Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Intermittency and Dissipation Regularity in Turbulence

Luigi De Rosa, Theodore D. Drivas, Marco Inversi, Philip Isett

TL;DR

The paper develops a geometric-analytic framework to analyze the Duchon–Robert energy-dissipation distribution $D$ for weak solutions of the incompressible Euler equations, establishing that velocity regularity in Besov spaces $B^\sigma_{p, ho}$ (with $σ o(0,1)$) yields negative Besov regularity for $D$, specifically $D\, extrm{in}\, B^{ rac{2σ}{1-σ}-1}_{p/3,\infty}$ locally. When $D$ is a Radon measure, the authors derive sharp dimensional constraints: $|D|$ is absolutely continuous with respect to a suitable Hausdorff measure $\, rak{H}^gamma$ provided $ rac{2σ}{1-σ}>1- rac{p-3}{p}(d+1-gamma)$, and provide pointwise bounds on $D$ on balls under positivity. The results yield intermittency-type implications, linking fractal dissipation geometries to lower Besov regularity below the Kolmogorov $1/3$ exponent and recovering Onsager-type singularity thresholds; the framework also yields local Navier–Stokes analogues, a refined energy decomposition, and connections to the Four-Fifths law. Collectively, the work advances quantitative control of dissipation concentrations and offers a principled route to compare theoretical predictions with experimental and numerical turbulence data. The findings have potential impact on the mathematical understanding of turbulence intermittency, regularity thresholds for conservation laws, and the design of convex-integrations constructions that capture dissipation patterns.

Abstract

We lay down a geometric-analytic framework to capture properties of energy dissipation within weak solutions to the incompressible Euler equations. For solutions with spatial Besov regularity, it is proved that the Duchon-Robert distribution has improved regularity in a negative Besov space and, in the case it is a Radon measure, it is absolutely continuous with respect to a suitable Hausdorff measure. This imposes quantitative constraints on the dimension of the, possibly fractal, dissipative set and the admissible structure functions exponents, relating to the phenomenon of "intermittency" in turbulence. As a by-product of the approach, we also recover many known "Onsager singularity" type results.

Intermittency and Dissipation Regularity in Turbulence

TL;DR

The paper develops a geometric-analytic framework to analyze the Duchon–Robert energy-dissipation distribution for weak solutions of the incompressible Euler equations, establishing that velocity regularity in Besov spaces (with ) yields negative Besov regularity for , specifically locally. When is a Radon measure, the authors derive sharp dimensional constraints: is absolutely continuous with respect to a suitable Hausdorff measure provided , and provide pointwise bounds on on balls under positivity. The results yield intermittency-type implications, linking fractal dissipation geometries to lower Besov regularity below the Kolmogorov exponent and recovering Onsager-type singularity thresholds; the framework also yields local Navier–Stokes analogues, a refined energy decomposition, and connections to the Four-Fifths law. Collectively, the work advances quantitative control of dissipation concentrations and offers a principled route to compare theoretical predictions with experimental and numerical turbulence data. The findings have potential impact on the mathematical understanding of turbulence intermittency, regularity thresholds for conservation laws, and the design of convex-integrations constructions that capture dissipation patterns.

Abstract

We lay down a geometric-analytic framework to capture properties of energy dissipation within weak solutions to the incompressible Euler equations. For solutions with spatial Besov regularity, it is proved that the Duchon-Robert distribution has improved regularity in a negative Besov space and, in the case it is a Radon measure, it is absolutely continuous with respect to a suitable Hausdorff measure. This imposes quantitative constraints on the dimension of the, possibly fractal, dissipative set and the admissible structure functions exponents, relating to the phenomenon of "intermittency" in turbulence. As a by-product of the approach, we also recover many known "Onsager singularity" type results.

Paper Structure

This paper contains 16 sections, 21 theorems, 119 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tools
  3. Besov spaces and mollification estimates
  4. Inequalities
  5. Radon measures
  6. Fractal dimensions
  7. Proof of the main results
  8. Corollaries
  9. The Euler equations
  10. The Navier--Stokes equations
  11. Discussion
  12. Intermittency in turbulence
  13. Comparison with experimental data
  14. Sharpness of the results & convex integration
  15. General open sets
  16. ...and 1 more sections

Key Result

Theorem 1.1

Assume that $u\in L^p_t B^\sigma_{p,\infty}$ is a weak solution to E for some $p\in [3,\infty]$ and $\sigma \in \left(0,1\right)$, with Duchon--Robert distribution $D$. Then $D\in B^{\frac{2\sigma}{1-\sigma}-1}_{\frac{p}{3},\infty}$ locally in space-time. If in addition $D$ is a real-valued Radon me

Figures (1)

  • Figure 1: Structure function exponents for $p\in[3,6]$. Blue dots are absolute structure function exponents measured from the JHU turbulence database. Red triangles are transverse exponents reported in iyer2020scaling. Dashed grey line corresponds to the Kolmogorov prediction of $\frac{p}{3}$. Solid grey line corresponds to our bound $\zeta_p^*$ with $\gamma = 3.85$ inferred from iyer2020scaling.

Theorems & Definitions (41)

  • Theorem 1.1: Dissipation regularity
  • Corollary 1.2: Intermittency
  • Proposition 1.3: Modified energy identity
  • Proposition 1.4: Mollification rates
  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3: Shinbrot S74*Lemma 4.2
  • Lemma 2.4
  • proof
  • Proposition 3.1
  • ...and 31 more