Instabilities in Incompressible Fluid Flows

This project investigates how instabilities develop in fluids—whether in the air, oceans, or even within jet engines—by analyzing the Euler equations, which are the fundamental equations that govern fluid motion. The principal investigator (PI) studies new scenarios in which a steady fluid flow can suddenly become unstable due to small disturbances. This includes studying both linear instabilities, which approximately describes the early stages of disruption, and nonlinear instabilities, which govern a precise notion of a fluid flow and can lead to a justification of an eventual breakdown into turbulence. The PI focuses on degenerate cases which can be quite difficult, such as axisymmetric flows in three dimensions—flows that are symmetrical around an axis but may become unstable when slightly disturbed in a non-symmetric way. These cases can be especially complicated near the axis of symmetry and pose major mathematical challenges. Another aspect of the project is concerned with the behavior of irregular flows, such as those arising from vortex sheets, where sharp changes in velocity occur. By improving our theoretical understanding of these instabilities, the project promotes the progress of science, since better models of fluid instability help inform a wide range of applications, from improving the efficiency of transportation and energy systems to enhancing weather prediction. The project investigates the phenomenon of instabilities of solutions to the three-dimensional (3D) incompressible Euler equations, and develops a theory linking linear and nonlinear instability theorems with the local well-posedness and ill-posedness theory of the equations. The project investigates both the case of “local instabilities”, where new functional frameworks are developed to capture multiscale the behaviour, as well as the case of “global instabilies” in the form of neutral limiting modes, which can be treated as perturbations of an eigenvalue problems. As a central example of local instabilities in the 3D case, the PI considers vortex columns, and establishes connections between multiscale instabilities and the upper and lower neutral limiting modes of vortex columns. Another aspect of the project investigates nonlinear instability arising from irregular velocity fields, such as velocity fields arising from vortex sheets. The PI investigates two-dimensional steady states in the form of parallel shear flow, and establishes a rigorous relationship between instabilities of shear flows with a localized inflection point and Kelvin-Helmholtz instabilities of a flat vortex sheet, as well as a theory of nonlinear instability of logarithmic spiral vortex sheets, based on the bicharacteristic approach. 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.