The investigators aim to advance our understanding of the phenomenon of turbulence, as well as to offer new insights into the problem of possible singularity formations in 3D viscous incompressible fluids described by the 3D Navier-Stokes equations (NSE). The project extends their recent results regarding existence and locality of energy cascades, and anomalous dissipation in physical scales of the flow. The main goals are, on one hand, to present Onsager-critical Morrey-type conditions preventing anomalous dissipation in 3D incompressible flows and to discover a set of universal scaling laws -- reminiscent of the fundamental Kolmogorov K41 laws -- that naturally transpire from their theory of turbulent cascades in physical scales, and on the other hand, to present mathematical evidence of a geometric scenario closing the scaling gap in the regularity problem, that is, of the criticality of the 3D NSE problem for large data. The key idea is to use a new ensemble-averaging process capable of detecting significant sign fluctuations of the physical density of interest at the given scale. This approach is highly local in space and is applicable to flows that do not satisfy homogeneity and isotropy assumptions. In addition, it provides a framework for the study of geometrically coherent structures in physical space.
In 2000, the Clay Mathematical Institute identified seven "Millennium Problems" in mathematics, solutions to which would have a broad impact on society in the 21st century. One of these problems is the task of providing a rigorous mathematical foundation (namely, addressing the possibilities of singularity formations) for the Navier-Stokes equations that describe the motion of fluids. Existence of singularities is a fundamental issue for mathematical modeling of any real-life phenomenon. In the case of Navier-Stokes equations this regularity problem is believed to be linked with the notion of turbulence. Despite successes in empirical modeling and computational techniques used to study turbulence, this phenomenon remains one of the unresolved fundamental problems of modern physics. Any measurable progress in understanding turbulence can have a broad impact in such areas as weather and climate modeling and engineering applications. On one hand, suppressing turbulence is essential in designing and engineering more efficient vehicles, wind turbines, pipeline systems. On the other hand, enhancing turbulent mixing is desired in a variety of nano-scale engineering designs, as well as in biomedical engineering -- one example being more efficient drug delivery systems. The project helps advance understanding of how to better predict and detect turbulent behavior, as well as possibly control it. The investigators engage graduate and undergraduate students in the project and present lectures on turbulence to local high school students.
