Almost global wellposedness of the 2-D full water wave problem
type: Item
instance of: scholarly article
title: Almost global wellposedness of the 2-D full water wave problem
zbMATH Open document ID: 1181.35205
author: Sijue Wu
published in: Inventiones Mathematicae
publication date: 2009-06-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/0910.2473
review text: This interesting paper is devoted to the problem of global in time existence and uniqueness of solutions of the \(2-D\) infinite depth full water wave equation. When surface tension is zero, the motion of the fluid is described by \[ \mathbf v_t+v\cdot \nabla v=-\mathbf k -\nabla P \quad \text{on } \Omega (t), \;t\geq 0,\tag{1} \] \[ \text{div } \mathbf v =0, \quad \text{curl }\mathbf v=0 \quad \text{on } \Omega (t), \;t\geq 0,\tag{2} \] \[ P=0 \quad \text{on } \Sigma (t),\tag{3} \] \[ (1, \mathbf v) \text{ is tangent to the free surface } (t,\Sigma (t)),\tag{4} \] where \(\mathbf v\) is the fluid velocity, \(P\) is the fluid pressure, \(\Sigma (t)\) (\(t\) - the time) is the free interface, \(\Omega (t)\) the region occupied by the fluid, and \(\mathbf k\) is the gravitational field (the unit vector pointing in the upward vertical direction). The author investigates the global in time behavior of solutions of the problem (1)--(4). It is known that there exists a solution defined in a time interval \([0,T/\epsilon ]\) for initial data of the form \(\epsilon\Psi \), where \(T\) depends only on \(\Psi \). An interesting result shown here is that for the same initial data there exists a unique solution in the interval \([0,T/\epsilon ]\). It turns out that there exists quantities \(\Theta \) that are the traces on the interface \(\Sigma (t)\) of some holomorphic or almost holomorphic functions in the air region \(\Omega (t)^c\). These quantities expressed in a new coordinate system satisfy equations of the form \(\partial_t^2\Theta -i\partial_{\alpha }\Theta =G\), where \(G\) consists of nonlinear terms of only cubic and higher orders. Using these equations one can prove existence and uniqueness of the solution to the problem (1)--(4) for the above stated initial data.
reviewed by: Dimitar A. Kolev
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 5566987
zbMATH Keywords: wellposedness
zbMATH Keywords: fluid dynamics
zbMATH Keywords: water wave
zbMATH Keywords: interface
zbMATH Keywords: inviscid fluid
zbMATH Keywords: incompressible fluid
zbMATH Keywords: irrotational fluid
MaRDI profile type: Publication
OpenAlex ID: W3099581489
arXiv ID: 0910.2473
cites work: The zero surface tension limit two‐dimensional water waves
cites work: Growth rates for the linearized motion of fluid interfaces away from equilibrium
cites work: Q3843473
cites work: Q4434430
cites work: Local smoothing effects for the water-wave problem with surface tension
cites work: On the motion of the free surface of a liquid
cites work: Well-posedness of the free-surface incompressible Euler equations with or without surface tension
cites work: L'intégrale de Cauchy définit un opératuer borne sur \(L^ 2 \)pour les courbes lipschitziennes
cites work: La solution des conjectures de Calderon
cites work: An existence theory for water waves and the boussinesq and korteweg-devries scaling limits
cites work: Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. (Calderón-Zygmund operators, para-accretive functions and interpolation)
cites work: Lectures on nonlinear hyperbolic differential equations
cites work: Q4824315
cites work: A shallow water approximation for water waves
cites work: Sur les ondes de surface de l'eau avec une justification mathématique des équations des ondes en eau peu profonde
cites work: A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves
cites work: Weighted L∞ and L1 estimates for solutions to the classical wave equation in three space dimensions
cites work: Uniform decay estimates and the lorentz invariance of the classical wave equation
cites work: Global existence of small amplitude solutions to nonlinear klein-gordon equations in four space-time dimensions
cites work: Q3733338
cites work: Well-posedness of the water-waves equations
cites work: Large time existence for 3D water waves and asymptotics
cites work: Well-posedness for the motion of an incompressible liquid with free surface boundary
cites work: Q4061809
cites work: FREE BOUNDARY PROBLEM FOR AN INCOMPRESSIBLE IDEAL FLUID WITH SURFACE TENSION
cites work: The long-wave limit for the water wave problem. I: The case of zero surface tension.
cites work: Normal forms and quadratic nonlinear Klein-Gordon equations
cites work: Geometry and a priori estimates for free boundary problems of the Euler's equation
cites work: A wave operator for a nonlinear Klein-Gordon equation
cites work: The Cauchy problem for nonlinear Klein-Gordon equations
cites work: Q4693724
cites work: Q3995410
cites work: The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I
cites work: Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains
cites work: Well-posedness in Sobolev spaces of the full water wave problem in 2D
cites work: Well-posedness in Sobolev spaces of the full water wave problem in 3-D
cites work: Gravity waves on the free surface of an incompressible perfect fluid of finite depth
cites work: On the free boundary problem of three-dimensional incompressible Euler equations
DOI: 10.1007/S00222-009-0176-8
description: scientific article