Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 7 / 1 / d / 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 $E(t)=\int |h(t,x,v){|}^{2}dxdv$ is finite and continuously differentiable, by differentiation under the integral sign. The transport equation and integration by parts in $x$ yieldThere is no boundary contribution because of compact support, and ${\mathrm{div}}_{x}a(v)=0$. Since $E(0)=0$, $h=0$ almost everywhere and then everywhere by continuity. The solution is unique in ${\mathcal{C}}_T$. The same calculation gives Lp conservation for incompressible transport at $p=2$.