How does an engine work when an airplane has $v=0$ on the ground? I mean, for a turbofan:
$$ T= \dot m_{air,h} (1+f) v_{e,h}+ \dot m_{air,c} v_{e,c}- (\dot m_{air,h}+ \dot m_{air,c} ) v_0+ (P_{e,h}-P_{amb} ) A_{e,h}+ (P_{e,c}-P_{amb} ) A_{e,c} $$

but
$$ \dot m_{air}=\rho Av_0 $$

do I have to impose  $\dot m_{air}$?
in this case how the thrust expression change (in particular the terms with $v_0$)?
To start the engine, the fan blades have to be spun up. This is typically handled by either blowing air through them from some outside source, and/or by using an APU to generate power to drive the shaft they're connected to.

Once the engine is running, it sucks in enough air to keep going on its own.