Why do they find it tolerable to live so far apart?
Twice in the film, Eve asks Adam to explain "Einstein's Theory of Entanglement". (Einstein isn't the only physicist who helped discover the theory, but I believe the film refers to him by name because he coined the phrase "spooky action at a distance," which Adam mentions in his explanation.) The metaphor is clear: just as entangled particles are affected by their entanglement instantaneously regardless of how far apart they are, Eve and Adam feel that their love "entangles" them so that they are inextricably bound together regardless of where they live.
Jim Jarmusch (the director) emphasized this point in an interview with The Guardian:
They live apart because they can, because it doesn't deprive them of time together. "If you live that long, separation for a year might feel like a weekend," says Jarmusch.... "It's not an obligation, it's an emotional connection."
But why Detroit?
Within the film itself, I believe the closest we come to getting an explanation for why Adam lives specifically in Detroit, but Eve doesn't, is in these two scenes:
- When Eve calls Adam, she shows pretty clear disgust for Detroit, saying "I can't believe you're making me do this" when she decides to book tickets to come to him after he expresses reluctance to visit Tangiers.
- When discussing the sites Adam and Eve can visit in Detroit, Adam mentions the old Motown Records building. Eve says, "I'm more of a Stax girl."
Based on the first scene, we can see that Eve dislikes Detroit; but the second scene seems to imply that Adam may feel more of a connection to the city because of his music.
If you're willing to consider footage that didn't make it into the final cut of the film, there's a deleted clip from the telephone conversation in which Eve asks Adam outright why he lives in Detroit. He responds that living there (as an immortal) is like watching time-lapse footage of a beautiful flower:
...it grows from a seed and pushes its way up, and...opens into an exquisite blossom. And then the flower starts to close. It shrivels, and it fades, and crumbles to dust. And all in one single century.