Abandoned

[The Squatter of Amon Rûdh]

We don't know about Rivendell, but we are told about Lothlórien in the appendices, since that's where Arwen went to die.

Then she said farewell to Eldarion, and to her daughters, and to all whom she had loved; and she went out from the city of Minas Tirith and passed away to the land of Lórien, and dwelt there alone under the fading trees until winter came. Galadriel had passed away and Celeborn also was gone, and the land was silent.

My guess would be that it was much the same with Rivendell. With the population so depleted I doubt that there was anyone to populate the lands after the Elves had departed; and I imagine the sort of general fading described above taking hold in all of the abandoned Elven realms.

Eventually, of course, these lands would have been resettled by Men or reverted to wilderness, being renamed or forgotten; the influence of the Elves fading with their memory. Eventually this would have become a matter of legend and folklore until Tolkien 'rediscovered' the stories of the Elves many centuries later.