log⁡F0∣)<∞\int F_0(1+|v|^2+|\log F_0|)<\infty on T3×R3\mathbb T^3\times\mathbb R^3 admitting two global entropy solutions with exact local conservation of mass, momentum and kinetic energy, differing on a set of positive measure, both in C([0,∞);L1)C([0,\infty);L^1), with ∫0T∫⟨v⟩k(Q++Q−)<∞\int_0^T\int\langle v\rangle^k(Q^++Q^-)<\infty for all T,kT,k. The companion (September 23) proves nonuniqueness in a class with local mass and total momentum conservation. The datum has unbounded amplitudes at shrinking spatial scales. It does not give nonuniqueness for a dense or generic set of data (the Gismondi-Golding-Novack conjecture), nor for other kernels, the whole space, or the Landau equation.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts: 'Nonuniqueness for the periodic hard-sphere Boltzmann equation' (September 23, 2026) gave the first construction in a renormalized class with local mass and total momentum conservation; the principal manuscript (October 5, 2026) keeps the concentrating-jet mechanism, cites the earlier one, and builds a new global continuation with exact local conservation of mass, momentum and energy.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript and Definition 1.1 were read against the uniqueness question. The solution class requires the renormalized equation, separately integrable normalized gain and loss, global energy and entropy-dissipation inequalities, and exact local balances for 1,v,∣v∣2/21,v,|v|^2/2; the datum has bounded velocity support and finite mass, energy and absolute entropy. The release has a Lean challenge for this family (BoltzmannNonuniqueness.json, declaration OAI.BoltzmannNonuniqueness.nonuniqueness, solution module present at the pinned commit) but it is not in the formalization catalogue, and its statement, read here, formalizes the September 23 companion's weaker class: local mass, total momentum, energy inequality, strong continuity only on an initial interval. The lean/docs page says the local-conservation refinement is outside it. So the entry's headline is not covered and the tier stays unreviewed; not rebuilt here. The result is one datum, hard spheres, periodic box only.

Sources

Changelog1 change

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