Analysis of Incompressible Flows with Rigid and Free Boundaries

刚性和自由边界不可压缩流动分析

• 批准号: 1907776
• 负责人: Huy Nguyen
• 金额: $ 17.37万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2022-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1907776&HistoricalAwards=false
• 关键词: Analysis; Incompressible; Flows; Rigid; Free

Interactions of fluids with boundaries are ubiquitous in nature, science and engineering, and are of great importance in practical applications ranging from biology to aerospace, oceanography and geoscience. The boundaries can be rigid as between an aircraft and the surrounding air or between water and the seabed or can be free as between water and the air or between water and oil in an oil reservoir. Despite being widely used in applications, many fundamental issues in the rigorous mathematical analysis of these interactions remain challenging. The presence of boundaries, especially free boundaries, makes the underlying partial differential equations modeling the phenomena highly nonlinear and nonlocal. This project will further the mathematical understanding of equations in which fluids interact with either rigid boundaries, free boundaries, or both. The problems studied in this project are motivated by physics (drop formation and fluid jets), economics (optimal transport), and engineering (reservoir engineering). The analytical understanding gained in this work will increase our understanding of the basic physical phenomena and mathematical models used in practice. This research will involve graduate students and postdoctoral scholars. This project will study several mathematical problems of current interest involving the interaction of fluids with rigid or free boundaries: 1. Well-posedness and global regularity of the Muskat problem with boundaries of low regularity and large variation. 2. Long-time behavior of an active vectorial transport equation arising in optimal transport and convective fluids. 3. Instabilities of small amplitude periodic traveling gravity waves in shallow water. 4. Finite-time and infinite-time pinch-off singularities for two models of drop formation. The project will combine tools from microlocal analysis, harmonic analysis, bifurcation theory, and spectral theory. Some of the tools and models developed will have applications to other problems in the theory and computational aspects of partial differential equations.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.

流体与边界的相互作用在自然界、科学和工程中普遍存在，并且在从生物学到航空航天、海洋学和地球科学的实际应用中具有重要意义。边界可以是刚性的，如在飞行器和周围空气之间或在水和海床之间，或者可以是自由的，如在水和空气之间或在水和储油层中的油之间。尽管在应用中被广泛使用，但这些相互作用的严格数学分析中的许多基本问题仍然具有挑战性。边界的存在，特别是自由边界，使得基本的偏微分方程建模的现象高度非线性和非局部。这个项目将进一步的数学方程的理解，其中流体与刚性边界，自由边界，或两者的相互作用。该项目研究的问题是由物理学（液滴形成和流体射流），经济学（最佳运输）和工程学（油藏工程）激发的。在这项工作中获得的分析理解将增加我们对基本物理现象和实际使用的数学模型的理解。这项研究将涉及研究生和博士后学者。 这个项目将研究几个当前感兴趣的数学问题，涉及流体与刚性或自由边界的相互作用：1。具有低正则性和大变化边界的Muskat问题的适定性和全局正则性。2.最佳输运和对流流体中主动矢量输运方程的长时间行为。3.浅水中小振幅周期重力行波的不稳定性。4.两种液滴形成模型的瞬时和无限时夹断奇点。该项目将结合联合收割机工具，从微观局部分析，谐波分析，分叉理论，频谱理论。开发的一些工具和模型将应用于偏微分方程的理论和计算方面的其他问题。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

Self-similar solutions for the Muskat equation

• DOI: 10.1016/j.aim.2022.108294
• 发表时间: 2021-09
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Eduardo Garc'ia-Ju'arez;Javier G'omez-Serrano;H. Nguyen;B. Pausader

Proof of Modulational Instability of Stokes Waves in Deep Water

深水中斯托克斯波调制不稳定性的证明

• DOI: 10.1002/cpa.22073
• 发表时间: 2023
• 期刊: Communications on Pure and Applied Mathematics
• 影响因子: 3
• 作者: Nguyen, Huy Q.;Strauss, Walter A.
• 通讯作者: Strauss, Walter A.

Global solutions for the Muskat problem in the scaling invariant Besov space

缩放不变贝索夫空间中 Muskat 问题的全局解

• 发表时间: 2022
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Huy Q. Nguyen

On well-posedness of the Muskat problem with surface tension

表面张力的 Muskat 问题的适定性

• DOI: 10.1016/j.aim.2020.107344
• 发表时间: 2020
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Nguyen, Huy Q.

The Vanishing Surface Tension Limit of the Muskat Problem

Muskat 问题的表面张力极限消失

• DOI: 10.1007/s00220-021-03980-9
• 发表时间: 2021
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Flynn, Patrick T.;Nguyen, Huy Q.
• 通讯作者: Nguyen, Huy Q.