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Dissipative continuous Euler flows

Camillo De Lellis, László Székelyhidi

TL;DR

This work constructs continuous, periodic weak solutions to the 3D incompressible Euler equations that dissipate kinetic energy, addressing Onsager-type questions through a convex integration framework. The authors implement an Euler–Reynolds scheme, iteratively reducing the Reynolds stress by adding high-frequency Beltrami-flow perturbations and a corrective term, while carefully controlling energy via Schauder estimates. A key geometric lemma enables representing any near-identity stress as averages of Beltrami-blocks, ensuring the flexibility needed to realize prescribed energy dissipation. The result demonstrates energy dissipation in the continuous category and highlights the non-uniqueness and intricate structure of Euler flows, with broader implications for turbulence-like constructions and $h$-principle phenomena in fluid dynamics.

Abstract

We show the existence of continuous periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy.

Dissipative continuous Euler flows

TL;DR

This work constructs continuous, periodic weak solutions to the 3D incompressible Euler equations that dissipate kinetic energy, addressing Onsager-type questions through a convex integration framework. The authors implement an Euler–Reynolds scheme, iteratively reducing the Reynolds stress by adding high-frequency Beltrami-flow perturbations and a corrective term, while carefully controlling energy via Schauder estimates. A key geometric lemma enables representing any near-identity stress as averages of Beltrami-blocks, ensuring the flexibility needed to realize prescribed energy dissipation. The result demonstrates energy dissipation in the continuous category and highlights the non-uniqueness and intricate structure of Euler flows, with broader implications for turbulence-like constructions and -principle phenomena in fluid dynamics.

Abstract

We show the existence of continuous periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy.

Paper Structure

This paper contains 20 sections, 17 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Onsager's Conjecture
  3. $h$-principle
  4. Comments on the proof
  5. Acknowledgements
  6. Setup and plan of the paper
  7. Construction of $v_1$
  8. Geometric preliminaries
  9. Beltrami flows
  10. The geometric lemma
  11. The maps $v_1$, $\mathring{R}_1$ and $p_1$
  12. The perturbation $w_o$
  13. The constants $\eta$ and $M$
  14. The correction $w_c$
  15. The pressure $p_1$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Assume $e: [0, 1]\to \mathbb{R}$ is a positive smooth function. Then there is a continuous vector field $v: \mathbb{T}^3 \times [0, 1]\to \mathbb{R}^3$ and a continuous scalar field $p:\mathbb{T}^3\times [0,1]\to \mathbb{R}$ which solve the incompressible Euler equations in the sense of distributions and such that

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1
  • Definition 2.1
  • Proposition 2.2
  • proof : Proof of Theorem \ref{['t:main']}
  • Proposition 3.1: Beltrami flows
  • proof
  • Lemma 3.2: Geometric Lemma
  • Remark 2
  • Proposition 3.3
  • ...and 28 more