  Several recent papers considered the high-friction limit for systems arising
in fluid mechanics. Following this approach, we rigorously derive the nonlocal
Cahn-Hilliard equation as a limit of the nonlocal Euler-Korteweg equation using
the relative entropy method. Applying the recent result by the first and third
author, we also derive rigorously the local degenerate Cahn-Hilliard equation.
The proof is formulated for dissipative measure-valued solutions of the
nonlocal Euler-Korteweg equation which are known to exist on arbitrary
intervals of time. Our work provides a new method to derive equations not
enjoying classical solutions via relative entropy method by introducing the
nonlocal effect in the fluid equation.
