Nonuniqueness for the periodic hard-sphere Boltzmann equation

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 L${}^1$, and their collision gains and losses are integrable.