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Let $v_i={\dot{x}}_i$ and use a fixed arbitrary material subvolume $B_0$ in the reference configuration, with outward unit normal $N_I$. The reference mass density ${\rho }_0$ is time independent. The global momentum conservation law isThe divergence theorem, differentiation under the integral, and arbitrariness of $B_0$ give

The global conservation of energy law includes both internal energy and kinetic energy:Here $q^0$ is outgoing heat flux per reference area. Localizing givesSubtract the scalar product of the local momentum conservation law with $v$. Since $v_{i,I}={\dot{F}}_{iI}$, this proves the Lagrangian internal energy balance

For positive temperature $\theta$, the global entropy inequality isIts local excess is a nonnegative production density $\gamma$. Expanding the divergence of $q^0/\theta$ proves the Lagrangian entropy inequality in the requested equality form:No constitutive assumption on the heat flux was needed for these balance identities.

For the reference-configuration jump balances, make the orientation convention explicit. Write $[a]=a^{+}-a^{-}$ and let $n^0$ point from the minus side to the plus side. Let $c$ be the actual interface speed in direction $n^0$. A signed level function $\varphi$ with the plus side $\varphi >0$ obeys ${\varphi }_t=-c|{\nabla }_X\varphi |$ on the interface. For $a=a^{-}+[a]H(\varphi )$, the singular time derivative is $-c[a]{\delta }_S$; the singular reference divergence of $b$ is $n_I^0[b_I]{\delta }_S$. Thus a local conservation law $a_t=\mathrm{Div}b+$ bounded sources has the Rankine-Hugoniot conditionThis is also the shrinking moving-pillbox calculation of the global law: finite volume supplies have no surviving surface term.

To match the displayed positive-sign momentum jump in the paper, use its speed parameter as $V=-c$, so the interface travels in direction $-V{n}^0$. Assuming continuous ${\rho }_0$ and no surface force or energy supply, the jump laws are thenand, from ${\partial }_t({\rho }_0\eta )+\mathrm{Div}(q^0/\theta )\ge {\rho }_{0}r/\theta$,The last inequality is nonnegative entropy production concentrated on the interface. Temperature and flux in the entropy term are the traces on their respective sides; one cannot take a common temperature outside the jump unless it is continuous.

If $V$ is instead defined as the interface speed in direction $n^0$, all three $V$ terms above acquire a minus sign. In particular the printed positive-sign momentum formula requires the opposite speed convention; changing just the order of the jump does not remove this convention issue. If reference density is discontinuous, replace ${\rho }_0[a]$ by the jump $[{\rho }_{0}a]$ of the complete conserved density.

An equivalent internal-energy jump follows by subtracting the kinetic-energy jump using $[|v{|}^2/2]=\{v_i\}[v_i]$, where $\{a\}=(a^{+}+a^{-})/2$:The averaged traction appears because $[P_{Ii}{v}_i]=\{P_{Ii}\}[v_i]+[P_{Ii}]\{v_i\}$; using a one-sided traction would generally be incorrect.