= Solution
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 $\mathbb R^3_x\times\mathbb R^3_v$, 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> gives
$$
\|f_t\|_p^p=\int_{\mathbb R^6}|f_0(z)|^pJ(t,z)\,dz.
$$
Thus the <Lp conservation for incompressible transport> is
$$
\boxed{\nabla_v\cdot F=0\quad\Longrightarrow\quad\|f_t\|_p=\|f_0\|_p,\quad1\leq p<\infty}.
$$
Without 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 $\|f_t\|_p=e^{3t/p}\|f_0\|_p$. The <essential supremum> is still conserved by a complete invertible flow, because composition does not change the range of values.
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