Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Principle of Maximum Entropy for the Navier-Stokes Equations

Gui-Qiang G. Chen, James Glimm, Hamid Said

Abstract

A principle of maximum entropy is proposed in the context of viscous incompressible flow in Eulerian coordinates. The relative entropy functional, defined over the space of $L^2$ divergence-free velocity fields, is maximized relative to alternate measures supported over the energy--enstrophy surface. Since thermodynamic equilibrium distributions are characterized by maximum entropy, connections are drawn with stationary statistical solutions of the incompressible Navier-Stokes equations. Special emphasis is on the correspondence with the final statistics described by Kolmogorov's theory of fully developed turbulence.

A Principle of Maximum Entropy for the Navier-Stokes Equations

Abstract

A principle of maximum entropy is proposed in the context of viscous incompressible flow in Eulerian coordinates. The relative entropy functional, defined over the space of divergence-free velocity fields, is maximized relative to alternate measures supported over the energy--enstrophy surface. Since thermodynamic equilibrium distributions are characterized by maximum entropy, connections are drawn with stationary statistical solutions of the incompressible Navier-Stokes equations. Special emphasis is on the correspondence with the final statistics described by Kolmogorov's theory of fully developed turbulence.
Paper Structure (7 sections, 2 theorems, 40 equations)

This paper contains 7 sections, 2 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Motivation for entropy maximization
  3. Principle of maximum entropy: classical formulation
  4. Entropy Maximization and the Physical Measure
  5. Principle of maximum entropy: formulation for the Navier-Stokes equations
  6. Main theorem for the principle of maximum entropy
  7. Connections and Remarks

Key Result

Lemma 2.1

Fix $t \in [0,T]$ and $e(t) \geq 0$. Then the following statements hold :

Theorems & Definitions (2)

  • Lemma 2.1
  • Theorem 2.2