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Random divergence-free drifts and the Onsager-Richardson threshold

Daniel W. Boutros, Camillo De Lellis, Svitlana Mayboroda

Abstract

We prove the absence of anomalous dissipation for passive scalars driven by random divergence-free autonomous vector fields in Hölder classes $C^α(\mathbb{T}^d)$ with $α> \frac{1}{3}$, without any regularity assumptions on the passive scalar except that the initial data is fixed and bounded. The random field has to satisfy very mild conditions in two dimensions, while in higher dimensions it is assumed to have a special geometric structure. The proof relies on dimension-theoretic arguments, as opposed to commutator estimates. A consequence of these results is that anomalous regularization does not occur for this class of random vector fields.

Random divergence-free drifts and the Onsager-Richardson threshold

Abstract

We prove the absence of anomalous dissipation for passive scalars driven by random divergence-free autonomous vector fields in Hölder classes with , without any regularity assumptions on the passive scalar except that the initial data is fixed and bounded. The random field has to satisfy very mild conditions in two dimensions, while in higher dimensions it is assumed to have a special geometric structure. The proof relies on dimension-theoretic arguments, as opposed to commutator estimates. A consequence of these results is that anomalous regularization does not occur for this class of random vector fields.
Paper Structure (9 sections, 7 theorems, 44 equations)

This paper contains 9 sections, 7 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. General theorem and reduction to the Morse-Sard property
  4. Proof of the Morse-Sard property
  5. Remarks
  6. Random vector fields
  7. Adding constant drifts
  8. Yaglom's law
  9. From the renormalization property to the absence of dissipation anomalies

Key Result

Theorem 1.2

Consider a probability measure $\mathbb P$ on the space of continuous divergence-free and average-free autonomous vector fields $v$ on $\mathbb T^2$. Assume that: Then $\mathbb{P}$-almost surely $v$ satisfies the DiPerna-Lions renormalization property (cf. Definition d:renormalization) and thus does not allow dissipation anomalies.

Theorems & Definitions (13)

  • Definition 1.1
  • Theorem 1.2: Two-dimensional case
  • Theorem 1.3: Three-dimensional case
  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['l:probabilistic']}
  • proof : Proof of Lemma \ref{['l:deterministic']}
  • ...and 3 more