Jacob Philip Bedrossian, Ph.D. - US grants

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.

Fluids that we interact with on a daily basis (such as water and air) display a remarkable variety of dynamics. At one extreme is chaos and turbulence, wherein the fluid forms complicated fractal-like patterns, the details of which cannot be exactly repeated in experiments due to being very sensitive to small changes in the initial conditions. Turbulence is observed often in fluids, such as in the wakes of aircraft, vehicles, and even obstacles such as buildings and bridges and in the large-scale dynamics of the ocean and atmosphere. At the opposite extreme is the motion of vortex filaments, the most notable examples being tornadoes and the wing-tip vortices commonly shed from wings and helicopter blades. Vortex filaments tend to move in a predictable manner and maintain their structural integrity for extended periods of time. The goal of this project is to develop a better mathematical understanding of these opposite phenomena in fluid mechanics. The accurate prediction of vortex filaments and turbulence is crucial in a variety of scientific and industrial applications, including in the design of air, land, and sea vehicles and in the understanding of complex fluid systems such as the climate and weather. Having a firm mathematical foundation could help other applied researchers obtain deeper insights and lead to better modeling. Further, an understanding of these extremes helps pave the way for a better understanding of the interactions and intermediate regimes, where flows have a mix of structure and chaos. Finally, overcoming the mathematical challenges to these questions will require innovations that will be of interest to the wider mathematical community. The research projects are also integrated with the training of graduate students and younger scientists in mathematics and STEM.

The PI will develop a more mathematically rigorous understanding of two behaviors observed in incompressible fluids at high Reynolds numbers: (1) the coherent motion of vortex filaments; (2) turbulence under "generic", statistically steady forcing. The fundamental questions motivating the PI are: (A) how accurate are the commonly used geometric evolution models such as the Local Induction Approximation (LIA) for the motion of a vortex filament in a fluid with vorticity concentrated on a smooth curve? (B) can we provide a proof for the experimentally observed positive Lyapunov exponents and anomalous dissipation (e.g., as the celebrated Kolmogorov 4/5 law) from the stochastically forced 3D Navier-Stokes in the high Reynolds number limit? These require a number of unexplored mathematical ideas and currently, there exists no clear way to attack them yet. Instead, the PI has identified several independently interesting problems to build necessary mathematical foundations. For (A), the PI and his collaborators will study vortex filaments in quantum fluids governed by the Gross-Pitaevskii equation. The quantum case is expected to be easier than the classical case due to the quantization of vorticity and slightly more amenable linearized operators. First, the PI and collaborators will study the stability of vortex solutions in 2d Gross-Pitaevskii, providing necessary ground for understanding the filament core. Next, the PI and collaborators will show that the LIA accurately describes the motion of nearly-straight, and potentially more general, quantum vortex filaments long enough to make useful predictions. For (B), the PI and his collaborators will: (1) develop novel qualitative theory for Lyapunov exponents in stochastic PDEs inspired by ideas from random dynamical systems; (2) develop better tools for quantitative hypoelliptic regularity and Lyapunov exponent estimation in high dimensional systems; and (3) study quantitative and nonlinear aspects of Lagrangian chaos, that is, how the chaotic dynamics of particles in a fluid can be translated into nonlinear dynamics of the fluid itself.

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.

This project will support participation of graduate students, post-doctoral researchers and early career researchers from the United States of America in one of the workshops "Branching systems, reaction-diffusion equations, and population models" to be held May 2–13, 2022 or "Unifying concepts in partial differential equations (PDEs) with randomness" to be held May 16–27, 2022 at the Centre de Recherches Mathematiques (CRM) in Montreal, Canada. The interplay between mathematical models and probability is rich and profound with applications far-reaching across all fields of science. This interplay can appear in many physical settings in materials science and fluid dynamics, but also in the modeling of various many-agent or many-particle dynamics, such as the self-organized motion of organisms or the spread of an infectious disease through a population. The Centre de Recherches Mathématiques is organizing an intensive thematic program on this topic, "Probabilities and PDEs" during the period January 2022 -- July 2022. There will be four workshops, each with mini courses specifically aimed at graduate students or early-career researchers in related areas, and many long-term visiting scientists as well. This award provides funding to support the participation of US-based graduate students, early career researchers, and under-represented minorities through travel and accommodation during one of the two workshops and accompanying mini-courses. This will provide the participants with a perfect opportunity to interact with each other and with leaders of the field and will help them to understand the major questions and learn the cutting-edge methods of the field.

Probability and PDEs have always been intertwined and modern research continues to intertwine them further. One common aspect concerns how introducing randomness into PDEs, through e.g. random initial data, random environments, random forcing will profoundly affect the long-term behavior, either because it provides a flexible and realistic approach to study physical models or because it is an inevitable feature of models of the underlying systems. Another aspect is how one can obtain deterministic PDEs from underlying random many-agent or many-problems, such as in classical kinetic theory or in many-agent problems such as in mathematical epidemiology. The program will bring together world experts and junior researchers in an intensive and focused program that will help to greatly further the understanding of these phenomena and this award will help train the next generation of US-based scientists in these methods. The thematic semester website is maintained at http://www.crm.umontreal.ca/2022/Probab22/index_e.php the Branching systems, reaction-diffusion equations, and population models workshop at http://www.crm.umontreal.ca/2022/Systemes22/index_e.php and the Unifying concepts in PDEs with randomness workshop at http://www.crm.umontreal.ca/2022/Concepts22/index_e.php.

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.