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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 $p=2$.

The finite-$p$ assertion needs the zero-divergence hypothesis from the preceding part, which is not repeated in this part's printed hypotheses. The norm is on ${\mathbb{R}}_x^3\times {\mathbb{R}}_v^3$, and finite conservation requires $f_0$ to belong to the corresponding Lp space in addition to its smoothness.

Writing ${\Phi }_t$ for the characteristic flow map, the general solution is $f_t=f_0\circ {\Phi }_t^{-1}$. A change of variables givesThus the Lp conservation for incompressible transport isWithout that condition the requested conclusion is false: the force in part (c), together with any nonzero smooth integrable initial datum given by a Gaussian function, gives $\Vert f_t{\Vert }_p=e^{3t/p}\Vert f_0{\Vert }_p$. The essential supremum is still conserved by a complete invertible flow, because composition does not change the range of values.