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Low regularity Cauchy theory for the water-waves problem: canals and wave pools. (English) Zbl 1348.35277

Lin, Fanghua (ed.) et al., Lectures on the analysis of nonlinear partial differential equations. Part 3. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-267-1/hbk). Morningside Lectures in Mathematics 3, 1-42 (2013).
Summary: In these notes, we are interested in the system of gravity water-waves equations (without surface tension). Our purpose is to try and understand the optimal regularity thresholds for the initial conditions. In terms of Sobolev embeddings, the initial surfaces we consider turn out to be only of \(C^{3/2}\) class and consequently have unbounded curvature, while the initial velocities are only Lipschitz. We also take benefit from our low regularity result and an elementary (though seemingly yet unnoticed) observation to solve a question raised by Boussinesq on the water-waves system in a canal. To analyse the water-waves system, we prove sharp microlocal results on the Dirichlet-Neumann operator in rough (Lipschitz) domains. After suitable para-linearizations, we show that the water-waves system can be arranged into an explicit symmetric system of quasilinear waves equation type. As an illustration of this reduction, we prove that in fact following the analysis by H. Bahouri and J.-Y. Chemin [Am. J. Math. 121, No. 6, 1337–1377 (1999; Zbl 0952.35073)] and Tataru for quasi-linear wave equations, using Strichartz estimates, the regularity threshold can be further lowered, which allows to solve the water-waves system.
For the entire collection see [Zbl 1342.35008].

MSC:

35Q86 PDEs in connection with geophysics
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
86A05 Hydrology, hydrography, oceanography
35B65 Smoothness and regularity of solutions to PDEs
35B44 Blow-up in context of PDEs

Citations:

Zbl 0952.35073
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