A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations Borys Alvarez-SamaniegoDavid Lannes 35F2535Q35Nash-MoserGreen-NaghdiSerre approximationsingular evolution equations We study the well-posedness of the initial value problem for a wide class of singular evolution equations. We prove a general well-posedness theorem under three simple assumptions: the first controls the singular part of the equation, the second the behavior of the nonlinearities, and the third one assumes that an energy estimate can be found for the linearized system. We allow losses of derivatives in this energy estimate and therefore construct a solution by a Nash-Moser iterative scheme. As an application to this general theorem, we prove the well-posedness of the Serre and Green-Naghdi equation and discuss the problem of their validity as asymptotic models for the water-waves equations. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3200 10.1512/iumj.2008.57.3200 en Indiana Univ. Math. J. 57 (2008) 97 - 132 state-of-the-art mathematics http://iumj.org/access/