If someone said "of course the equation is well-posed because it is elliptic, etc" it would teach the reader that the statement is not only true, but that there is a well known train of thought for proving the result.
In this sense there is no claim that the proof is short or contains only simple steps, but it does say that that among a certain class of people familiar with the area that it is in fact a well-walked path.
I briefly consulted a dictionary and found these two definitions for "of course" 1. "certainly; definitely" 2. "in the usual or natural order of things"
I am suggesting that "of course" is useful in the second instance as a pedagogical indicator.
I am not claiming that this is always the useage usage employed in mathematical writing but it is a good one if used in the right circumstances.