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

By Borys Álvarez-Samaniego, David Lannes. 2007

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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 assumptions easy to check: 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.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/022cc84ea987d5d9e2331546037a9f60c96a1713

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Large time existence for 3D water-waves and asymptotics (2007, methodology): “on 5.1 some simple suﬃcient conditions on the initial data and the bottom parameterization bwhich ensure that it is satisﬁed. Remark 5.4. Onealso hasthefollowing stability property(see Corollary 1 in [2]): let T>0 and (Upapp)p∈P = (ζ app p ,ψ app p )p∈P, bounded in Ys+P T (see Deﬁnition 4.2), be ”
The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations (2007, result): “Proof. The first point of the theorem is a direct consequence of Proposition 4. Thanks to Proposition 2, we now that (u e,µ ,ζ e,µ )e,µ is consistent with the Green‐ Naghdi equations (3), so that the second point of the theorem and the error estimate follow at once from the well-posedness and stabil”