A travelling wave would satisfy your conditions

PDE-constrained optimization is a very big and well-developed field. See here for a quick overview: archive.siam.org/meetings/op08/Heinkenschloss.pdf

No. Incompressible means $\nabla.a=0$, since in that case, there is no concentration (aka compression) or expansion.

Ah, thats just confusion due to notation. I meant origin of real line ($x=0$, spatial origin, as $t\rightarrow \infty$), not $t=0$.

There is no mass creation. However, large $\nabla. a$ means rapid concentration, e.g. into origin for the 1d case I described. The density or measure in that case will become increasingly concentrated on origin. Whether $\nabla .a$ being unbounded leads to finite time blowup, I am not sure.

See my edit that explains this point.

In general (including multi-dimensional case), $\nabla. a$ gives the local "expansion" or "concentration" depending upon whether it is positive or negative.