What is the cause for new swamp cooler's fan stopping after 15 seconds
15 minutes sounds like a control system issue. Being that it's a swamp cooler, my initial guess would be that there is a problem with the water flow or return. The system will let the fan run without water flow for 15 minutes, but no longer.

Could also be heat related; your motor is overheating and tripping a protection device intended to keep it from catching on fire. it takes 15 minutes for the temperature to rise to that point. The proof of that would be if you IMMEDIATELY restart it after it shuts down, it runs for LESS than 15 minutes the second time, meaning it took less time to over heat because it was already hot. If it still shuts down in 15 minutes, it's a timer of some sort.