Material mass balance implies local continuity

Let $I\subseteq \mathbb{R}$ be an open interval containing $0$, let $B\subseteq {\mathbb{R}}^3$ be a bounded open reference region with $C^1$ boundary (locally a $C^1$ graph with the region on one side), and let $U$ be an open neighborhood of $\bar{B}$. Let $\chi \in C^2(I\times U;{\mathbb{R}}^3)$ have orientation-preserving diffeomorphic time slices (bijections onto open images with $C^2$ inverses and positive Jacobian determinant), and satisfy $\chi (0,a)=a$. Here $C^r$ means all coordinate partial derivatives through order $r$ are continuous; write ${\mathrm{\Omega }}_t={\chi }_t(B)$ and $u(t,\chi (t,a))={\mathrm{\partial }}_t\chi (t,a)$. Let ${\rho }_0\in C^1(\bar{B})$ and $\rho \in C^1(\mathcal{O})$ be positive mass densities, where $\mathcal{O}=\{(t,x):t\in I,x\in {\chi }_t(U)\}$. Suppose that for every nonempty bounded open $V$ with compact closure in $B$ and $C^1$ boundary locally a graph with the region on one side and every $t\in I$, ${\int }_{{\chi }_t(V)}\rho (t,x)dx={\int }_V{\rho }_0(a)da\mathrm{.}$ Then the pointwise continuity equation ${\mathrm{\partial }}_t\rho +{\mathrm{div}}_x(\rho u)=0$ holds throughout the balanced physical region ${\mathcal{O}}_B:=\{(t,x):t\in I,x\in {\chi }_t(B)\}$. The larger set $\mathcal{O}$ is only the ambient regularity domain; no balance outside ${\chi }_t(B)$ is assumed or concluded. This is a conditional deduction in the smooth material model and asserts no extension to weak fields or vacuum.

Single-component nonpolar continuum and material balances

Smooth transport, Jacobian and advected reference density

C1 material regions have compatible Jordan volume integrals

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

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