Effective Equations for a Compressible Liquid-Vapor Flow Model with Highly Oscillating Initial Density

TL;DR

This paper derives and justifies an effective model for highly oscillating compressible liquid-vapor flows, capturing the macroscopic behavior and mixing dynamics through Young measures and a kinetic equation.

Contribution

It introduces a new effective model combining deterministic hydrodynamics and a kinetic equation for Young measures, applicable to highly oscillatory initial data in liquid-vapor flows.

Findings

Proves convergence of solutions to the effective model.

Characterizes Young measures via cumulative distribution functions.

Rewrites the kinetic equation into a more accessible form.

Abstract

We derive and justify a new effective model for a compressible viscous liquid-vapor flow on a spray-like scale, i.e., for settings with a large number of phase boundaries. As a model on the detailed scale, we start from a parabolic relaxation of the Navier-Stokes-Korteweg system. We consider a sequence of initial data where the sequence of initial densities is assumed to be highly oscillating mimicking the high number of phase boundaries initially. Then, we consider a sequence of finite energy weak solutions corresponding to the sequence of initial data. Anticipating that the effective equations are found in the limit of infinitely many initial phase changes, we interpret the densities as Young measures and prove the convergence of the sequence of solutions to the effective model. The effective model consists of a deterministic part for the fluid's hydrodynamic quantities and a kinetic equation for the limit Young measure encoding the mixing dynamics. By characterizing the Young measure with the corresponding cumulative distribution function, we rewrite the kinetic equation for the Young measure into a kinetic equation for the cumulative distribution function such that the resulting equations are accessible by standard approximation methods.