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.

与边界的流体相互作用在性质，科学和工程上无处不在，并且在从生物学到航空航天，海洋学和地球科学等实用应用中非常重要。边界可以像飞机和周围的空气之间，水与海床之间一样僵硬，也可以像水与空气之间的水和油和油之间的水和油之间的自由。尽管被广泛用于应用中，但对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，尤其是自由边界，使基本的部分微分方程对高度非线性和非局部性现象进行建模。该项目将进一步对流体与刚性边界，自由边界或两者相互作用的方程式的数学理解。该项目中研究的问题是由物理学（掉落形成和流体喷气机），经济学（最佳运输）和工程（水库工程）的动机。这项工作中获得的分析理解将增加我们对实践中使用的基本物理现象和数学模型的理解。这项研究将涉及研究生和博士后学者。 该项目将研究当前兴趣的几个数学问题，涉及流体与刚性或自由边界的相互作用：1。穆斯卡特问题的适当性和全球规律性，具有低规律性和较大变化的边界。 2。在最佳传输和对流流体中产生的主动矢量传输方程的长期行为。 3。小幅度周期性旅行重力波在浅水中的不稳定性。 4。有限的时间和无限的时间捏合奇异性，用于两种模型的落下形成。该项目将结合微局部分析，谐波分析，分叉理论和光谱理论的工具。所开发的一些工具和模型将在部分微分方程的理论和计算方面中应用于其他问题。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛的影响评估的评估来支持的。

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.

On Global Stability of Optimal Rearrangement Maps

论最优重排图的全局稳定性

• DOI: 10.1007/s00205-020-01552-0
• 发表时间: 2020
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Nguyen, Huy Q.;Nguyen, Toan T.
• 通讯作者: Nguyen, Toan T.