Mathematical Analysis of Water Waves and Other Fluid Models

水波和其他流体模型的数学分析

• 批准号: 1907584
• 负责人: Ming Chen
• 金额: $ 19万
• 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: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1907584&HistoricalAwards=false
• 关键词: Mathematical; Analysis; Water; Waves; Other

Despite their ubiquity and importance in physics and engineering, water waves remain very difficult to analyze or to predict, largely because of the presence of the unknown, time-varying fluid domain and the nonlinear way the spatiotemporal patterns affect the transport of energy and fluid particles. While many analytical tools currently exist for studying water waves of small amplitude, this project focuses on developing new machinery to construct large-amplitude waves and to investigate their ability to persist under small perturbations. Another objective of this project is to understand how energy evolves in response to the interaction between a fluid and a solid. Results from this project will advance the mathematical theory of water waves and contribute to understanding of other mathematical models in fluid mechanics and other related branches of applied science and engineering. This research also involves training and collaboration with graduate students and postdoctoral researchers.This project will develop a rigorous study of some nonlinear partial differential equations arising from water waves and other fluid models, extending some of the analytic techniques to related contexts, and integrating the research to the education of undergraduate and graduate students. Specifically, the proposed research addresses three separate directions regarding the existence and qualitative properties of the solutions to the water wave problem as well as other physically significant systems: (1) using a novel global bifurcation theoretic machinery to construct large-amplitude front-type solutions and solitary wave solutions to a two-phase fluid system in a channel, and extend this global theory to handle traveling waves that evolve according to more general symmetry groups; (2) proving the spectral stability of solitary waves to some long-wave water wave model which are indefinite energy saddles; (3) establishing exact energy equality for a fluid-interaction model and deriving sufficient conditions for the weak inviscid limit of Navier-Stokes solutions in domains with boundary. The main ingredients and techniques involved in the study include methods from elliptic partial differential equations, bifurcation and degree theory and stability analysis, together with energy estimates and Fourier analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

尽管它们在物理学和工程学中无处不在和重要性，但水波仍然很难分析或预测，主要是因为存在未知的时变流体域以及时空模式影响能量和流体粒子传输的非线性方式。虽然目前存在许多用于研究小振幅水波的分析工具，但该项目的重点是开发新的机械来构建大振幅波，并研究它们在小扰动下持续存在的能力。该项目的另一个目标是了解能量如何响应流体和固体之间的相互作用。该项目的成果将推进水波的数学理论，并有助于理解流体力学和应用科学和工程的其他相关分支中的其他数学模型。本研究还包括与研究生和博士后研究人员的培训和合作，本项目将对由水波和其他流体模型产生的一些非线性偏微分方程进行严格的研究，将一些分析技术扩展到相关背景，并将研究融入本科生和研究生的教育中。具体来说，拟议的研究涉及三个独立的方向，关于水波问题以及其他物理上重要的系统的解决方案的存在性和定性性质：（1）利用一种新的全局分叉理论机制构造了通道中两相流体系统的大振幅波前型解和孤立波解，（2）证明了一类长波水波模型孤立波的谱稳定性，这些长波水波模型是不确定的能量鞍;（3）建立了流体相互作用模型的精确能量等式，导出了有边界区域内Navier-Stokes解弱无粘极限的充分条件。 该研究涉及的主要成分和技术包括椭圆偏微分方程、分叉和度理论和稳定性分析的方法，以及能量估计和傅立叶分析。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

$W^{1,\infty}$ 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.

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

Orbital Stability of Internal Waves

内波的轨道稳定性

• DOI: 10.1007/s00220-022-04332-x
• 发表时间: 2022
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Chen, Robin Ming;Walsh, Samuel
• 通讯作者: Walsh, Samuel

A Kato-Type Criterion for Vanishing Viscosity Near Onsager’s Critical Regularity

接近 Onsager 临界正则性的加藤型粘度消失准则

• DOI: 10.1007/s00205-022-01822-z
• 发表时间: 2022
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Chen, Robin Ming;Liang, Zhilei;Wang, Dehua
• 通讯作者: Wang, Dehua

Global ill-posedness for a dense set of initial data to the isentropic system of gas dynamics

气体动力学等熵系统的一组密集初始数据的全局不适定性

• DOI: 10.1016/j.aim.2021.108057
• 发表时间: 2021
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Chen, Robin Ming;Vasseur, Alexis F.;Yu, Cheng
• 通讯作者: Yu, Cheng