Local mass balance: conservation on every material subregion implies ${\mathrm{\partial }}_t\rho +\mathrm{div}(\rho u)=0$

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as used by the transport theorem. Let $I\subseteq \mathbb{R}$ be a nonempty open interval, let $B\subseteq {\mathbb{R}}^3$ be a material region, let $\chi :I\times O\to {\mathbb{R}}^3$ be a motion of $B$ on $I$ with spatial velocity $u$ (Continuum motions, spatial velocity, material acceleration and swept regions), and let $J_0\subseteq I$ be an open interval with $\overline{J_0}\subseteq I$. Define the swept region

$\mathrm{\Sigma }:=\{(t,x):t\in \overline{J_0},x\in {\chi }_t(\bar{B})\}\mathrm{.}$

Write ${\mathcal{D}}_{\chi }:=\{(t,x):t\in I,x\in {\chi }_t(O)\}$ for the open domain of the spatial velocity $u$ (Continuum motions, spatial velocity, material acceleration and swept regions). Let $U\subseteq {\mathcal{D}}_{\chi }$ be open and contain the swept region $\mathrm{\Sigma }$. Let $\rho :U\to (0,\mathrm{\infty })$ be of class $C^1$ ($C^k$ Euclidean maps and diffeomorphisms); call it a density field. A material subregion is a nonempty bounded open $B^{\mathrm{\prime }}\subseteq B$ with $\overline{B^{\mathrm{\prime }}}\subseteq B$ (Material regions, densities and their mass); for such $B^{\mathrm{\prime }}$ the same $\chi$ is a motion of $B^{\mathrm{\prime }}$.

(i) Localization. Assume for every material subregion $B^{\mathrm{\prime }}$ and every $t\in J_0$

$$\begin{array}{cccc}& \frac{d}{dt}{\int }_{{\chi }_t(B^{\mathrm{\prime }})}\rho (t,x)dx=0,& & \text{(CM)}\end{array}$$

It is sufficient to assume this identity only for those pairs $(t,B^{\mathrm{\prime }})$ for which ${\chi }_t(B^{\mathrm{\prime }})$ is an open Euclidean ball with closure inside ${\chi }_t(B)$. Under either version of the conservation hypothesis,

${\mathrm{\partial }}_t\rho (t,x)+\mathrm{div}\left(\rho u\right)(t,x)=0$

for every $(t,x)\in J_0\times {\chi }_t(B)$, with the material derivative and divergence of Euclidean spatial calculus: gradient, divergence and material derivative.

(ii) Converse. Conversely, if ${\mathrm{\partial }}_t\rho +\mathrm{div}(\rho u)=0$ on an open neighbourhood of $\mathrm{\Sigma }$ in $U$, then (CM) holds for every material subregion $B^{\mathrm{\prime }}$ and every $t\in J_0$.

• Positivity of $\rho$ is part of the density reading and is not used by the implication (i); only $C^1$ regularity on $U$ and the conservation hypothesis enter. The converse also uses only the stated regularity.

• The two implications are conditional on the adopted conservation law (CM) or on the local equation; neither direction asserts that a physical density is conserved. Localization needs the full stated neighbourhood family: all small current balls suffice. Conservation on only one chosen region is insufficient.

• The statement is local in the current region: it concludes the equation at every point of ${\chi }_t(B)$ for $t\in J_0$, and says nothing on $\mathrm{\partial }{\chi }_t(B)$ or at the endpoints of $I$, where the transport theorem is not asserted.