= Solution
Fix an arbitrary finite $T$ and use the <Banach space> of <bounded continuous functions> $C_b([0,T]\times\mathbb R^{2d})$ with the <supremum norm>. Set $A=\|k_1\|_\infty\|k_2\|_1$. For $f$ in this space, the velocity <integral>
$$
H_f(t,x)=\int k_2(v_*)f(t,x,v_*)\,dv_*
$$
is bounded by $\|k_2\|_1\|f\|_\infty$. If $(t_n,x_n)\to(t,x)$, the integrands converge pointwise and are bounded by the integrable function $\|f\|_\infty k_2$. The <dominated convergence theorem> proves continuity. Thus the <rank-one gain on bounded continuous functions>, $Kf=k_1H_f$, is continuous and bounded. The same theorem applied to the time <integral> shows that $\tau$ maps this space into itself, including at $t=0$.
Time ordering gives the <factorial bound for a Volterra iterate>:
$$
\|(\tau^n h)(t)\|_\infty\leq\frac{A^nt^n}{n!}\sup_{s\leq t}\|h(s)\|_\infty.
$$
Consequently the <Volterra series for the linear Boltzmann equation>
$$
\boxed{f=\sum_{n=0}^\infty\tau^nF(f_0)}
$$
converges uniformly on the whole finite time slab, so its sum is continuous and bounded and satisfies the <fixed point> equation. Tracking the damping factors gives the sharper pointwise bound $\|f(t)\|_\infty\leq e^{(A-1)t}\|f_0\|_\infty$. This proves existence for every finite $T$, without requiring $AT<1$ or uniform continuity of $f_0$. Boundedness on finite slabs does not assert a uniform bound over infinite time.
Back to article page
