Long Time Existence Results for Solutions of Water Waves Equations

Abstract

We present in this talk various results, obtained during the last years by several authors, about the problem of long time existence of solutions of water waves and related equations, with initial data that are small, smooth, and decaying at infinity. After recalling some facts about local existence theory, we shall focus mainly on global existence theorems for gravity waves equations proved by Ionescu-Pusateri, Alazard-Delort and Ifrim-Tataru. We shall describe some of the ideas of the proofs of these theorems, and mention as well related results.From joint work with Thomas Alazard. Partially supported by the ANR project 13-BS01-0010-02 "Analyse asymptotique des équations aux dérivées partielles d'évolution". MSC2010: primary 76B15; secondary 35Q35, 35B40, 35S50. 2209

Modeling shallow water waves

“…) Compared to the KdV-BBM family, the inclusion of new nonlinear terms of size O(εµ) (as well as O(ε 2 ) and O(ε 3 ) in ( 75)) restores the possibility of wave breaking, as shown in [51] for (73) and (75) (and this could likely be extended to ( 75)); we recall that wave breaking for v means that v remains bounded but that ∂ x v blows up in finite time (and a similar definition holds of course for ζ). This wave breaking is shown to occur on a O(1/ε) time scale, which is the same as for the Burgers equations ( 61) and (62).…”

“…) . Let us for instance seek an equation on v and an algebraic expression for ζ in terms of v. Since the SGN equations are a O(µ) perturbation of the NSW equations, right going waves are expected to be a O(µ) perturbation of the Burgers equation (62), that is,…”

“…We describe below two other formulations: one is Zakharov's Hamiltonian formulation [190] whose well-posedness was proved in [115] (see also [3] for the low regularity Cauchy problem and [5,91] for uniform bounds in several asymptotic regimes), as well as a formulation in (ζ, Q), where Q is the horizontal discharge, that proves very useful to derive and understand the mechanism at stake in shallow water asymptotic models. For other recent mathematical advances on the water waves equations, such as long time/global existence, we refer to the surveys [62,95].…”

“…Since the SGN equations are a O(µ) perturbation of the NSW equations, right going waves are expected to be a O(µ) perturbation of the Burgers equation (62), that is,…”