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Several recent papers considered the high-friction limit for systems arisingin fluid mechanics. Following this approach, we rigorously derive the nonlocalCahn-Hilliard equation as a limit of the nonlocal Euler-Korteweg equation usingthe relative entropy method. Applying the recent result by the first and thirdauthor, we also derive rigorously the local degenerate Cahn-Hilliard equation.The proof is formulated for dissipative measure-valued solutions of thenonlocal Euler-Korteweg equation which are known to exist on arbitraryintervals of time. Our work provides a new method to derive equations notenjoying classical solutions via relative entropy method by introducing thenonlocal effect in the fluid equation.

From nonlocal Euler-Korteweg to local Cahn-Hilliard via the high-friction limit