On vorticity and expansion-rate of fluid flows, conditional law duality and their representations

TL;DR

This paper develops a probabilistic framework for modeling viscous compressible fluid flows using vorticity and expansion-rate variables, introducing duality of conditional laws and a Feynman-Kac formula for elliptic operators.

Contribution

It introduces a novel approximation model for compressible flows with slowly varying density and establishes probabilistic tools like duality of conditional laws and a Feynman-Kac formula for elliptic operators.

Findings

Formulation of motion equations in terms of vorticity and expansion-rate.

Development of a probabilistic duality of conditional laws.

Formulation of a random vortex method for viscous compressible flows.

Abstract

By using a formulation of motion equations for a viscous (compressible) fluid flow in terms of the vorticity and the rate of expansion as the main fluid dynamical variables, an approximation model is established for compressible flows with slowly varied (over the space) fluid density. The probabilistic tools and the main ingredient such as the duality of conditional laws and the forward type Feynman-Kac formula are established for elliptic operators of second order, in order to formulate the corresponding random vortex method for a class of viscous compressible fluid flows, based on their approximation motion equations.