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Stabilization by transport noise and enhanced dissipation in the Kraichnan model

Benjamin Gess, Ivan Yaroslavtsev

TL;DR

The paper establishes that transport noise can stabilize linear SPDEs with potentially unstable deterministic parts on compact manifolds and can yield exponential mixing with a rate that can be made arbitrarily large by increasing the noise amplitude. It achieves this via a two-pronged approach: (i) a Lyapunov/ergodicity analysis of the Lagrangian two-point motion under Hölder-coefficient noise, including a constructed Lyapunov function and Harris-type ergodicity; and (ii) a transfer to the Eulerian scalar via stochastic mixing techniques, yielding exponential decay in $H^{-s}$ and thus strong mixing for the passive scalar. The Kraichnan model is shown to satisfy the required (A)–(C) conditions (even in irregular regimes), with explicit discussion of isotropy on $\mathbb{T}^d$ and a four-mode stabilization example in 2D; a Hörmander-based extension broadens applicability to finite-mode or smooth-coefficient cases. The results provide a rigorous mechanism for enhanced dissipation in turbulent transport and give practical criteria for designing noise to achieve fast convergence to equilibrium. The work thus connects stochastic transport, Lyapunov spectra, and hypoelliptic regularity to yield quantitative stabilization and mixing in turbulent-like transport problems.

Abstract

Stabilization and sufficient conditions for mixing by stochastic transport are shown. More precisely, given a second order linear operator with possibly unstable eigenvalues on a smooth compact Riemannian manifold, it is shown that the inclusion of transport noise can imply global asymptotic stability. Moreover, it is shown that an arbitrary large exponential rate of convergence can be reached, implying enhanced dissipation. The sufficient conditions are shown to be satisfied by the so-called Kraichnan model for stochastic transport of passive scalars in turbulent fluids. In addition, an example is given showing that it can be sufficient to force four modes in order to induce stabilization.

Stabilization by transport noise and enhanced dissipation in the Kraichnan model

TL;DR

The paper establishes that transport noise can stabilize linear SPDEs with potentially unstable deterministic parts on compact manifolds and can yield exponential mixing with a rate that can be made arbitrarily large by increasing the noise amplitude. It achieves this via a two-pronged approach: (i) a Lyapunov/ergodicity analysis of the Lagrangian two-point motion under Hölder-coefficient noise, including a constructed Lyapunov function and Harris-type ergodicity; and (ii) a transfer to the Eulerian scalar via stochastic mixing techniques, yielding exponential decay in and thus strong mixing for the passive scalar. The Kraichnan model is shown to satisfy the required (A)–(C) conditions (even in irregular regimes), with explicit discussion of isotropy on and a four-mode stabilization example in 2D; a Hörmander-based extension broadens applicability to finite-mode or smooth-coefficient cases. The results provide a rigorous mechanism for enhanced dissipation in turbulent transport and give practical criteria for designing noise to achieve fast convergence to equilibrium. The work thus connects stochastic transport, Lyapunov spectra, and hypoelliptic regularity to yield quantitative stabilization and mixing in turbulent-like transport problems.

Abstract

Stabilization and sufficient conditions for mixing by stochastic transport are shown. More precisely, given a second order linear operator with possibly unstable eigenvalues on a smooth compact Riemannian manifold, it is shown that the inclusion of transport noise can imply global asymptotic stability. Moreover, it is shown that an arbitrary large exponential rate of convergence can be reached, implying enhanced dissipation. The sufficient conditions are shown to be satisfied by the so-called Kraichnan model for stochastic transport of passive scalars in turbulent fluids. In addition, an example is given showing that it can be sufficient to force four modes in order to induce stabilization.

Paper Structure

This paper contains 13 sections, 34 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. Further comments on the literature
  3. Preliminaries
  4. Stochastic Mixing
  5. Lyapunov exponents and moment Lyapunov functions for SDEs with Hölder coefficients
  6. Construction of a Lyapunov function for the two-point motion
  7. Ergodicity of the two-point motion
  8. Proof of Theorem \ref{['thm:H-sconvergensofHsinvartext']}
  9. Stablilization by transport noise
  10. Regular Kraichnan model on $\mathbb T^d$
  11. Hörmander conditions
  12. Acknowledgements
  13. Heat kernel regularity and positivity on manifolds

Key Result

Theorem 1.1

Let $(\sigma_k)_{k\geq 1}$ satisfy the conditions (A)--(C) and let $T$ be a second-order strictly elliptic operator defined by eq:INTROTda. Then there exist $A_0>0$ and $\gamma_0>0$ such that for any $\gamma\in(0, \gamma_0]$ and $A\geq A_0$ there is $D^{A,\gamma}:\Omega\to \mathbb R_+$ with $\sup_{

Theorems & Definitions (76)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • ...and 66 more