 Solution

Formally the Jacobian determinant of the characteristic flow map satisfies $J^{\prime }=\mathrm{div}B({\Phi }_t)J=0$, $J(0)=1$, so $J=1$. Changing variables $z={\Phi }_t(z_0)$ givesTaking the $p$th root proves $\boxed{\Vert f(t){\Vert }_{L^p}=\Vert f_I{\Vert }_{L^p}}$ for $1\le p<\infty$. Equivalently multiply the equation by $p{f}^{p-1}$ and integrate the divergence term, with vanishing boundary terms. Volume preservation gives the result without needing such boundary decay for general integrable data.