Steady stratified water waves and asymptotic models

稳态分层水波和渐近模型

• 批准号: 1613375
• 负责人: Ming Chen
• 金额: $ 19.37万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1613375&HistoricalAwards=false
• 关键词: Steady; stratified; water; waves; asymptotic

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.

陈DMS-1613375 水波理论是一个令人着迷的话题。 对流体的第一个全面的数学理解可以追溯到18世纪50年代的欧拉，以他的名字命名的方程仍然是今天数学家和工程师使用的基本模型。 对这些方程进行精确和严格的数学分析，可以理解它们所代表的物理系统的定性结构和行为。 在这个项目中，研究人员研究：海洋内部孤立波，这被认为是沿海海洋动力学的重要组成部分;波流相互作用的详细动力学，这可以帮助设计近海工程结构以及潜艇航行;以及近似模型理论，揭示了海啸波的各个方面。 该项目产生的数学结果有望促进对海浪和洋流的基本理解。 研究生也参与了该项目的工作。 本项目的主要工作是：（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孤立波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）适定性（包括存在性、唯一性、和对初值的连续依赖性）的整体弱解的存在性. 本研究所发展的方法可望对其他非线性水波方程的求解有所帮助。 研究生也参与了该项目的工作。

项目成果

Global bifurcation of solitary waves to the Boussinesq abcd system

Boussinesq abcd 系统的孤立波全局分叉

• DOI: 10.1016/j.jde.2021.11.019
• 发表时间: 2022
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: Chen, Robin Ming;Jin, Jie
• 通讯作者: Jin, Jie

The Shallow-Water Models with Cubic Nonlinearity

• DOI: 10.1007/s00021-022-00685-4
• 发表时间: 2022-04
• 期刊: Journal of Mathematical Fluid Mechanics
• 影响因子: 1.3
• 作者: R. Chen;Tianqiao Hu;Yue Liu

$W^{1,\infty}$ instability of $H^1$-stable peakons in the Novikov equation

诺维科夫方程中 $H^1$ 稳定峰值的 $W^{1,infty}$ 不稳定性

• DOI: 10.4310/dpde.2021.v18.n3.a1
• 发表时间: 2021
• 期刊: Dynamics of Partial Differential Equations
• 影响因子: 1.3
• 作者: Chen, Robin Ming;Pelinovsky, Dmitry E.
• 通讯作者: Pelinovsky, Dmitry E.