Smooth vs. Physical Solutions of the Navier-Stokes Equation

J. Glimm, J. Petrillo, M. C. Lee

Published 2022-12-30Version 1

Smooth solutions for the Navier-Stokes equations of the class considered by Leray and Hopf exist. They describe the mean state of a turbulent flow but do not describe a turbulent state itself. They fail the admissibility criterion of a maximum rate of entropy production. They have zero entropy, zero turbulent fluctuations and are laminar. For the same general initial conditions, the physically admissible solution of the turbulent flow exists as a Young measure, with a stochastic formulation. An understanding of the proofs is offered by renormalized perturbation series, which is constructed as a modification of the renormalized perturbation series of quantum field theory. This expansion gives a statistical description of the lowest energy fully turbulent state, usually referred to as fully developed turbulence. This state is statisticly stationary and translation invariant. We also construct fully turbulent nonstationary and nontranslation invariant turbulent states which decay into the lowest energy fully developed turbulent state. Topologically, the moments are interpreted in terms of energy and vorticity surfaces in $\mathbb{R}^3$ in the form of spheres and possibly knotted tori of arbitrary genus. Considerations of phase transitions within turbulent flow lead to a conjectured unified picture of phase transitions within fluids, Yang-Mills fields with quarks and quantum general relativity.