Global Analysis of a Model for Capillary Water Waves in Two Dimensions

Abstract

In this paper we prove a global regularity result for a quadratic quasilinear model associated to the water waves system with surface tension and no gravity in dimension 2 (the capillary waves system). The model we consider here retains most of the difficulties of the full capillary water waves system, including the delicate time resonance structure and modified scattering. It is slightly simpler, however, at the technical level, and our goal here is to present our method in this simplified situation. The analysis of the full system is presented in a preprint by the authors (2014). © 2016 Wiley Periodicals, Inc.

Absence of Shocks for One Dimensional Euler–Poisson System

“…The details are presented in Section 4.1. The estimate for || ξ N 1 +10 w|| L ∞ is carried out in Section 4.2 as an adaptation of the proof in [18,19,20,21]. Through precise frequency decompositions and stationary phase analysis, we notice that a phase correction is needed to the leading order term and thus leads to the modified scattering behavior (Proposition 4.5).…”

“…we see xr and xu satisfy 20) where N ′ j = N ′ j (r, u) (j = 1, 2) are linear and quadratic terms not including xr and xu,…”

“…Our work is inspired by recent work of [1,19,20,21] on water waves system, which depends on a delicate interplay between higher energy estimates and a low order L ∞ estimate. It is well-known that due to poor decay rate of t −1/2 , the classical energy estimate with quadratic nonlinearity is impossible to close, and it is necessary to perform the energy estimate in a new system with a cubic nonlinearity.…”

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

“…It turns out that the difficulties are still manageable. The ideas of proving the L 1 -type decay estimate are mainly based on the works of Ionescu-Pusateri [20][21][22]24]. To improve the understanding of the L 1 decay estimate, many ideas are combined.…”

“…Now it is easy to see that they are all homogeneous of degree 5 2 . As in [22], we need to modify the phase of the profile first to successfully close the argument and see the modified scattering property. We define From (4.16), we can see that the modified phase is only effective for the cubic term I C;C; .t; /, which is the only one that has nonempty space-time resonance set.…”