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Sep 20, 2012 at 12:18 comment added Bazin @Elena You have plenty of Eulerian results (I mean for the vector field PDE) for that type of vector fields. In particular a MR search on Ambrosio as author and keyword "nearly incompressible" or "$BV$ vector field" will provide plenty of references.
Sep 17, 2012 at 17:43 comment added Elena Yudovina Thank you. I was specifically fishing for results where $f$ has jump discontinuities -- in my case, the functions $f$ tend to be piecewise-nice but may, for example, depend on which of the coordinates of $X$ is positive (with, potentially, a jump at the boundary).
Sep 8, 2012 at 7:58 history answered Bazin CC BY-SA 3.0