Bifurcation, Stability, and Non-uniqueness in Ideal Fluids

理想流体中的分岔、稳定性和非唯一性

基本信息

批准号：
2205910
负责人：
Ming Chen
金额：
$ 25.33万
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：
2022
资助国家：
美国
起止时间：
2022-09-01 至 2025-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2205910&HistoricalAwards=false
关键词：
Bifurcation Stability Non uniqueness Ideal

项目摘要

This project seeks to promote and advance the mathematical theory of free-surface water waves and compressible fluids, which host a wide range of interesting physical phenomena and whose dynamics is governed by the Euler equations. The main goals are to develop new or extend existing analytic techniques to establish existence and stability theory for steady water waves, and to investigate uniqueness properties for weak solutions to one-dimensional system of compressible gases. Progress in this project will enhance our understanding of the mathematics of ideal fluids and develop novel mathematical tools that can provide insight into truly nonlinear phenomena in partial differential equations. This research will also involve training and collaboration with graduate students and postdoctoral researchers.This project will bring new perspectives and develop novel approaches to make progress on some fundamental problems in the study of ideal fluids. Specifically, the PI will use a novel global bifurcation theoretic machinery to construct large-amplitude solitary water waves in presence of localized disturbances coming from either submerged objects or the bottom topography. The second topic of this project concerns stability of solitary water waves. The PI will study the spectral stability of multimodal gravity-capillary internal solitary waves and prove nonlinear transverse instability of gravity-capillary internal solitary waves. The final track of the project consists of the study of the non-uniqueness of entropy solutions to the compressible isentropic Euler system. The main ingredients and techniques involved in the study include bifurcation method, complex analysis, stability analysis, convex integration machinery, and the theory of hyperbolic conservation laws.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.

该项目旨在促进和推进自由表面水波和可压缩流体的数学理论，这些理论包含各种有趣的物理现象，其动力学由欧拉方程控制。主要目标是开发新的或扩展现有的分析技术，以建立稳定水波的存在性和稳定性理论，并研究一维可压缩气体系统弱解的唯一性。该项目的进展将增强我们对理想流体数学的理解，并开发新颖的数学工具，可以深入了解偏微分方程中真正的非线性现象。这项研究还将涉及与研究生和博士后研究人员的培训和合作。该项目将带来新的视角并开发新的方法，以在理想流体研究的一些基本问题上取得进展。具体来说，PI 将使用一种新颖的全局分岔理论机制，在存在来自水下物体或底部地形的局部扰动的情况下构建大幅值的孤立水波。该项目的第二个主题涉及孤立水波的稳定性。 PI将研究多模态重力-毛细管内孤立波的谱稳定性，并证明重力-毛细管内孤立波的非线性横向不稳定性。该项目的最后一个轨道包括研究可压缩等熵欧拉系统熵解的非唯一性。该研究涉及的主要内容和技术包括分岔法、复分析、稳定性分析、凸积分机制和双曲守恒定律理论。该奖项体现了 NSF 的法定使命，并通过利用基金会的智力评估进行评估，认为值得支持。优点和更广泛的影响审查标准。