Nonuniqueness with local conservation for the hard-sphere Boltzmann equation

1 · The essentials

An equation can describe how a gas changes without always selecting just one possible future. The main manuscript shows this for a model in which particles collide like hard spheres, in three dimensions with repeating spatial boundaries. It constructs one starting distribution of particles that leads to two different evolutions, both defined for all future times. The starting distribution has a bound on particle speeds and finite amounts of mass, energy, and a quantity measuring disorder. Both evolutions obey exact local accounting rules for mass, momentum, and kinetic energy, so the difference is not explained by a failure of those rules. Both also change continuously in an overall measure of the difference between distributions. The main result has a Lean formalization, although the supplied description does not specify which individual claims it covers.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine two different schedules for moving goods through a repeating network of warehouses, starting from the same inventory. Both schedules can obey exact accounting at every location without being the same schedule. This illustrates why accounting rules alone need not select a unique evolution; it does not reproduce the gas equation.

Why it matters

The result shows that exact local conservation does not, by itself, restore uniqueness in the class of solutions considered. The two evolutions also meet entropy and continuity requirements, making the distinction more specific than simply finding two unrestricted solutions. The companion manuscripts separate weaker conservation and time-continuity conclusions from the stronger conclusions of the main manuscript.

2 · Going further

The problem concerns the hard-sphere Boltzmann equation on $T^{3} \times R^{3}$ : position is periodic in three dimensions, while velocity ranges over three-dimensional space. The main manuscript constructs a nonnegative initial density with bounded velocity support and finite mass, energy, and absolute entropy. From this same density, it obtains two distinct global entropy solutions satisfying exact local conservation of mass, momentum, and kinetic energy.

Both solutions in the main manuscript are strongly continuous in $L^{1}$ . Their collision gain and loss terms are integrable with every polynomial velocity weight on every bounded time interval. These are additional controls on the solutions, not a uniqueness conclusion: the construction still yields two different evolutions. The abstract describes the argument at the level of constructing a shared initial density and two solutions; it does not explain the construction mechanism or its intermediate steps.

The companion manuscript also constructs two distinct global solutions from the same nonnegative initial density with the stated support and finiteness properties, but describes them as renormalized solutions. It states local mass conservation, total momentum conservation, and global energy and entropy-dissipation inequalities. Its strong $L^{1}$ continuity and integrability of collision gains and losses are stated on a common initial interval. By contrast, the main manuscript states exact local conservation of all three quantities, strong $L^{1}$ continuity globally, and polynomially weighted collision integrability on every bounded time interval.

Useful notions

Initial data and time evolution
Particle distributions in position and velocity
Local versus total conservation
Integrability and $L^{1}$ continuity
Entropy conditions for solutions

What was known before / what is new

Before

The companion manuscript states nonuniqueness among global renormalized solutions for the periodic hard-sphere Boltzmann equation. Its two solutions conserve local mass and total momentum and satisfy global energy and entropy-dissipation inequalities, with additional continuity and collision-integrability properties on a common initial interval. The supplied material gives no broader historical account or chronology.

New

The main manuscript shows nonuniqueness while requiring exact local conservation of mass, momentum, and kinetic energy. It also states strong $L^{1}$ continuity for both global entropy solutions and integrability of their collision gains and losses with every polynomial velocity weight on every bounded time interval. A Lean formalization exists for the main result.

Limits and assumptions

The setting is the three-dimensional periodic hard-sphere Boltzmann equation, and the construction concerns one particular nonnegative initial density, not every admissible density. Bounded velocity support is stated for the initial density; the supplied abstracts do not state that this bound persists. Nonuniqueness is asserted in the specified solution classes and does not settle uniqueness in other classes. The main result has a Lean formalization, but the supplied material does not delineate its coverage of individual claims or companion results. No further open problems are identified in the supplied text.

Mini glossary

Nonuniqueness: The existence of two distinct solutions with the same initial data.
Global solution: A solution defined for all future times.
Local conservation: Conservation expressed through balance at each location, rather than only through totals over the whole space.
Bounded velocity support: A property meaning that the density is zero outside a bounded set of velocities.
Strong $L^{1}$ continuity: Continuity in time measured by the integral of the absolute difference between distributions.
Collision gain and loss terms: The parts of the collision expression accounting for additions to and removals from the particle distribution.

Keywords : Boltzmann equation · Nonuniqueness · Local conservation

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/363.md

Manuscripts in this family

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Nonuniqueness with local conservation for the hard-sphere Boltzmann equation

05/10/2026

To be verified

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 GitHub folder

Nonuniqueness for the periodic hard-sphere Boltzmann equation

23/09/2026

To be verified

We prove nonuniqueness for the periodic hard-sphere Boltzmann equation by constructing two distinct global renormalized solutions with the same nonnegative initial density on $T^{3} \times R^{3}$ . 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 GitHub folder