= Spatially homogeneous Boltzmann equation
{title2=$\partial_tf=Q(f,f)$}
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, $\|Q(f,g)\|_1\leq C\|f\|_1\|g\|_1$ 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.
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