On the water-wave equations with surface tension

By Thomas Alazard, Nicolas Burq, Claude Zuily. 2009

212 citations recorded by the source corpus.

The purpose of this article is to clarify the Cauchy theory of the water waves equations as well in terms of regularity indexes for the initial conditions as for the smoothness of the bottom of the domain (namely no regularity assumption is assumed on the bottom). Our main result is that, following the approach developped by T. Alazard and G. Metivier in [1], after suitable paralinearizations, the system can be arranged into an explicit symmetric system of Schrodinger type. We then show that the smoothing effect for the (one dimensional) surface tension water waves proved by H. Christianson, V. M. Hur, and G. Staffilani in [9], is in fact a rather direct consequence of this reduction, which allows also to lower the regularity indexes of the initial data, and to obtain the natural weights in the estimates.

Subfields: Atmospheric Science, Earth-Surface Processes, Oceanography

View Paper: https://www.semanticscholar.org/paper/a95be4f8770ef9206a500baae49a6cae4f5b7db8

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Citations in the corpus

On the Cauchy problem for gravity water waves (2012, methodology): “1 Given ρ ∈ [0, 1] and m ∈ R, m ρ (Rd) denotes the space of locally bounded functions a(x, ξ) onRd ×(Rd\0), which areC∞ with respect to ξ for ξ = 0 and such that, for all α ∈ Nd and all ξ = 0, the function x → ∂ ξ a(x, ξ) belongs to W ρ,∞(Rd) and there exists a constant Cα such that,”
Two Dimensional Water Waves in Holomorphic Coordinates (2014, unknown): “Without being exhaustive, we list some of the more recent references [1, 2, 4, 5, 7, 8, 13, 15, 18, 24].”
Paralinearization of the Muskat Equation and Application to the Cauchy Problem (2019, methodology): “With regards to the existence, we construct solutions to the Muskat equation as limits of solutions to a sequence of approximate nonlinear systems, following here [1, 2, 32, 33].”
Finite time singularities for the free boundary incompressible Euler equations (2011, background): “[1] took advantage of the dispersive properties of the equations to lower the regularity of the initial data.”
A Paradifferential Approach for Well-Posedness of the Muskat Problem (2019, background): “esult on the continuity of the Dirichlet-Neumann map leads to direct application for the Muskat problem. This is especially relevant in view of the recent intensive work in the context of water-waves [2, 3, 7, 31, 47]. The Muskat problem is the natural parabolic analog of the water-wave problem and ”
Global solutions and asymptotic behavior for two dimensional gravity water waves (2013, methodology): “We refer the reader to [2, 3, 19] for the study of other dispersive properties of the water waves equations (Strichartz estimates and smoothing effect of surface tension).”
Two dimensional water waves in holomorphic coordinates II: global solutions (2014, background): “same problem in three or higher space dimensions, and also on related problems with surface tension, vorticity, ﬁnite bottom, etc. Without being exhaustive, we list some of the more recent references [1, 2, 4, 5, 7, 8, 15, 17, 20, 25]. An essential choice in any approach to this problem is that of t”