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Stability and invariant measure asymptotics in a model for heavy particles in rough turbulent flows

David P. Herzog, Hung D. Nguyen

Abstract

We study a system of Skorokhod stochastic differential equations (SDEs) modeling the pairwise dispersion (in spatial dimension $d=2$) of heavy particles transported by a rough self-similar, turbulent flow with Hölder exponent $h\in (0,1)$. Under the assumption that $h>0$ is sufficiently small, we use Lyapunov methods and control theory to show that the Markovian system is nonexplosive and has a unique, exponentially attractive invariant probability measure. Furthermore, our Lyapunov construction is radially sharp and gives partial confirmation on a predicted asymptotic behavior with respect to the Hölder exponent $h$ of the invariant probability measure. A physical interpretation of the asymptotics is that intermittent clustering is weakened when the carrier flow is sufficiently rough.

Stability and invariant measure asymptotics in a model for heavy particles in rough turbulent flows

Abstract

We study a system of Skorokhod stochastic differential equations (SDEs) modeling the pairwise dispersion (in spatial dimension ) of heavy particles transported by a rough self-similar, turbulent flow with Hölder exponent . Under the assumption that is sufficiently small, we use Lyapunov methods and control theory to show that the Markovian system is nonexplosive and has a unique, exponentially attractive invariant probability measure. Furthermore, our Lyapunov construction is radially sharp and gives partial confirmation on a predicted asymptotic behavior with respect to the Hölder exponent of the invariant probability measure. A physical interpretation of the asymptotics is that intermittent clustering is weakened when the carrier flow is sufficiently rough.

Paper Structure

This paper contains 34 sections, 21 theorems, 264 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The equations
  3. History and results
  4. Remarks on reflective SDEs
  5. Organization of paper
  6. Notation, Mathematical Setting and Main Results
  7. Heuristics
  8. Minorization
  9. Lyapunov definition
  10. Differences in the reflected setting
  11. Change of coordinates and decomposition of the phase space
  12. Dynamics along a fixed large ray in the $(u,v)$-plane
  13. Dynamics at large radius near the negative $u$-axis
  14. Dynamics at bounded radius and small $z$
  15. The construction in $\mathcal{R}_\text{O}$
  16. ...and 19 more sections

Key Result

Proposition 2.1

There exists a constant $0<c_*< 1$ sufficiently small such that for all $\kappa_1> 0$ and $\kappa_2>0$ satisfying the locally defined solution $(\mathbf{x}_t, k_t)$ is nonexplosive; that is, for all initial conditions $\mathbf{x}=(x, y, z) \in \mathcal{O}$ we have

Figures (2)

  • Figure 1: Large-time average $\frac{1}{t}\int_0^tU_s\text{d} s$ with initial condition $U_0=V_0=0$ and noise parameters $\kappa_1=1$, $\kappa_2=1+2h$.
  • Figure 2: The outer subregions $\mathcal{R}_i$, $i=0,1,2,3$ in $(u,v)-$plane.

Theorems & Definitions (54)

  • Proposition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • Proposition 2.7: Minorization
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 44 more