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An elementary approach to mixing and dissipation enhancement by transport noise

Dejun Luo, Bin Tang, Guohuan Zhao

Abstract

We investigate the mixing properties of solutions to the stochastic transport equation $d u= \circ d W \cdot\nabla u$, where the driving noise $W(t,x)$ is white in time, colored and divergence-free in space. Furthermore, we prove the dissipation enhancement in the presence of a small viscous term. Applying our results, we also derive the mixing properties for a regularized stochastic 2D Euler equation.

An elementary approach to mixing and dissipation enhancement by transport noise

Abstract

We investigate the mixing properties of solutions to the stochastic transport equation , where the driving noise is white in time, colored and divergence-free in space. Furthermore, we prove the dissipation enhancement in the presence of a small viscous term. Applying our results, we also derive the mixing properties for a regularized stochastic 2D Euler equation.
Paper Structure (15 sections, 16 theorems, 185 equations, 1 figure)

This paper contains 15 sections, 16 theorems, 185 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mixing by transport noise
  3. Dissipation enhancement and Capiński's conjecture
  4. Mixing for regularized stochastic 2D Euler equations
  5. Preliminaries
  6. Notions and functional setting
  7. Choice of noise
  8. Basic results for \ref{['STE-Ito']} and a discrete Poincaré type inequality
  9. Mixing for stochastic transport equation
  10. Mixing in the average sense
  11. Mixing under $\mathbb{P}$-almost sure sense
  12. Positive Sobolev norm estimate
  13. Dissipation enhancement and Capiński's conjecture
  14. Mixing for regularized stochastic 2D Euler equations
  15. The proof of Theorem \ref{['thm: Fourier-coefficient-exp-decay']} ($d \geq 3$)

Key Result

Theorem 1.1

Assume $u_0 \in L^2(\mathbb{T}^d)$ and $\theta \in \ell^2(\mathbb{Z}_0^d)$, then the solution $u$ to equation STE-Ito is exponentially mixing in the averaged sense:

Figures (1)

  • Figure 1: The decomposition of $Q_1$ for $l=(2,1)$

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 27 more