= Conserved first moment of a draining porous current
{title2=$D=\int_0^\infty xh\,dx$}
For $h_t=\kappa(hh_x)_x$ 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.
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