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)

Manuscripts (2)

Nonuniqueness with local conservation for the hard-sphere Boltzmann equation

October 5, 2026 · source · cite as \cite{OAI:Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation-October-5-2026}

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.

Nonuniqueness for the periodic hard-sphere Boltzmann equation

September 23, 2026 · source · cite as \cite{OAI:Nonuniqueness-for-the-periodic-hard-sphere-Boltzmann-equation-September-23-2026}

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.

Result 363 · Partial differential equations. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.