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Application of Onsager's Variational Principle to the Navier-Stokes Equations

Hamid A. Said

TL;DR

The paper addresses when Leray–Hopf solutions to the 3D incompressible Navier–Stokes equations satisfy global and local energy equalities by leveraging an Onsager-inspired variational principle. It constructs a sequence of minimization problems for functionals $\mathcal{K}^n(v)$ under an enstrophy-ball constraint, coupled with coarse-grained velocity fields and Reynolds stresses. The main result shows that if the weak limit $v_*$ of the minimizers is nonzero, then the Leray–Hopf solution obeys the local energy equality (and hence the global energy equality), with the Reynolds stress vanishing in the limit; physically, the Poisson relations act as a constitutive law for turbulent fluxes. The analysis illuminates a link between energy conservation and a nontrivial variational limit, offering a turbulence-informed perspective on energy dissipation mechanisms and suggesting extensions to energy-prescribed ensembles and higher-order variational approaches.

Abstract

In this note we propose a basic $L^2$-based approach for studying the global and local energy equalities of the incompressible 3D Navier-Stokes equations in the standard energy class on $\mathbb{T}^3 \times (0,T]$. Motivated by L. Onsager's principle of least dissipation of energy (1931), we give a new sufficient condition for the energy equalities in terms of the limit of a sequence of minimizers. In particular, we show that the equalities are attained if this limit is non-vanishing. We observe that the indeterminacy in the vanishing case is reminiscent of turbulence-driven energy transfer dominating other transport processes.

Application of Onsager's Variational Principle to the Navier-Stokes Equations

TL;DR

The paper addresses when Leray–Hopf solutions to the 3D incompressible Navier–Stokes equations satisfy global and local energy equalities by leveraging an Onsager-inspired variational principle. It constructs a sequence of minimization problems for functionals under an enstrophy-ball constraint, coupled with coarse-grained velocity fields and Reynolds stresses. The main result shows that if the weak limit of the minimizers is nonzero, then the Leray–Hopf solution obeys the local energy equality (and hence the global energy equality), with the Reynolds stress vanishing in the limit; physically, the Poisson relations act as a constitutive law for turbulent fluxes. The analysis illuminates a link between energy conservation and a nontrivial variational limit, offering a turbulence-informed perspective on energy dissipation mechanisms and suggesting extensions to energy-prescribed ensembles and higher-order variational approaches.

Abstract

In this note we propose a basic -based approach for studying the global and local energy equalities of the incompressible 3D Navier-Stokes equations in the standard energy class on . Motivated by L. Onsager's principle of least dissipation of energy (1931), we give a new sufficient condition for the energy equalities in terms of the limit of a sequence of minimizers. In particular, we show that the equalities are attained if this limit is non-vanishing. We observe that the indeterminacy in the vanishing case is reminiscent of turbulence-driven energy transfer dominating other transport processes.

Paper Structure

This paper contains 11 sections, 4 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Onsager's principle and statement of main result
  3. Method of proof
  4. Variational analysis
  5. Mollification
  6. Minimization problem
  7. Passage to the limit
  8. Discussion
  9. Principle of least dissipation of energy
  10. Overview of theory
  11. Proof of Proposition \ref{['prop3.1']}

Key Result

Theorem 1.1

Define $J^n(x,t)$ as in equation sigma, and let $v^n_*$ be the unique solution to the problem min-prob-intro. Then there exists a subsequence (not relabeled) and $v_* \in L^2(0,T;V)$ such that $v^n_* \rightharpoonup v_*$ in $L^2(0,T;V)$. If $v_* \neq 0$, then any Leray-Hopf solution $u \in L^\infty(

Theorems & Definitions (13)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 2.1
  • Remark 4
  • Proposition 2.1
  • Remark 5
  • Corollary 2.1
  • Proposition 3.1
  • ...and 3 more