Analysis of Incompressible Flows with Rigid and Free Boundaries

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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.