From kinetic mixtures to compressible two-phase flow: A BGK-type model and rigorous derivation

Seung Yeon Cho; Young-Pil Choi; Byung-Hoon Hwang; Sihyun Song

doi:10.3934/krm.2026010

Article Contents

2026, Volume 22: 147-196. Doi: 10.3934/krm.2026010

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From kinetic mixtures to compressible two-phase flow: A BGK-type model and rigorous derivation

1.
Department of Mathematics, Gyeongsang National University, Republic of Korea
2.
Department of Mathematics, Yonsei University, Republic of Korea
3.
Department of Mathematics Education, Sangmyung University, Republic of Korea

Received: November 18, 2025

Revised: February 22, 2026

Published online: March 18, 2026

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Abstract

We propose a BGK-type kinetic model for a binary gas mixture, designed to serve as a kinetic formulation of compressible two-phase fluid dynamics. The model features species-dependent adiabatic exponents, and the relaxation operator is constructed by solving an entropy minimization problem under moments constraints. Starting from this model, we derive the compressible two-phase Euler equations via a formal Chapman–Enskog expansion and identify dissipative corrections of Navier–Stokes type. We then rigorously justify the Euler limit using the relative entropy method, establishing quantitative convergence estimates under appropriate regularity assumptions. Finally, we present numerical experiments based on an implicit-explicit Runge–Kutta method, which confirm the asymptotic preserving property and demonstrate the convergence from the BGK model to the isentropic two-phase Euler system in the hydrodynamic regime.

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Mathematics Subject Classification: Primary: 76P05, 76T17; Secondary: 35B40.

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Figure 1. Riemann Problem I: Comparison of species densities (left panels) and bulk velocity of mixtures (right panels)

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Figure 2. Riemman problem II: Comparison of species densities (left panels) and bulk velocity of mixtures (right panels) when $ \gamma_1 = 2 $ and $ \gamma_2 = 7/5 $

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Figure 3. Riemman problem II: Comparison of species densities (left panels) and velocity (right panels) when $ \gamma_1 = 3 $ and $ \gamma_2 = 7/5 $

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