Well, the Elves had been slowly deserting Rivendell for many years at the time of the War of the Ring. However, it was not completely abandoned at the end of the Third Age, as Celeborn came north and dwelt there along with his grandsons Elladan and Elrohir. After they left, however, it was likely utterly foresaken, since no one could really lay claim to it.
I would say most likely it either was utterly forsaken like Elendil the tall said or Aragorn would claim it in re-uniting the ancient man kingdom of the North.
now I don't want to get annoying, but we are talking here about lore, and not about more or less likely things. So mouthofsauron, we don't have any proof that Rivendell was united with Arnor, and anyway, it was mose certainly not done by Aragorn. "The King of Gondor and Lord of all the Western Lands', said Gandalf; 'and he has taken back all his ancient realm." His ancient realm meaning of course Arnor. The border of Arnor in the times of old had been to the east was the River Loudwater, which as we know passed near Rivendell, but Rivendell was on the other side. So there would be no point why Rivendell should become part of the kingdom. there is one letter on which many base this theory. letter #131: "We are to see the overthrow of the last incarnation of Evil, the unmaking of the Ring, the final departure of the Elves, and the return in majesty of the true King, to take over the Dominion of Men, inheriting all that can be transmitted of Elfdom in his high marriage with Arwen daughter of Elrond, as well as the lineal royalty of Numenor." Now the term "elfdom" is used by many to describe Rivendell, Lorien, Mirkwood, and all other territories inhabited by the elves, but as we know not all came in the Reunited Kingdom, so we can be sure it was the same for Rivendell. And the best proof for that is that Aragorn married Arwen, which had no right at all for Rivendell. The ones that ruled where her older brothers Elladan and Elrohir, which of course were not married to Aragorn, and did not give him right over Rivendell. So it is to assume that Rivendell was an ally of the reunited Kingdom, but not at all a part of it.
But if all the elves have passed west, or all but for the wood Elves then there would be no 'Elf' claim to Rivendell. As Elendil said it would likely be utterly deserted unless you can quote directly otherwise?
there is no quote for that, as many other things Tolkien left that riddle unanswered, just as the fate of Lorien, it was eventually deserted. we can make no assumptions as we know nothing for sure, and this is after all lore, so all we know is that until the year 120 IV, the last date we know about, no land of the Elves was added to the Reunited Kingdom.
If Tolkien left it unsaid, as part of a piece of fiction, I think you have to just imagine and decide for yourself, (and just not use your decision in any fact-based quiz or argument, hehe). We take Lord of the Rings pretty seriously, but it is a story, not just a long piece of historic lore to be looked at. I think it would have been left empty eventually, I always remember a small quote from near the very end of LoTR, when the last of the company passes through Rivendell and meets Bilbo again. It mentioned Frodo, or someone else looking out, and there being a small cobweb in the corner of a window, and it just made me think "Oh! It's over..." and I felt sad, because it was like Rivendell had come to the end of its story-life. Maybe someone did come and occupy it though, who knows?
Now I might sound very annoying saying this but this is a lore forum and whether we want it or not we can't make up ideas here. We have however a fanfic forum where we can post our own opinions about possible future of ME in the Fourth Age.
Well, if there's no obvious lore supporting, you do have to go into lore-based speculation. You can do that without going into fan-fic though, right? I apologise for bringing that in here if it's not the right place though, sorry! : )
no no prob, I don't want you think I am a lore obsessed freak, but still as long as we don't know anything we shouldn't speculate, not here at least... Of course it is nice to think about what happened, I also do that, but still each of us has his own opinion