Two Dimensional Water Waves in Holomorphic Coordinates

Abstract

This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem as a quasilinear dispersive equation, we establish two results: (i) local well-posedness in Sobolev spaces, and (ii) almost global solutions for small localized data. Neither of these results are new; they have been recently obtained by Alazard-Burq-Zuily [1], respectively by Wu [23] using different coordinates and methods. Instead our goal is improve the understanding of this problem by providing a single setting for both problems, by proving sharper versions of the above results, as well as presenting new, simpler proofs. This article is self contained.For the derivation of the above equations, we refer the reader to Appendix A. In the real case these equations originate in [17]. The changes needed for the periodic case are also described in the same Appendix A. There are also other ways of expressing the equations, for instance in Cartesian coordinates using the Dirichlet to Neumann map associated to the water domain, see e.g. [1] . Here we prefer the holomorphic coordinates due to the simpler form of the equations; in particular, in these coordinates the Dirichlet to Neumann map is given in terms of the standard Hilbert transform.It is convenient to work with a new variable, namelyThe equations becomeThese equations are considered either in R × R or in R × S 1 .As the system (1.1) is fully nonlinear, a standard procedure is to convert it into a quasilinear system by differentiating it. Observing that almost no undifferentiated functions appear in (1.1), one sees that by differentiation we get a self-contained first order quasilinear system for (W α , Q α ). To write this system we introduce the auxiliary real function b, which we call the advection velocity, and is given byThe reason for this will be immediately apparent. Using b, the system (1.1) is written in the form

Two dimensional water waves in holomorphic coordinates II: global solutions

“…We consider this problem expressed in position-velocity potential holomorphic coordinates, and prove that small localized data leads to global solutions. This article is a continuation of authors' earlier paper [11]. …”

“…Instead, the present article relies on holomorphic coordinates, which were originally introduced by Nalimov [16]; these are briefly described below. In the earlier article [11], using holomorphic coordinates, we revisited this problem in order to provide a new, self-contained approach, which considerably simplified and improved on many of the results mentioned above. Our results included:…”

“…Now we recall the function spaces introduced in [11]. The system (1.3) is a well-posed linear evolution in the spaceḢ 0 of holomorphic functions endowed with the L 2 ×Ḣ 1 2 norm.…”

“…which solve the linearized equations, see [11]. However, these are not the correct diagonal variables; instead the diagonal variables are AS(W, Q), where the diagonalization operator A is given by A(w, q) := (w, q − Rw).…”

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

“…Global existence of the 2D gravity water waves for small initial data was first proved by Ionescu-Pusateri [21], and a similar result was proved independently by Alazard-Delort [3] in Eulerian coordinates. More recently Hunter-Ifrim-Tataru [16] used holomorphic coordinates to give a different proof of the almost global existence result; then Ifrim-Tataru [19] extended it to global existence in holomorphic coordinates. Ionescu-Pusateri [24] proved the global existence of the 2D capillary waves system for small initial data without the momentum condition on the associated profile in the Eulerian coordinates.…”