= Solution
Formally the <Jacobian determinant> of the <characteristic flow map> satisfies $J'=\operatorname{div}B(\Phi_t)J=0$, $J(0)=1$, so $J=1$. Changing variables $z=\Phi_t(z_0)$ gives
$$
\int_{\mathbb R^{2d}}|f(z,t)|^p\,dz=\int_{\mathbb R^{2d}}|f_I(z_0)|^p\,dz_0.
$$
Taking the $p$th root proves \b[$\boxed{\|f(t)\|_{L^p}=\|f_I\|_{L^p}}$] for $1\leq p<\infty$. Equivalently multiply the equation by $pf^{p-1}$ and integrate the divergence term, with vanishing boundary terms. Volume preservation gives the result without needing such boundary decay for general integrable data.
Back to article page
