Result 363, Partial differential equations

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.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

The Boltzmann equation describes how a gas evolves through particle motion and collisions. The manuscript claims that even exact local conservation laws and entropy constraints can allow two different futures from the same initial state.

What changes?

The manuscript reports two distinct global entropy solutions, meaning evolutions satisfying entropy constraints, for the three-dimensional hard-sphere Boltzmann equation with periodic spatial boundaries. This equation tracks particle density by position and velocity in a gas with hard-sphere collisions. Both start from one nonnegative density with bounded velocity support and finite mass, energy, and absolute entropy. Both are strongly continuous in L1, so density changes continuously in integrated absolute difference, and exactly conserve mass, momentum, and kinetic energy locally.

What does that help mathematicians do?

The claimed construction rules out uniqueness based on these conditions alone, not uniqueness for every initial density or in more restrictive solution classes. The collision gain and loss terms are also integrable with every polynomial velocity weight on each bounded time interval. Thus, these integrability conditions do not remove the ambiguity either. Researchers seeking uniqueness would need additional restrictions that distinguish or exclude the constructed evolutions.

Are there practical applications?

Its immediate value is foundational for gas dynamics: it tests whether the equation and stated admissibility conditions determine a single evolution. The reported example identifies a limit of that mathematical framework. It does not establish competing outcomes for a laboratory gas or a numerical simulation, but clarifies what these solution conditions alone can guarantee.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Nonuniqueness with local conservation for the hard-sphere Boltzmann equation

October 5, 2026 78 pages

We prove nonuniqueness for the three-dimensional periodic hard-sphere Boltzmann equation among global entropy solutions satisfying exact local conservation of mass, momentum, and kinetic energy. We construct one nonnegative initial density with bounded velocity support and finite mass, energy, and absolute entropy that gives rise to two distinct such solutions. Both are strongly continuous in L1, and their collision gain and loss terms are integrable with every polynomial velocity weight on every bounded time interval.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation-October-5-2026,
  author = {{OpenAI}},
  title = {{Nonuniqueness with local conservation for the hard-sphere Boltzmann equation}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation-October-5-2026/paper.pdf}{OAI:Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation-October-5-2026}},
  year = {2026}
}

Nonuniqueness for the periodic hard-sphere Boltzmann equation

September 23, 2026 65 pages

We prove nonuniqueness for the periodic hard-sphere Boltzmann equation by constructing two distinct global renormalized solutions with the same nonnegative initial density on T3×R3\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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/363.md.

Nonuniqueness with local conservation for the hard-sphere Boltzmann equation

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves nonuniqueness for the periodic hard-sphere Boltzmann equation by constructing two distinct global renormalized solutions on T3×R3\mathbb T^3\times\mathbb R^3 with the same nonnegative initial density. The data have 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 L1L^1, with integrable collision gains and losses. These are the conditions of the selected periodic result; later local-conservation refinements are outside it.

Comparator links

Result Comparator statement
Two periodic hard-sphere Boltzmann solutions with the same data BoltzmannNonuniqueness.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.