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Nonuniqueness for the periodic hard-sphere Boltzmann equation
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Abstract
We prove nonuniqueness for the periodic hard-sphere Boltzmann equation by constructing two distinct global renormalized solutions with the same nonnegative initial density on $\mathbb T^3\times\mathbb R^3$. This density has bounded velocity support and finite mass, energy, and absolute entropy. Both solutions conserve local mass and total momentum and satisfy the global energy and entropy-dissipation inequalities. On a common initial interval, they are strongly continuous in L1, and their collision gains and losses are integrable.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- OpenAI, Nonuniqueness with local conservation for the hard-sphere Boltzmann equation (OpenAI Math Release preprint, 5 October 2026). This companion reuses the September concentrating-jet mechanism and claims a new global continuation with exact local mass, momentum and energy conservation and polynomially weighted collision integrability on every bounded time interval. [AI-generated per source] [cites earlier version]
Comments and discussion
- John Evans, Comment on Boltzmann nonuniqueness with exact local conservation (LinkedIn, October 2026). In notes described as ChatGPT-assisted genealogy rather than an independent proof check, Evans compares the jet-and-seed construction and its later conservation refinement with instability methods and concentration/gluing strategies. [family-level]
Formalization
- OpenAI math release, Lean scope note for family 363 (lean/docs/363.md, release of 2026-10-06). The scope note names this paper as the one the formalization accompanies, describing the formalized statement as nonuniqueness of global renormalized solutions with local mass and total momentum conservation, global energy and entropy-dissipation inequalities, and strong L^1 continuity on a common initial interval; it excludes the October local-conservation refinements. [family-level]
- OpenAI math release, OAI/MathematicalPhysics/Boltzmann/Main.lean (openai/math, commit adc7f12). States the theorem
BoltzmannNonuniqueness.nonuniqueness: twoAdmissibleGlobalevolutions from one datum with bounded velocity support, both satisfyingStrongEarly T, which differ at some time in [0,T]; the proof is imported fromUnconditionalMainProof. [family-level] - OpenAI math release, ComparatorChallenges/BoltzmannNonuniqueness.lean (openai/math). The comparator statement linked from the scope note: a self-contained Mathlib statement of the admissibility class and the nonuniqueness theorem, with the proof left as
sorryin this file. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Lu, Wennberg, Solutions with increasing energy for the spatially homogeneous Boltzmann equation (Nonlinear Analysis: Real World Applications 3(2), 243–258, 2002). The paper contrasts its result with Wennberg's homogeneous nonuniqueness examples, which permit increasing energy; this later Lu–Wennberg work on energy-increasing homogeneous solutions is not in the bibliography.
- Bodineau, Gallagher, Saint-Raymond, Simonella, Derivation of the Boltzmann equation from hard-sphere dynamics (after Y. Deng, Z. Hani, and X. Ma) (arXiv:2602.04407, Séminaire Bourbaki Exposé 1247; v2 2026-10-14). This survey states that uniqueness of DiPerna–Lions renormalized solutions "is not known to date" and that it is unknown whether they satisfy the equation distributionally, which bears on the paper's framing and its remark that both solutions obey the unrenormalized equation on the early interval.
- Modena, Székelyhidi, Non-uniqueness for the transport equation with Sobolev vector fields (arXiv:1712.03867, 2017/2018). An analogous nonuniqueness result for linear transport with Sobolev fields, the other main DiPerna–Lions renormalized framework.
Digests and expositions
- Latent Seminar, Boltzmann nonuniqueness and exact local conservation (6 October 2026). This exposition explains the shared concentrating-jet, nonconcentration and branching mechanisms and the later bath-and-retirement continuation argument, while stating that it does not independently verify the complete proofs. [family-level]
Elsewhere
- Emergent Mind: family 363
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Nonuniqueness-for-the-periodic-hard-sphere-Boltzmann-equation-September-23-2026,
author = {{OpenAI}},
title = {{Nonuniqueness for the periodic hard-sphere Boltzmann equation}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Nonuniqueness-for-the-periodic-hard-sphere-Boltzmann-equation-September-23-2026/paper.pdf}{OAI:Nonuniqueness-for-the-periodic-hard-sphere-Boltzmann-equation-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/363.md. The Lean statements have not been checked by people against the paper's claims.
Family 363: Nonuniqueness with local conservation for the hard-sphere Boltzmann equation
Constructs two distinct global entropy solutions of the three-dimensional periodic hard-sphere Boltzmann equation from the same nonnegative initial density, with bounded velocity support and finite mass, energy and absolute entropy. Both are strongly continuous in L1 and satisfy exact local conservation of mass, momentum and kinetic energy.
Other papers in this family: