Finite depth gravity water waves in holomorphic coordinates

具有常涡量的二维孤立水波，第一部分：深重力情形

Two dimensional solitary water waves with constant vorticity, Part I: the deep gravity case

该研究在无限深常涡量二维纯重力水波场景下，采用全纯坐标与不动点论证，构造出临界速度附近、类缩放Benjamin-Ono孤子的光滑孤立波，并给出其渐近展开式。

关于含点涡的重力-毛细水波

On the gravity-capillary water waves with point vortices

本文研究含点涡的二维深水重力-毛细水波，在全纯坐标下推导先验能量估计，证明初值在 $\mathcal{H}^s\times \Omega_t^N$（$s>\frac{3}{2}$）时解的唯一性，并指出对称双涡纯毛细水波解具有延长的三次方寿命。

二维空间拟周期重力水波的局部适定性

Local well-posedness of spatially quasiperiodic gravity water waves in two dimensions

本文首次证明了二维重力水波方程在空间拟周期初始条件下的局部适定性，通过全纯坐标和希尔伯特变换计算狄利克雷-涅曼算子，并利用Littlewood-Paley分解建立乘积和交换估计。

Low regularity solutions for gravity water waves

We prove local well-posedness for the gravity water waves equations without surface tension, with initial velocity field in $H^s$, $s > \frac{d}{2} + 1 - μ$, where $μ= \frac{1}{10}$ in the case $d = 1$ and $μ= \frac{1}{5}$ in the case $d \geq 2$, extending previous results of Alazard-Burq-Zuily. The improvement primarily arises in two areas. First, we perform an improved analysis of the regularity of the change of variables from Eulerian to Lagrangian coordinates. Second, we perform a time-interval length optimization of the localized Strichartz estimates.

一种关于重力水波存在性的极小极大变分方法

A min-max variational approach to the existence of gravity water waves

该研究通过将山路引理应用于二维水波方程相关Alt-Caffarelli泛函的奇异扰动，在物理坐标系中建立了重力水波的存在性，框架适用于流体平衡问题及伯努利型自由边界问题的极小极大解构造。

The NLS approximation for two dimensional deep gravity waves

This article is concerned with infinite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. Our goal is to study this problem with small wave packet data, and to show that this is well approximated by the cubic nonlinear Schrödinger equation (NLS) on the natural cubic time scale.

A remark on the two dimensional water wave problem with surface tension

We consider the motion of a two-dimensional interface between air (above) and an irrotational, incompressible, inviscid, infinitely deep water (below), with surface tension present. We propose a new way to reduce the original problem into an equivalent quasilinear system which are related to the interface's tangent angle and a quantity related to the difference of tangential velocities of the interface in the Lagrangian and the arc-length coordinates. The new way is relatively simple because it involves only taking differentiation and the real and the imaginary parts. Then if assuming that wav…

柱坐标下分层赤道流

Stratified equatorial flows in cylindrical coordinates with surface tension

本文研究了柱坐标下具有自由表面的无粘不可压缩流体在赤道区域的稳定流动，考虑了重力和表面张力作用下的波传播，并证明了自由表面的正则性结果。

带涡量和旋度的轴对称毛细水波的分岔分析

Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl

该研究针对带一般涡量、旋度及表面张力的定常轴对称水波，通过变量替换克服坐标奇异性，将问题转化为适用于Rabinowitz全局分岔定理的形式，构造出从平坦层流分岔的局部与全局解曲线。

Corrections to the KdV approximation for water waves

In order to investigate corrections to the common KdV approximation for surface water waves in a canal, we derive modulation equations for the evolution of long wavelength initial data. We work in Lagrangian coordinates. The equations which govern corrections to the KdV approximation consist of linearized and inhomogeneous KdV equations plus an inhomogeneous wave equation. These equations are explicitly solvable and we prove estimates showing that they do indeed give a significantly better approximation than the KdV equation alone.

作为 Babenko 方程解的尖峰 Stokes 波

Peaked Stokes waves as solutions of Babenko's equation

本文证明深水极限下尖峰 Stokes 波的性质可由 Babenko 方程恢复，并通过在最大高程附近将其改写为不动点问题开展奇点局部分析，排除了全纯坐标中的角点奇点。

Exact Pollard-like internal water waves

In this paper we construct a new solution which represents Pollard-like three-dimensional nonlinear geophysical internal water waves. The Pollard-like solution includes the effects of the rotation of Earth and describes the internal water wave which exists at all latitudes across Earth and propagates above the thermocline. The solution is provided in Lagrangian coordinates. In the process we derive the appropriate dispersion relation for the internal water waves in a stable stratification and discuss the particles paths. An analysis of the dispersion relation for the constructed model identifi…

The pressure in a deep-water Stokes wave of greatest height

In this paper we investigate the qualitative behaviour of the pressure function beneath an extreme Stokes wave over infinite depth. The presence of a stagnation point at the wave-crest of an extreme Stokes wave introduces a number of mathematical difficulties resulting in the irregularity of the free surface profile. It will be proven that the pressure decreases in the horizontal direction between a crest-line and subsequent trough-line, except along these lines themselves where the pressure is stationary with respect to the horizontal coordinate. In addition we will prove that the pressure st…