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On the Endpoint Regularity in Onsager's Conjecture

Philip Isett

Abstract

Onsager's conjecture states that the conservation of energy may fail for $3D$ incompressible Euler flows with Hölder regularity below $1/3$. This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question with further connections to turbulence theory. In this work, we construct energy non-conserving solutions to the $3D$ incompressible Euler equations with space-time Hölder regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents $[0,1/3)$. Our construction improves the author's previous result towards the endpoint case. To obtain this improvement, we introduce a new method for optimizing the regularity that can be achieved by a convex integration scheme. A crucial point is to avoid power-losses in frequency in the estimates of the iteration. This goal is achieved using localization techniques of \cite{IOnonpd} to modify the convex integration scheme. We also prove results on general solutions at the critical regularity that may not conserve energy. These include a theorem on intermittency stating roughly that energy dissipating solutions cannot have absolute structure functions satisfying the Kolmogorov-Obukhov scaling for any $p > 3$ if their singular supports have space-time Lebesgue measure zero.

On the Endpoint Regularity in Onsager's Conjecture

Abstract

Onsager's conjecture states that the conservation of energy may fail for incompressible Euler flows with Hölder regularity below . This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question with further connections to turbulence theory. In this work, we construct energy non-conserving solutions to the incompressible Euler equations with space-time Hölder regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents . Our construction improves the author's previous result towards the endpoint case. To obtain this improvement, we introduce a new method for optimizing the regularity that can be achieved by a convex integration scheme. A crucial point is to avoid power-losses in frequency in the estimates of the iteration. This goal is achieved using localization techniques of \cite{IOnonpd} to modify the convex integration scheme. We also prove results on general solutions at the critical regularity that may not conserve energy. These include a theorem on intermittency stating roughly that energy dissipating solutions cannot have absolute structure functions satisfying the Kolmogorov-Obukhov scaling for any if their singular supports have space-time Lebesgue measure zero.

Paper Structure

This paper contains 17 sections, 15 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Regularity of kinetic energy at the critical exponent
  3. Singularities of dissipative solutions with critical regularity
  4. Stability of local energy dissipation in a critical class
  5. Notation
  6. The Main Lemma
  7. The Improved Convex Integration Lemma
  8. Modifying the Convex Integration
  9. Localizing the Convex Integration Construction
  10. Estimates for Components of the Construction
  11. The Error Terms
  12. Solving the Symmetric Divergence Equation
  13. The Parametrix Expansion
  14. Iterating the Main Lemma
  15. Heuristics and deriving the optimization problem for the parameters
  16. ...and 2 more sections

Key Result

Theorem 1

There exists $(v,p)$ a weak solution to the incompressible Euler equations that has non-empty, compact support in time on $\mathbb{R} \times \mathbb{T}^3$ and belongs to the class $v \in \bigcap_{\alpha < 1/3} C_{t,x}^\alpha$. Moreover, one may arrange that $v$ also satisfies an estimate of the form for some constants $C$ and $B$ and for all $(t,x) \in \mathbb{R} \times \mathbb{T}^3$ and all $|\De

Theorems & Definitions (30)

  • Theorem 1
  • Theorem 2: Intermittency Theorem
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • proof : Proof of Theorem \ref{['prop:singPos']}
  • proof : Proof of Theorems \ref{['prop:endIntDissMsr']} and \ref{['prop:locOnsSingCrit']}
  • Theorem 6
  • proof
  • Definition 4.1
  • ...and 20 more