  We consider the 3D Boltzmann equation with the constant collision kernel. We
investigate the well/ill-posedness problem using the methods from nonlinear
dispersive PDEs. We construct a family of special solutions, which are neither
near equilibrium nor self-similar, to the equation, and prove that the
well/ill-posedness threshold in $H^{s}$ Sobolev space is exactly at regularity
$s=1$, despite the fact that the equation is scale invariant at
$s=\frac{1}{2}$.
