NSF-UEFISCDI: Linear and Nonlinear Stability of Physical Flows

The main goal of the project is to study the dynamics of solutions of several important partial differential equations. These equations describe some of the classical systems of Applied Mathematics and Physics, such as incompressible fluids, geophysical flows, plasma, and water waves, and stability or instability of their solutions is a fundamental question, with both theoretical and practical applications. We aim to analyze these solutions rigorously and to recover quantitative and qualitative information about their behavior as mathematical theorems. The project will establish links and collaborations between researchers at leading American and Romanian universities, as part of the new NSF-UEFISCDI initiative, support the training and development of early-career researchers and students, with a focus of enhancing diversity, and disseminate the research widely at conferences and summer schools. The project will investigate the long-time dynamics of solutions of several evolution PDE, such as the Euler and the Navier-Stokes equations, geophysical flows, the Vlasov-Poisson system, and water waves models, mainly in dimensions 2 and 3. These equations have important explicit solutions, and the main questions to be considered have to do with the long-time stability of these solutions. Specific major problems will be investigated, concerning the global stability of shear flows and vortices for the 2D Euler and Navier-Stokes equations, the instability of mountain waves, the complex dynamics of oceanic flows, and the global stability of homogeneous solutions of the Vlasov-Poisson system. This is a very active field, with many recent methods and results. It is likely that other important problems will emerge and will become feasible during the course of the investigation. 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.