Uniqueness conjecture for the annihilated Boltzmann hierarchy with hard-sphere interactions

Assume that the collision kernel $\mathsf{B}$ is given by

\mathsf{B}(v-v_{*},\omega)=\frac{1}{2pi}\left|\left(v-v_{*}\right) \cdot omegaright|,

for $(v,v_{*})in \mathbb{R}^{3}timesmathbb{R}^{3}$ and $omegain \mathbb{S}^{2}$, and let $\mathcal{Q}$ be the corresponding Boltzmann operator. Let $f_{0}$ be a non-negative probability distribution satisfying the normalization condition. For the unique solution $f(t)$ of the Boltzmann equation with initial datum $f(0)=f_{0}$, define

$$nu_{k}(t)=f(t)^{\otimes k},\qquad tgeq 0,\qquad kgeq 1.$$

Uniqueness conjecture. The sequence $bm{nu}^{\infty}={nu_{k}}_{k}$ is the unique solution in $L^{\infty}([0,\infty),\mathcal{X})$ to the annihilated Boltzmann hierarchy in the sense of the stated weak-solution definition, with initial data $nu_{k}(0)=f_{0}^{\otimes k}$ for every $kgeq 1$. This uniqueness would yield propagation of chaos for hard-sphere interactions, extending the result beyond bounded collision kernels; the obstruction is the lack of a suitable uniqueness proof for the annihilated Boltzmann hierarchy, since the De Finetti approach fails in this setting.