Is there an adverb for "quickly at first, slowly later"?

Let's face it, there is no unclumsy adverb for this. Go with a verb. The stream dies off, loses momentum, dries up are all possible. So is your very own tapers off, and the peters out suggested in a comment. The pressure drops down. Custom-tailored, more colorful metaphors are possible as well, like loses dedication, gets lazy, gives up, or any number of others. Try as you might, you won't get that kind of flexibility with an adverb.