# Is the kinetic equation for turbulent gas-particle flows ill-posed?

## TL;DR

This paper challenges previous claims that the kinetic equation for turbulent gas-particle flows is ill-posed, demonstrating that proper coupling and diffusion considerations show it is well-posed and clarifying the limitations of the GLM PDF approach.

## Contribution

It corrects the misconception that the kinetic equation is ill-posed by highlighting the importance of phase space coupling and diffusion tensor properties, and compares it with the GLM PDF approach.

## Key findings

- The kinetic equation is well-posed when phase space coupling is considered.
- Extra positive diffusion outweighs negative diffusion in the model.
- Limitations of the GLM PDF approach are identified and discussed.

## Abstract

This paper is about well-posedness and realizability of the kinetic equation for gas-particle flows and its relationship to the Generalized Langevin Model (GLM) PDF equation. Previous analyses claim that this kinetic equation is ill-posed, that in particular it has the properties of a backward heat equation and as a consequence, its solutions will in the course of time exhibit finite-time singularities. We show that the analysis leading to this conclusion is fundamentally incorrect because it ignores the coupling between the phase space variables in the kinetic equation and the time and particle inertia dependence of the phase space diffusion tensor. This contributes an extra $+ve$ diffusion that always outweighs the contribution from the$-ve$ diffusion associated with the dispersion along one of the principal axes of the phase space diffusion tensor. This is confirmed by a numerical evaluation of analytic solutions of these $+ve$ and $-ve$ contributions to the particle diffusion coefficient along this principal axis. We also examine other erroneous claims and assumptions made in previous studies that demonstrate the apparent superiority of the GLM PDF approach over the kinetic approach. In so doing we have drawn attention to the limitations of the GLM approach which these studies have ignored or not properly considered, to give a more balanced appraisal of the benefits of both PDF approaches.