Johnny Depp and Vanessa Paradis: Separate Continents

Vanessa Paradis and Johnny Depp, in 2010

Hahn-Nebinger-Orban/Abaca

With their nearly 14-year relationship at a breaking point, Johnny Depp and Vanessa Paradis have been 5,000 miles apart this week.

French actress-singer Paradis, 39, has spent the last few days in Paris, where on Friday she shopped in the Marais district and went for a fitting at Chanel's headquarters on Rue Cambon.

While Paradis is expected at Monday's Paris premiere of her film Caf de Flore, Depp, 48, has been keeping a low profile in Los Angeles. He was spotted sporting a fedora and driving a luxury sports car Thursday, and he showed up later that night at the Hollywood Museum, according to sources.

He's due in New Mexico next month to start production on his new movie, The Lone Ranger.

Earlier in the week, a solo Paradis was spotted browsing in an art gallery and buying DVDs at the department store FNAC, where she picked out a box set of films by classic comedy director Billy Wilder highlighted by The Seven-Year Itch, about a married man tempted to stray due to the charms of Marilyn Monroe.

Paradis's initial home base in Paris was the Hotel Costes, where Depp fell in love with her at first sight in 1998. "I saw that back and that neck, and then she turn! ed aroun d and I saw those eyes and boom! My life as a single man was done," Depp told the Daily Mail last year. Later in the week, Paradis switched hotels.

Reps for Depp and Paradis have still not commented on the state of their relationship. As PEOPLE reports in this week's cover story, the never-wed couple parents to Lily-Rose, 12, and Jack, 9 have been leading separate lives, with some sources in their circle saying they have already split.

For more on the state of the Depp-Paradis relationship, pick up this week's PEOPLE, on newsstands now