| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | January 20, 2022 |
| Latest Amendment Date: | January 20, 2022 |
| Award Number: | 2205734 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
Pedro Embid
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | October 1, 2021 |
| End Date: | October 31, 2023 (Estimated) |
| Total Intended Award Amount: | $173,696.00 |
| Total Awarded Amount to Date: | $77,090.00 |
| Funds Obligated to Date: |
FY 2021 = $59,350.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
3112 LEE BUILDING COLLEGE PARK MD US 20742-5100 (301)405-6269 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
MD US 20742-1000 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | APPLIED MATHEMATICS |
| Primary Program Source: |
01002122DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
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.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
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 main goal of the proposal is to advance our understanding of interactions of fluids with boundaries, including both rigid and free boundaries. This goal was achieved through studying various nonlinear and nonlocal partial differential equations arising in different contexts in which boundaries are present and active in different manners.
The PI studied the Muskat problem arising in petroleum engineering. This problem involves the free boundary between oil and water, and rigid boundaries (e.g., cap rock) in an oil reservoir. The PI proved the existence and uniqueness of solutions to the Muskat problem in great generality, allowing for boundaries of large variations and for various physical effects, including surface tension and its vanishing limit. The PI also proved the existence of self-similar solutions - those that respect the natural scaling of the problem.
The PI studied the water wave problem in oceanography. The emphasis was placed on Stokes waves (1847), which are among the first exact solutions to the fully nonlinear water wave problem. They travel in a fixed direction with a constant speed and a periodic profile. Although stability of traveling surface waves has been investigated extensively for water waves and many of its approximate models, stability of the classical Stokes waves still remains open in various aspects. The PI developed a method to rigorously prove that small-amplitude Stokes waves in deep water are unstable when perturbed by small long waves. This resolved a long standing open problem.
The PI studied a coupled system of partial differential equations of transport type introduced to capture optimal rearrangement maps on bounded domains as the infinite time limits of solutions to the system. The PI rigorously justified that optimal rearrangement maps with strictly convex potentials can be obtained via this system. This is a stability analysis for a nonlocal and nonlinear system in domains with physical boundaries.
The PI has dissiminated the results obtained from the projects at various national and international conferences, workshops, and seminars. The PI organized weekly research seminars at Brown and UMD. At UMD, he also organized a weekly Research Interaction Team seminar designed to foster interaction between faculty, students, and postdocs in his field. The PI organized an international summer school in fluid dynamics, which attracted students from different countries, including female students. Some results from the project were lectured at the summer school.
The project involved a graduate student (at Brown) and an undergraduate student (at UMD), each has a joint paper with the PI. After the project, the undergraduate student applied for math PhD programs and successfully received admissions (from Berkeley, Courant, Brown, Duke, etc).
Last Modified: 02/26/2024
Modified by: Huy Q Nguyen
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