A scale-by-scale description of the energy transfer is required for understanding many physical and mathematical aspects of fluid motion. These include derivation of turbulence laws from the governing Navier-Stokes system of equations, or Euler equations in the inertial subrange of scales; regularity problems and problems of long time asymptotic behavior. The conventional analytical methods traditionally used to approach the problems of turbulence are limited to subcritical regularity regimes not consistent with empirical observations. The long standing Onsager's conjecture states however that in spite of these limitations the Euler equations allow for particular singular solutions sharing common scaling properties of a homogeneous isotropic turbulence. The principal goal of the project is award developing analytical tools to study such solutions. We will obtain new frequency local estimates on the energy flux through dyadic scales; examine the Onsager-critical smoothness of solutions in the range of Besov spaces; use local energy estimates to study solutions in a variety of intermittency regimes including the classical Kolmogorov and fully intermittent regimes. We will revisit some of the fundamental regularity criteria for Leray-Hopf solutions of the Navier-Stokes equations to improve the known sufficient conditions for the energy equality.
Turbulence is an intricate physical process of complex motion of fluid elements constantly stirred by a mixing force. Turbulent wakes behind cars, planes, ships, etc. are a common everyday phenomenon. A better understanding of this phenomenon leads to finding more effective designs and energy saving solutions. Due to its complexity a turbulent motion of fluid is usually studied by statistical methods involving averaging of observed quantities over a large number of experimental data or long periods of time. This research is directed to finding particular individual realizations of turbulent motion directly from the governing equations. The project will help to gain a new prospective on the so-called Onsager's conjecture, stated in 1949, which questions the ability of the classical equations of fluid motion to describe turbulence.
Turbulent motion of inviscid or almost inviscid fluid, like water or antifreeze, is a complicated process of constant mixing of fluid particles which to a naked eye defies any order or law. At the top of the process stands a source of energy, for example, heat from the sun or stirring force of a paddle, which gets transferred gradually from large scale structures, such as swirling patches, into smaller structures, like rapidly rotating eddies. This energy transfer, from a practical point of view, is the most important feature of turbulence since if uncontrolled it results in loss of energy causing so called anomalous dissipation. This dissipation is responsible for many inefficiencies, such as creation of extra drag force behind a vehicle or a wing of an airplane. This project focused on fluids in their outmost irregular regimes, as occurring in turbulence, idealizing some features or geometric configurations of irregularities to make them more amenable to analysis. The project created mathematical foundation and analytical tools suitable for studying these fluids and the process of anomalous dissipation they produce giving it precise mathematical meaning. The outcome of the methodology adapted in the project was development of mathematical completeness, or regularity, theory for basic equations of fluid motion in both viscous and inviscid cases. Along this direction the principal investigator found new regularity criteria which revealed close connections between the modern rogorous mathematical theory of turbulence and its statistical or phenomenological counterpart known since yearly 40's. The project resulted in creation various educational materials such as video records, lecture notes, and survey papers written on the introductory level suitable for a wider audience. The development of human resources under the project involved training of graduate, undergraduate, and high school students, focused on developing analytical skills and general mathematical education. Organization of regional and national conferences under this project helped promote public awareness of the problems raised in the project while facilitated participation of minority groups in the process.
