Nonuniqueness with local conservation for the hard-sphere Boltzmann equation

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 \(L^1\), and their collision gain and loss terms are integrable with every polynomial velocity weight on every bounded time interval.