Steady stratified water waves and asymptotic models

ChenDMS-1613375水波理论是一个引人入胜的话题。对流体的第一个全面的数学理解可以追溯到17世纪50年代的欧拉，以他的名字命名的方程式至今仍是数学家和工程师使用的基本模型。对这些方程进行精确而严谨的数学分析，使理解它们所代表的物理系统的定性结构和行为成为可能。在这个项目中，调查员研究了：海洋内孤立波，这被认为是沿海海洋动力学的非常重要的组成部分；波流相互作用的详细动力学，这可有助于近海工程结构的设计以及海底航行；以及揭示海啸波各方面情况的近似模型理论。该项目产生的数学结果有望促进对海浪和海流的基本理解。研究生被包括在该项目的工作中。在本项目中，(1)建立了大振幅分层孤立波的存在性；(2)研究了分层流体中表面和内部孤立波对流线密度的连续依赖性；(3)研究了一些长波水波模型和等离子体模型中孤立波的稳定性；(4)研究了全水波渐近模型方程光滑解的破裂机制；(5)研究了一类拟线性色散方程整体弱解的适定性(包括存在性、唯一性和对初值的连续依赖性)。本项目所开发的技术可望对其他非线性水波方程也有一定的借鉴意义。研究生被包括在该项目的工作中。

ChenDMS-1613375 The theory of water waves is a fascinating topic. The first comprehensive mathematical understanding of a fluid goes back to Euler in the 1750s, and the equations named after him continue to be the basic model used by mathematicians and engineers today. Precise and rigorous mathematical analysis of these equations makes it possible to understand the qualitative structures and behaviors of the physical systems they represent. In this project, the investigator studies: oceanic internal solitary waves, which are recognized to be highly significant components of the dynamics of the coastal sea; the detailed dynamics of wave current interactions, which can help in design of offshore engineering structures as well as in submarine navigation; and the theory of approximate models that shed light on aspects of tsunami waves. The mathematical results generated from this project are expected to advance the fundamental understanding of ocean waves and currents. Graduate students are included in the work of the project. In this project, (1) existence of large-amplitude stratified solitary waves is established; (2) continuous dependence of surface and internal solitary waves on the streamline densities in a stratified fluid is examined; (3) the stability of solitary waves in some long-wave water wave models as well as a plasma model is studied; (4) the breakdown mechanism of smooth solutions for asymptotic model equations of full water waves is studied; (5) the well-posedness (including existence, uniqueness, and continuous dependence on initial data) of global weak solutions to a class of quasilinear dispersive equations is investigated. The techniques developed in carrying out this project are expected to be useful for other nonlinear water wave equations. Graduate students are included in the work of the project.