U 1_{\Omega_0} (\Phi (X)) \tilde{\Psi}(X) \left( \frac{d}{d t} f (\tilde{x}^\varepsilon( X, t ),t) \right) \sqrt{J^\varepsilon (X,t)} { \ }d X\\
+ \int_U 1_{\Omega_0} (\Phi (X)) \tilde{\Psi}(X) f (\tilde{x}^\varepsilon( X, t ),t) \left( \frac{ \partial }{\partial t} \sqrt{J^\varepsilon (X,t)} \right) { \ }d X\\
= \int_{\Omega^\varepsilon (t)} \{ D_t^\varepsilon f + ({\rm{div}}_{\Gamma^\varepsilon} v^\varepsilon ) f  \} (x,t){ \ }d \mathcal{H}_{x}^2.
\end{multline*}
Therefore we have \eqref{eq318}. Similarly, we see \eqref{eq317}.
 \end{proof}
Lemma \ref{lem39} attacks Proposition \ref{prop13}.
\begin{proof}[Proof of Proposition \ref{prop13}]
We only prove the assertion $(\mathrm{ii})$ of Proposition \ref{prop13}. Fix $t \in (0,T)$ and $\Omega^\varepsilon ( t) \subset \Gamma^\varepsilon (t)$. Since the flow map $\hat{x}^\varepsilon (\xi, t)$ is bijective, there is $\Omega_0 \subset \Gamma_0$ such that
\begin{equation*}
\Omega^\varepsilon (t) = \{ x \in \mathbb{R}^3; { \ }x = \hat{x}^\varepsilon (\xi, t ),{ \ } \xi \in \Omega_0 \}.
\end{equation*}
From Lemma \ref{lem39}, we see that
\begin{equation*}
\frac{d }{d t}\int_{\Omega^\varepsilon (t)} \rho^\varepsilon (x,t) { \ }d \mathcal{H}_{x}^2 = \int_{\Omega^\varepsilon (t)} \{ D_t^\varepsilon \rho^\varepsilon + ( {\rm{div}}_{\Gamma^\varepsilon} v^\varepsilon ) \rho^\varepsilon \} (x,t) { \ } d \mathcal{H}^2_{x}.
\end{equation*}
Since $\Omega^\varepsilon (t)$ is arbitrary, we conclude that
\begin{equation*}
D_t^\varepsilon \rho^\varepsilon + ({\rm{div}}_{\Gamma^\varepsilon} v^\varepsilon) \rho^\varepsilon = 0 \text{ on } \mathcal{S}^\varepsilon_T.
\end{equation*}
Therefore we have the continuity equation for the fluid on $\Gamma^\varepsilon (t)$. Similarly, we can prove Proposition \ref{prop13}.
 \end{proof}
See also \cite{Bet86}, \cite{GSW89}, \cite{DE07}, \cite{KLG}, and \cite{SSO07} for the proof of Proposition \ref{prop13}.


\subsection{Representation of Density, Kinetic Energy, Dissipation Energies, Work for the Fluid on an Evolving Surface}\label{subsec34}

In this subsection we study the representation of the kinetic energy, dissipation energies, and work for the fluid on an evolving surface by applying the Riemannian metric induced by a flow map. From now we follow the convention:
\begin{convention}\label{conv310}
Assume that $\rho = \rho (x,t)$, $\rho^\varepsilon = \rho^\varepsilon (x,t)$, $\rho_0 = \rho_0 (x)$, $\sigma = \sigma (x,t)$, $e = e (x,t)$, $\theta = \theta (x,t)$, $h = h (x,t)$, $s = s (x,t)$, $e_F = e_F (x,t)$, $F = F (x,t) = { }^t (F_1, F_2, F_3)$, $\mu = \mu (x,t)$, $\lambda = \lambda (x,t)$, $\kappa = \kappa (x,t)$, $\nu = \nu (x,t)$, $C = C (x,t)$, $C_\theta = C_\theta (x,t)$, $Q_\theta = Q_\theta (x,t)$, $Q_C = Q_C (x,t)$, $\mathcal{F}_1 = \mathcal{F}_1(x,t) $, $\mathcal{F}_2 = \mathcal{F}_2 (x,t)$ are smooth functions. Moreover, $p \in C^1 ( (0, \infty ))$ or $p \in C^1 ( [0, \infty ))$.
\end{convention}

Based on Proposition \ref{prop13}, we assume that $ \rho $ and $\rho^\varepsilon$ satisfies
\begin{equation*}
\begin{cases}
D_t \rho + ({\rm{div}}_\Gamma v) \rho =0 & \text{ on } \mathcal{S}_T,\\
\rho |_{t = 0} = \rho_0 & \text{ on }\Gamma (0),
\end{cases}
\end{equation*}
\begin{equation*}
\begin{cases}
D_t^\varepsilon \rho^\varepsilon + ({\rm{div}}_{\Gamma^\varepsilon} v^\varepsilon ) \rho^\varepsilon =0 & \text{ on } \mathcal{S}^\varepsilon_T,\\
\rho^\varepsilon |_{t = 0} = \rho_0 & \text{ on }\Gamma (0).
\end{cases}
\end{equation*}


The aim of this subsection is to prove the following three propositions.
\begin{proposition}[Representation of Energy densities $(\mathrm{I})$]\label{prop311}
Set
\begin{align*}
\tilde{\rho}_0 (X) = & \rho_0 (\tilde{x}(X,0)) \sqrt{J(X,0)},\\
\mathcal{K}(e_K)  =& \frac{1}{2} \frac{\tilde{\rho}_0 (X) }{\sqrt{J (X,t)}} \tilde{x}_t (X, t) \cdot \tilde{x}_t (X,t),\\
\mathcal{K}(e_A) =& \frac{\tilde{\rho}_0 (X) }{\sqrt{J (X,t)}} \left\{ \frac{1}{2} \tilde{x}_t (X, t) \cdot \tilde{x}_t (X,t) + e (\tilde{x} (X,t), t) \right\},\\
\mathcal{K}(e_B) =&\frac{1}{2}  \frac{\tilde{\rho}_0 (X) }{\sqrt{J (X,t)}} \{ \tilde{x}_t (X, t) \cdot \tilde{x}_t (X,t)\} - p \left( \frac{\tilde{\rho}_0 (X) }{\sqrt{J (X,t)}} \right),\\
\mathcal{K}(e_{W_2}) =&\frac{\tilde{\rho}_0 (X) }{\sqrt{J (X,t)}}  \tilde{x}_t (X,t) \cdot F(\tilde{x} (X,t), t).
\end{align*}
Then
\begin{align*}
\int_{\Gamma (t)} \left\{ \frac{1}{2} \rho | v |^2 \right\} (x,t) { \ }d \mathcal{H}_{x}^2 = \int_U \tilde{\Psi} (X) \mathcal{K}(e_K) \sqrt{ J (X,t)  } { \ }d X,\\
\int_{\Gamma (t)} \left\{ \frac{1}{2} \rho | v |^2 + \rho e \right\} (x,t) { \ }d \mathcal{H}_{x}^2 = \int_U \tilde{\Psi} (X) \mathcal{K}(e_A)\sqrt{ J (X,t)  } { \ }d X,\\
\int_{\Gamma (t)} \left\{ \frac{1}{2} \rho | v |^2 - p (\rho) \right\} (x,t) { \ }d \mathcal{H}_{x}^2 = \int_U \tilde{\Psi} (X) \mathcal{K}(e_B)\sqrt{ J (X,t)  } { \ }d X,\\
\int_{\Gamma (t)} \{ \rho F \cdot v \} (x,t) { \ }d \mathcal{H}_{x}^2 = \int_U \tilde{\Psi} (X) \mathcal{K}(e_{W_2}) \sqrt{ J (X,t)  } { \ }d X.
\end{align*}
Here
