Lp conservation for incompressible transport ($\Vert f_0\circ {\Phi }_t^{-1}{\Vert }_p=\Vert f_0{\Vert }_p$)

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.