= Solution
The <chain rule> gives $d[f(t,X(t),V(t))]/dt=0$ along each <characteristic curve>. Pulling the initial datum back by the inverse <characteristic flow map> therefore gives
$$
\boxed{f(t,x,v)=f_{\mathrm{in}}(x-ta(v),v).}
$$
Direct differentiation also gives $f_t=-a(v)\cdot\nabla_xf$, so this is a classical solution of the <transport equation>. Composition with a <smooth function> makes $f$ smooth jointly in $(t,x,v)$; nonnegative initial data remain nonnegative.
If $K=\operatorname{supp}f_{\mathrm{in}}$, then $\operatorname{supp}f(t)=S_{0,t}(K)$. A continuous image of a <compact set> is compact. Thus <compact support> is preserved at every finite time; there is no claim of one fixed compact support for all $t\geq0$.
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