Consider the continuity equation

$$\partial_t u(t,x) + \mathrm{div}(a(t,x)u(t,x)) = 0,$$ where $u: [0,T]\times \mathbb{R}^N \to \mathbb{R}$ is the solution and $a:[0,T]\times \mathbb{R}^N \to \mathbb{R}^N$ is a vector field.

I've head many times that assumptions on $\mathrm{div}\,a$, for example that it is bounded, amount to asking some requirements on the "concentration of mass" transported by the equation.

I'm not sure what that means (neither heuristically, nor rigorously) and I would appreciate some insight (or detailed references) on this point.

Previous posts with closely related question and framework are Role of absolute continuity of divergence of BV function in proof of renormalization property and Prove that the flow of a divergence-free vector field is measure preserving.