CAREER: Inviscid Limits and Stability at High Reynolds Numbers

University Of Maryland, College Park, College Park MD

Investigators

Abstract

In many applications of fluid mechanics, such as those arising in atmosphere and ocean sciences, aerospace engineering, and high-energy or fusion plasma physics, understanding the dynamics of fluids and plasmas where dissipative forces (e.g., friction) are weak is crucial. For example, the stability of the layer of air over a wing, and the dissipation of energy nearby, can have major implications for the aircraft, such as drastically changing the fuel efficiency. The focus of the project is to better understand dissipation and stability of equilibrium configurations and related problems in this regime using mathematical analysis. The project helps lay the foundation for a wider mathematical theory on mixing, dissipation, and stability in fluid mechanics. As it requires both pure and applied mathematical innovations, it also presents an excellent opportunity to train new, versatile researchers who are fluent in both scientific applications and sophisticated mathematical analysis. Graduate students are included in the work of the project. The project aims to further elucidate basic questions of nonlinear stability, direct cascades, and the dissipation of enstrophy or energy at small scales in fluids and to expand the rigorous mathematical theory for understanding these phenomena. The investigator and his collaborators focus on fundamental questions that will have broad impact due to intrinsic interest, as will the new tools developed to solve them. Three general areas are studied: (A) analysis tools for understanding enhanced dissipation and transient unmixing; (B) linear and nonlinear stochastically forced problems in order to understand statistically stationary direct cascades in mathematically accessible settings; (C) estimating the subcritical transition thresholds and understanding instabilities for laminar flows, such as pipe flow, in infinite and finite regularity. Finally, natural extensions to hypoelliptic problems, such as collisionless limits in kinetic theory, may also be considered. The work is primarily mathematical analysis; however, computer experiments may be performed in order to provide preliminary insights and to provide accessible training opportunities for undergraduates in applied mathematics. Graduate students are included in the research activities.