Onsager's conjecture and the energy of singular flows

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

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.