If you keep mowing them, they eventually starve and die. They cannot live off of the root ball forever. So far it looks like it takes two years for that to happen. Every time they throw out shoots it consumes the root ball's food. Cutting off the shoots before they can photosynthesize more food will do the trick. It also helps to cut them down after they flower and before their resources get sucked back into the root ball in the fall.

For scotch broom, they'll also grow back if you cut them off at ground level. But I found if you cut them off right after they flower and start going to seed, they don't grow back. Apparently they throw so many resources into the flower they can't recover.

We also have a huge problem with english ivy, which looks nice but kills the trees.