Nonuniqueness with local conservation for the hard-sphere Boltzmann equation

What OpenAI claims

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)

Papers (2)

1. Nonuniqueness with local conservation for the hard-sphere Boltzmann equation

   5 October 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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.

2. Nonuniqueness for the periodic hard-sphere Boltzmann equation

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's 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 𝕋³×ℝ³. 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.