The ivy will do surprisingly well in just water. It will grow but I'm not sure how much with just a test tube to hold all its roots. There is a complex relationship between plant roots and plant top - neither can survive without the other. And there is a balancing act going on to maintain a proportional relationship, each doing its fair share to keep the whole plant alive and healthy. Eventually, the roots will not be able to keep up their end of the bargain and the plant will need some nutrients to survive. Right now, the ivy is living on stored energy, probably impatiently waiting for roots to send up some food.
In theory, you could keep a plant like Ivy alive for years with a bigger jar and a drop of liquid fertilizer once a week.
