Passive Advection and the Degenerate Elliptic Operators $M_n$

TL;DR

This paper establishes bounds for the heat kernels and Green's functions of degenerate elliptic operators in the Kraichnan model, advancing understanding of stationary states at zero diffusivity.

Contribution

It provides new estimates for the operators $M_n$, crucial for analyzing the stationary states in turbulent passive advection models.

Findings

Upper bounds for heat kernels of $M_n$

Bounds for Green's functions of $M_n$

Insights into zero diffusivity stationary states

Abstract

We prove estimates for the stationary state $n$-point functions at zero molecular diffusivity in the Kraichnan model. This is done by proving upper bounds for the heat kernels and Green's functions of the degenerate elliptic operators $M_n$ that occur in the Hopf equations for the $n$-point functions.