I'm having problems knowing where to go with this question:
"Suppose that a certain material initially fills an open region $O \subseteq \mathbb{R}^3$, with $\Omega := [−1, 1]^3 \subset O$, so in particular the cube centered at zero with volume $8$ is filled with the material at time $t = 0$. Suppose that the particle paths $\varphi t : O \to \mathbb{R}^3$ are such that $$ \Omega t :=\varphi^t([−1,1]^3)=\{(x_1,x_2,x_3)^T ∈[−1,1]\times [−1+t,1+t]\times [−(1+t),1+t]\}~\text{for}~t \geq 0. $$ Determine whether or not $\rho(x, t)$ can be the mass density function of the material if $\rho=\rho_k$ for $k\in\{1,2\}$ with $$ \rho_1(x, t) = 3t(x_1 + 1)(x_2 + 1)(x_3^2)/4(1+t), \\ \rho_2(x, t) = 3(x_1 + 1)(x_2 + 1)(x_3^2)/4(1 + t)2. $$ I've tried calculating the time derivative of the volume integral of each $\rho$ and setting to zero, which in both cases just yields a function of $t$ which is not zero for all $t$, which seems unlikely that they're both not suitable for a mass density function. I'm also unable to use $\frac{\partial\rho}{\partial t} + \nabla \cdot (\rho\nu) = 0$ as I'm not given $\nu$.
Could someone please explain how I can go about solving this, or is my initial method correct?
Thanks, Dan