Intermittent Solutions of the Navier-Stokes Equations: From Onsager's Conjecture to Turbulence

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This project is to study some fundamental open questions concerning the Navier-Stokes equations of fluid motion. These equations, widely used by physicists and engineers for real-life applications, were introduced almost two centuries ago to describe the motion of viscous fluids, such as air or water. However, some essential questions, such as the existence and uniqueness of classical solutions, are still not answered. Therefore, a notion of weak solutions, whose existence is known, has been extensively used by mathematicians. The PI will demonstrate a limitation of this notion by constructing weak solutions with various unphysical properties. On the other hand, the developed technique will allow the PI to study physical solutions that are expected to describe turbulence. Turbulence, often referred to as the last unsolved problem in classical physics, is a fundamental and ubiquitous hydrodynamical and aerodynamical phenomenon occurring in nature, in engineering, and in environmental applications. This research will involve graduate students and postdocs.Kolmogorov's theory of turbulence is based on the assumption that eddies are densely packed at each length scale, which has been invalidated in numerous experiments and numerical simulations. It turns out that the eddies are packed somewhat loosely, a phenomenon called intermittency. In this project the PI will study how the intermittency can be defined mathematically and measured experimentally; how the intermittency affects regularity properties of weak solutions to the 3D NSE and their ability to satisfy the energy equality; how one can derive rigorous bounds on intermittency and justify scaling turbulence laws for solutions of the Navier-Stokes equations (NSE). For instance, the PI will analyze how the intermittency affects the ability of solutions to anomalously loose or gain energy. As a result, intermittent wild solutions with various anomalous properties will be constructed.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.