= Lp conservation for incompressible transport
{c}
{title2=$\|f_0\circ\Phi_t^{-1}\|_p=\|f_0\|_p$}
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.
Back to article page
