My classes said that the balance of the diaphragm and the abs was necessary to control the air. Suck air in and totally relax the diaphragm with no abs and you get a sigh. Continue the diaphragm being tight and push with the abs and you get a controlled expiration. Relax the diaphragm and push with the abs and you get a very rapid exhalation that is good for nothing but blowing out candles. So the answer is both the diaphragm and the abs must be balanced to get the proper air control to blow an instrument that is free blowing. Play an oboe and all bets are off. Play a flute and you will discover the principles above very quickly.