Analysis of Free Boundaries: Contact Lines and Viscous Traveling Waves

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Fluids with free boundaries occur universally in nature. At a human scale, we encounter them daily in our cups of coffee or on the surface of a pond. Important examples at smaller scales include the flow of blood and air through our cardiovascular and respiratory systems, while examples at larger scales include the Earth's ocean and atmosphere or even the hot plasma on the surface of a star. Since free boundaries are so common and occur at so many scales, it is important to understand the role they play in fluid mechanics. The main scientific goal of this project is to contribute to the understanding of such free boundaries through the mathematical study of the partial differential equations that describe their dynamics. This involves not only the application of existing mathematical tools to these problems, but also the development of new tools and techniques, which in turn may be useful in the study of other problems. The project also aims to contribute to the development of the next generation of researchers at multiple levels: at the undergraduate level through the project's Summer Analysis Program, which will provide undergraduate research opportunities to eight students; and at the graduate level through graduate research and mentoring.This project aims at studying fundamental questions of well-posedness and stability for two long-standing open problems in interfacial mechanics. The first is the contact line problem, which deals with the dynamics of a triple interface between a viscous fluid, a solid, and a vapor phase. This component of the project aims to verify the soundness of a recently proposed continuum model of contact lines. This has the potential to make a serious impact in the science of triple-phase junctions, to open many new lines of research, and to further the applications of contact line dynamics. From an analytic perspective, it will also create and further develop bridges between energy methods, the functional calculus of differential operators, and elliptic regularity theory, which will be useful in many other contexts. The second problem is the viscous traveling wave problem. While the existence of traveling wave solutions to the free boundary, incompressible Euler equations has been known for nearly a century, progress on the corresponding Navier-Stokes problem only began recently with the work of the PI and collaborators. The proposal aims to build on this work to construct traveling wave solutions in more general contexts and to study their dynamical stability and vanishing viscosity limits. It is important to account for the viscous case because, while many fluids have small viscosity (or more precisely, the fluid configuration has large Reynolds number), small does not mean zero, so all fluids experience some viscous effects. Developing the viscous theory also opens the possibility of connecting the viscous and inviscid cases through vanishing viscosity limits, which could potentially yield insight into the zoo of known inviscid solutions. In particular, it could lead to a selection mechanism for physically relevant inviscid solutions. This work will also contribute fundamental new tools in the theory of function spaces, which will be useful in other traveling wave problems.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.