CAREER: New Mechanisms for Stability, Regularity and Long Time Dynamics of Partial Differential Equations

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

The project focuses on mathematical analysis of nonlinear partial differential equations that are inspired by fluid dynamics and wave propagation. Understanding the dynamics of incompressible fluids, such as water and air at subsonic speed, is important for a variety of applications, ranging from the design of airplanes, boats and motors, to the study of oceans and the atmosphere. Coherent structures, such as vortices (eddies) and shear flows, are prominent features in fluid dynamics. The formation, stability, and evolution of coherent structures are critical fluid phenomena to understand in order to reduce drag, oscillation, and instability in scientific and engineering applications. The PI will develop new, innovative mathematical methods to analyze the dynamic properties of physically important coherent structures, which can resolve theoretical difficulties as well as provide powerful mathematical tools for practical applications. The PI will also study the interaction of radiation and particles in the context of wave maps, which have a deep connection to the classical field theories from mathematical physics. The proposed projects provide an ideal training ground for junior researchers in applying cutting edge mathematical analysis to study sophisticated physical phenomena in fluid dynamics and wave propagation. Graduate students will be actively involved in these research projects. The PI and collaborators aim to develop new methods that can effectively combine precise spectral and Fourier analysis in the context of nonlinear asymptotic stability problems of fluid dynamics. In many physical problems, the analysis of large coherent structures requires precise spectral analysis for the linearized flow, while Fourier analysis has proved indispensable in uncovering delicate nonlinear interactions. Thus, the techniques developed in the project may have a wider range of applications in other technically challenging perturbative problems. The PI will also study simpler models of fluid equations in an effort to understand the interaction and balance between vorticity stretching and vorticity transportation effects, which play a fundamental role in the regularity theory of three-dimensional Euler equations. For the wave maps equation, the main goal is to extend the "channel of energy" argument for outgoing waves to this technically challenging model to study the decoupling of radiation from solitons in a non-perturbative regime. These projects provide a wide range of problems for graduate students, who will learn to use tools from spectral analysis, Fourier analysis, dynamical systems, and numerical simulation, in the study of physically significant 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.