Spatially homogeneous Boltzmann equation (${\partial }_{t}f=Q(f,f)$)

When the velocity distribution is independent of position and there are no forces, the free transport equation contributes zero and only collisions evolve $f$. For a bounded cutoff kernel, $\Vert Q(f,g){\Vert }_1\le C\Vert f{\Vert }_1\Vert g{\Vert }_1$ gives local existence and uniqueness by the Banach fixed-point theorem. Positivity and mass conservation bound the norm and permit global continuation in this particular model. Stronger kernels require moment or weighted estimates.