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
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10.1512/iumj.2008.57.3200
10.1512/iumj.2008.57.3200
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Indiana Univ. Math. J. 57 (2008) 97 - 132
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