You could take a look at the shapebuilder tool, live paint or the pathfinder panel.
Or Object>Path>Divide Objects Below.
Thank you! Dividing objects doesn't actually work in this case, but possibly I'm doing it wrong somehow. Anyway, I made it with live paint tool.
For my part you are welcome, M.
With Divide Objects Below you need to select each white path (and nothing else), then apply.