= Solution
For two solutions with the same initial datum, let $h=f_1-f_2$. Their uniform <compact support> on $[0,T]$ puts $h$ and $h_t$ in one compact region of <phase space>. Smoothness there implies that $E(t)=\int |h(t,x,v)|^2\,dx\,dv$ is finite and continuously differentiable, by <differentiation under the integral sign>. The <transport equation> and <integration by parts> in $x$ yield
$$
E'(t)=-2\int h\,a(v)\cdot\nabla_xh\,dx\,dv
=-\int a(v)\cdot\nabla_x(h^2)\,dx\,dv=0.
$$
There is no boundary contribution because of <compact support>, and $\operatorname{div}_x a(v)=0$. Since $E(0)=0$, $h=0$ almost everywhere and then everywhere by <continuity>. \b[The solution is unique in $\mathcal C_T$.] The same calculation gives <Lp conservation for incompressible transport> at $p=2$.
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