Related papers: A New Structure for the 2D water wave equation: En…
The two dimensional gravity water wave problem concerns the motion of an incompressible fluid occupying half the 2D space and flowing under its own gravity. In this paper we study long-term regularity of solutions evolving from small but…
The gravity water waves equations describe the evolution of the surface of an incompressible, irrotational fluid in the presence of gravity. The classical regularity threshold for the well-posedness of this system requires initial velocity…
This is the second part of our work initiating the rigorous study of wave turbulence for water waves equations. We combine energy estimates, normal forms, and probabilistic and combinatorial arguments to complete the construction of…
In this paper we prove global regularity for the full water waves system in 3 dimensions for small data, under the influence of both gravity and surface tension. This problem presents essential difficulties which were absent in all of the…
This article represents the second installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on global solutions for small and localized data. Such…
We consider the 2D gravity water waves equation on an infinite domain. We prove a local wellposedness result which allows interfaces with corners and cusps as initial data and which is such that the time of existence of solutions is uniform…
As a starting point of studying the long time behavior of the $3D$ water waves system in the flat bottom setting, in this paper, we try to improve the understanding of the Dirichlet-Neumann operator in this setting. As an application, we…
Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKE) for water waves models. This problem has received intense attention in recent years in the context of…
We consider two-dimensional steady periodic gravity waves on water of finite depth with a prescribed but arbitrary vorticity distribution. The water surface is allowed to be overhanging and no assumptions regarding the absence of stagnation…
We consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for suitably small and localized perturbations of a…
We prove local well-posedness for the gravity water waves equations without surface tension, with initial velocity field in $H^s$, $s > \frac{d}{2} + 1 - \mu$, where $\mu = \frac{1}{10}$ in the case $d = 1$ and $\mu = \frac{1}{5}$ in the…
This paper is devoted to the proof of a global existence result for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description in physical coordinates of the solution,…
We prove an asymptotic stability result for the water wave equations linearized around small solitary waves. The equations we consider govern irrotational flow of a fluid with constant density bounded below by a rigid horizontal bottom and…
This paper is devoted to the stabilization of the incompressible Euler equation with free surface. We study the damping of two-dimensional gravity waves by an absorbing beach where the water-wave energy is dissipated by using the variations…
In this paper, we establish the transverse linear asymptotic stability of one-dimensional small-amplitude solitary waves of the gravity water-waves system. More precisely, we show that the semigroup of the linearized operator about the…
This article represents the first installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on sharp cubic energy estimates. Precisely, we introduce and…
We construct large families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed. A Riemann-Hilbert problem approach is used to recast the governing equations…
We consider the two dimensional gravity water wave equation in a regime where the free interface is allowed to be non-$C^1$. In this regime, only a degenerate Taylor inequality $-\frac{\partial P}{\partial \bf n}\ge 0$ holds, with…
We study a fundamental model in fluid mechanics--the 3D gravity water wave equation, in which an incompressible fluid occupying half the 3D space flows under its own gravity. In this paper we show long-term regularity of solutions whose…
We show existence of global solutions for the gravity water waves equation in dimension 3, in the case of small data. The proof combines energy estimates, which yield control of L^2 related norms, with dispersive estimates, which give decay…