On the Cauchy problem for gravity water waves

Abstract

We are interested in the system of gravity water waves equations without surface tension. Our purpose is to study 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+ǫ -class for some ǫ > 0 and consequently have unbounded curvature, while the initial velocities are only Lipschitz. We reduce the system using a paradifferential approach.Many results have been obtained on the Cauchy theory for System (1.5), starting from the pioneering works of Nalimov [44], Shinbrot [50], Yoshihara [57], Craig [25]. In the framework of Sobolev spaces and without smallness assumptions on the data, the well-posedness of the Cauchy problem was first proved by Wu for the case without surface tension (see [52,53]) and by Beyer-Günther in [12] in the case with surface tension. Several extensions of their results have been obtained by different methods (see [22,31,32,33,35,41,54,55,59] for recent results and the surveys [11,29,36] for more references). Here we shall use the Eulerian formulation. Following Zakharov [58] and Craig-Sulem [28], we reduce the analysis to a system on the free surface Σ(t) = {y = η(t, x)}. If ψ is defined by ψ(t, x) = φ(t, x, η(t, x)), then φ is the unique variational solution of ∆ x,y φ = 0 in Ω, φ| y=η = ψ, ∂ n φ = 0 on Γ. 3 7

Two dimensional water waves in holomorphic coordinates II: global solutions

“…Extensive work was also done on the 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].…”

Global Infinite Energy Solutions for the 2D Gravity Water Waves System

“…Recall that the the local well-posedness for finite energy initial data was obtained first in [32] and then in [2]. We remark that, with minor modifications, local well-posedness also holds for the initial data with infinite energy, which is small in the H N;p space.…”

“…As the system (1.3) lacks symmetries, one fails at the beginning of the energy estimate due to the quasilinear nature. Thanks to the work of Alazard-Métivier [4], the works of Alazard-Burq-Zuily [1,2], and the work of the Alazard-Delort [3], their paralinearization method helps us to see the good structures inside the system (1.3) and to find the good substitution variables U 1 and U 2 . The system of equations satisfied by U 1 and U 2 has requisite symmetries to avoid losing derivatives when doing the energy estimate.…”

“…To prove the local well-posedness in the infinite energy setting, we pay attention to the low frequency and redo the argument in [2], which can be summarized in three main steps as follows:…”

“…With minor modifications in the argument in [2], we can show the local wellposedness for the infinite energy initial data because of three observations as follows:…”