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We study the well-posedness of the initial value problem for a wide class ofsingular evolution equations. We prove a general well-posedness theorem underthree assumptions easy to check: the first controls the singular part of theequation, the second the behavior of the nonlinearities, and the third oneassumes that an energy estimate can be found for the linearized system. Weallow losses of derivatives in this energy estimate and therefore construct asolution by a Nash-Moser iterative scheme. As an application to this generaltheorem, we prove the well-posedness of the Serre and Green-Naghdi equation anddiscuss the problem of their validity as asymptotic models for the water-wavesequations.

A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations