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Sobolev estimates for two dimensional gravity water waves. (English) Zbl 1360.35002

Astérisque 374. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-821-3/pbk). viii, 241 p. (2015).
The authors study irrotational water waves, subject to gravity. The fluid region is two-dimensional and has infinite depth. The work uses the Craig-Sulem-Zakharov formulation of the gravity water waves problem, which includes the Dirichlet-to-Neumann operator in the evolution equations. In the present work, the main result is an estimate for a Sobolev energy, valid for small solutions. The smallness condition ensures that a requisite change of variables is close to the identity.
The proof is based on the vector fields method of Klainerman: in addition to estimating the unknowns in \(L^{2}\)-based Sobolev spaces, the authors define an operator \(Z=t\partial_{t}+2x\partial_{x},\) and make estimates in \(L^{2}\) for spatial derivatives of powers of \(Z\) as well. The proofs make extensive use of paradifferential calculus. Two main ingredients of the proofs throughout the work are paralinearization, after which quadratic estimates can be made, and normal form transformations, to compensate for the quadratic terms.
Throughout, the authors have made a very delicate analysis. They go on to use their results in [Ann. Sci. Éc. Norm. Supér. (4) 48, No. 5, 1149–1238 (2015; Zbl 1347.35198)] to prove existence for all time of solutions with small data, and to provide asymptotic formulas which indicate the waves do not scatter at infinity. While estimates of a similar character to those in the present work have appeared elsewhere, such as in works by D. Lannes [J. Am. Math. Soc. 18, No. 3, 605–654 (2005; Zbl 1069.35056)], S. Wu [Invent. Math. 177, No. 1, 45–135 (2009; Zbl 1181.35205)], and P. Germain et al. [Commun. Pure Appl. Math. 68, No. 4, 625–687 (2015; Zbl 1314.35100)], the authors here prove very precise versions of the estimates for use in their global existence theorem.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35S50 Paradifferential operators as generalizations of partial differential operators in context of PDEs
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