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Scalar advection preserves every finite Lp norm if its complete differentiable characteristic flow preserves volume and the initial datum belongs to the corresponding Lp space. In general the $p$th power of the norm is the initial $|f_0{|}^p$ weighted by the phase-space flow Jacobian. Completeness alone preserves the essential supremum, but does not preserve finite-$p$ norms for compressible transport.

Let $A(t)=D_{(x,v)}(X,V)$ along one characteristic curve. Differentiating the characteristic equation gives $\dot{A}=D_{z}b(t,X,V)A$, $A(0)=I_6$, with $b=(V,F)$. The Liouville formula for a fundamental matrix then gives the phase-space flow JacobianIndeed $\mathrm{tr}{D}_{z}b={\mathrm{div}}_{x,v}(v,F)={\nabla }_v\cdot F$, since $v$ is independent of $x$. Differentiation proves $\dot{J}=({\nabla }_v\cdot F)J$. If that divergence vanishes, $J(0)=1$ gives $\boxed{J\equiv 1}$ and the flow preserves Lebesgue measure on phase space.