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

Authors: David Lannes and Borys Alvarez-Samaniego

Issue: Volume 57 (2008), Issue 1, 97-132

Abstract: 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.