Do liquids continuously boil in a pressure cooker? First, I understand that the boiling point of water is increased as the pressure is increased. At 15 PSI, the boiling point is about 250F. 
My question, which seems like nobody can answer with certainty, is this: Is there a continuous boil within the pressure cooker once it reaches temp/pressure?
I understand that there needs to be a boil at some point to create the steam. But once it reaches that pressure, then what? An electric PC will turn the heating element off until the pressure drops. A user will turn the heat down on an analog PC. Does the liquid just boil constantly while the heat is off? It probably maintains temp & pressure for a lot longer in a sealed environment. In my mind, if there were a continuous boil, that steam can only build so much until it exceeds the pressure that the PC is capable of maintaining and it begins to vent. So, sure you could have a constant boil, but that means it would also be constantly venting, right?
 A: The way my older pressure cooker works is that, once it reaches operating temperature, it boils all the time. It has a pressure relief valve that has a weight that is placed over a conical tube sticking up in the lid. The weight has different diameter holes drilled partway through it, so that as larger a hole is placed over the conical vent tube, it will go down farther to have a larger sealing surface, so less steam pressure will lift it to release steam. This changes the relief pressure so it can be set to boil at different temperature / pressures. It is made to be used on a cook stove. If a lot of steam is coming out of the vent (rapid boil), you can reduce heat. If no steam is coming out of the vent you can increase heat, so that ideally, a little steam comes out so you know it is boiling. If it is not boiling, it may be below desired temperature.
A: It seems likely that the lid of a pressure cooker would be at a slightly lower temperature than the heated bottom. This could allow the steam from a very slow boil to condense without venting.
A: First of all, boiling is not required to produce steam. In a stove+PC case, you will make it boil as the fastest way to produce steam.
If you were to close that PC right after it starts boiling, putting your stove to its max power makes it just produce steam but not rising its temperature. It will only rise gradually, as pressure builds up. If you keep the max power, water should keep boiling, just at a gradually higher temperature, at least until the valve starts releasing/controlling pressure, when you usually decide to apply about half the heating power.
The wheight-valve system acts as a barostat, keeping a constant pressure in the same way we seize the boiling point of water to ensure a constant temperature is kept. Keeping the stove power high enough (not usually its max anymore) to keep a stream of steam (i.e. a raised valve) ensures the pressure is kept constant, at a value decided by design (diameter and weight of the valve). This means boiling point inside is kept constant. To keep the stream going out you usually would require keep water boiling, although a fine balance of the stove power should allow keeping temperature right below the boiling point, to prevent a fast loss of water, waste of energy, etc.
About electric PC, I can only guess they try to do this control. Once they measure a pressure just below their safety valve one, they should reduce the heating power to keep water slightly under the boiling point (greater than 100°C), without wasting too much steam or too much power, which would otherwise be invested in producing steam faster, not in cooking your food.
So, in an electronics-controlled cookware, boiling points per se are pointless: you have thermostats! (well, they keep setting your maximum possible  temperature, but that's all) Traditionally, boiling is just used as a temperature reference, in order to make cooking times measurements reliable. You should get the same result boiling something at 98°C for 10 min in a mountain than at the same temperature and time at sea level (where water won't be boiling).

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*For your last specific question, with no heat input, it will keep T and P (ideal isolation and sealing), but boiling is not possible. At boiling point, all the energy input is invested in the phase change, but this change requires heat input.

