Well/Ill-Posedness Bifurcation for the Boltzmann Equation with Constant Collision Kernel

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 \(H^{s}\) $H^s$ Sobolev space is exactly at regularity \(s=1\) $s=1$ , despite the fact that the equation is scale invariant at \( s=\frac{1}{2}\) $s=\frac{1}{2}$ .