Steady solutions and stability results for problems in fluid dynamics

The study of the behavior of fluids is one the most foundational problems in the physical world. Fluids arise everywhere, from the planet’s oceans to the blood flowing through one's veins. Although the mathematical study of such problems goes back many centuries, researchers have barely made a dent in the magnitude of the work that needs to be done and many fundamental problems are still wide open. The aim of this project is to tackle some of these problems, which can be modeled by partial differential equations. These are highly physical with many applications to other fields such as physics, biology and engineering. As part of this award, the principal investigator (PI) also mentors undergraduate students by providing them with opportunities to learn new mathematical methods whilst solving original problems with physical relevance. The project will investigate two distinct types of problems: on the one hand, the PI will study the global stability of solutions to dynamical problems. More specifically, the global stability of the sphere to the three-dimensional Peskin problem (modeling the flow of blood through the heart valves), and the global stability to the full two-dimensional Muskat problem (modeling the interaction of two immiscible fluids propagating through porous medium) will be investigated. On the other hand, the investigator introduces a definitive construction of large-amplitude steady solutions to the water wave problem (modeled by the fundamental Euler equations), which can have either overhanging wave profiles or contain other interesting physical singularities. The strategy to obtain these will rely on a deeper understanding of various parameters to be able to determine exactly when such objects can occur. The study of each situation is in itself already an important problem. 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.