Reference-configuration jump balances

Let an interface move at signed speed $c$ in a reference unit normal direction $n$, and set $[a]=a^{+}-a^{-}$ with $n$ pointing from minus to plus. For continuous reference density and no surface sources, distributional momentum conservation, conservation of energy and the Lagrangian entropy inequality giveFor a level-set interface $\varphi =0$, ${\varphi }_t=-c|\nabla \varphi |$, the singular part of ${\partial }_{t}a$ is $-c[a]{\delta }_S$ and that of $\mathrm{Div}b$ is $n\cdot [b]{\delta }_S$, proving these formulas. Using $V=-c$ reverses the speed terms. Density jumps require jumps of the complete conserved densities.