 Conserved first moment of a draining porous current

For $h_t=\kappa (h{h}_x{)}_x$ with a drained boundary $h(0,t)=0$ and a finite current vanishing at its nose, integration by parts gives $\dot{D}=-\kappa [h^2{]}_0^{\infty }/2=0$. The outlet can have finite volume flux because its contribution to the first-moment flux is multiplied by $x=0$. A positive prescribed boundary height instead gives $\dot{D}=\kappa h(0,t{)}^2/2$, so drainage is essential to this invariant.