Transport across a moving control boundary

Assume countable choice: every countable family of nonempty sets has a choice function. Let $I\subseteq \mathbb{R}$ be an open interval, let $C_0\subseteq {\mathbb{R}}^3$ be a bounded open region with $C^1$ boundary (locally a $C^1$ graph with the region on one side), and let $U_C$ be an open neighborhood of $\overline{C_0}$. Let $X\in C^2(I\times U_C;{\mathbb{R}}^3)$ have orientation-preserving diffeomorphic time slices onto open images (bijections with $C^2$ inverses and positive Jacobian determinant). Here $C^r$ means all coordinate partial derivatives through order $r$ are continuous, and set $C(t)=X_t(C_0)$. Let $g$ be a scalar $C^1$ field on an open neighborhood of $\{(t,x):x\in \overline{C(t)},t\in I\}$. Define the boundary velocity by $w(t,X(t,\xi ))={\mathrm{\partial }}_{t}X(t,\xi )$ and let $n$ be the outward Euclidean unit normal on $\mathrm{\partial }C(t)$. Use Euclidean surface measure, whose chart density is $\sqrt{\det (D{Y}^{\mathsf{T}}DY)}$. Then $\frac{d}{dt}{\int }_{C(t)}g(t,x)dx={\int }_{C(t)}{\mathrm{\partial }}_{t}g(t,x)dx+{\int }_{\mathrm{\partial }C(t)}g(t,x)w(t,x)\cdot n(t,x)d{S}_x\mathrm{.}$ For vector-valued $g$, the identity holds componentwise. No condition equating $w$ with a fluid velocity is required.

Smooth transport, Jacobian and advected reference density

Divergence on a bounded C1 Euclidean domain

Powers, Lecture Notes on Intermediate Fluid Mechanics, §§4.6.1–4.6.4.

Mei, MIT OpenCourseWare, Kinematic Transport Theorem, §1.3.