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Incompressible Limit of Strong Solutions to the Diffuse Interface Model for Two-phase Flows

Yinghua Li, Manrou Xie

TL;DR

This work rigorously justifies the incompressible limit for both compressible Navier–Stokes–Cahn–Hilliard (NSCH) and Navier–Stokes–Allen–Cahn (NSAC) diffuse-interface models with density/phase-dependent viscosities under periodic boundary conditions, by showing that as the Mach number $M=\varepsilon^2\to0$, strong solutions $(\rho^\varepsilon,u^\varepsilon,\phi^\varepsilon)$ converge to the incompressible counterparts $(1,u,\phi)$. The authors develop uniform-in-$\varepsilon$ energy estimates via quadratic cancellations of singular terms, construct a contraction mapping for local-in-time solutions, and pass to the limit to obtain the incompressible NSCH/NSAC systems. They further obtain quantitative convergence rates using a modulated-energy method, e.g., $\|u^\varepsilon-u\|^2+\|\phi^\varepsilon-\phi\|_1^2\le C\varepsilon$ and $\|\rho^\varepsilon-1\|_s^2\le C\varepsilon^2$, with corresponding bounds on higher derivatives. The results bridge compressible and incompressible diffuse-interface models, ensuring consistent interfacial dynamics and providing precise error scales for practical modeling in multi-phase flows.

Abstract

This paper is concerned with the incompressible limit problem for strong solutions of compressible two-phase flow models under periodic boundary conditions, where the Navier-Stokes equations are nonlinearly coupled with either Cahn-Hilliard equations or Allen-Cahn equations. The viscosity coefficients are allowed to depend both on the density and the phase field variable. We establish rigorous convergence of both local and global strong solutions of compressible systems to their incompressible systems as the Mach number tends to zero.This theoretical framework establishes an essential linkage between compressible and incompressible phase field models, demonstrating that both formulations exhibit consistent physical fidelity in capturing interfacial flow dynamics.Furthermore, we provide some convergence rate estimates of the solutions.

Incompressible Limit of Strong Solutions to the Diffuse Interface Model for Two-phase Flows

TL;DR

This work rigorously justifies the incompressible limit for both compressible Navier–Stokes–Cahn–Hilliard (NSCH) and Navier–Stokes–Allen–Cahn (NSAC) diffuse-interface models with density/phase-dependent viscosities under periodic boundary conditions, by showing that as the Mach number , strong solutions converge to the incompressible counterparts . The authors develop uniform-in- energy estimates via quadratic cancellations of singular terms, construct a contraction mapping for local-in-time solutions, and pass to the limit to obtain the incompressible NSCH/NSAC systems. They further obtain quantitative convergence rates using a modulated-energy method, e.g., and , with corresponding bounds on higher derivatives. The results bridge compressible and incompressible diffuse-interface models, ensuring consistent interfacial dynamics and providing precise error scales for practical modeling in multi-phase flows.

Abstract

This paper is concerned with the incompressible limit problem for strong solutions of compressible two-phase flow models under periodic boundary conditions, where the Navier-Stokes equations are nonlinearly coupled with either Cahn-Hilliard equations or Allen-Cahn equations. The viscosity coefficients are allowed to depend both on the density and the phase field variable. We establish rigorous convergence of both local and global strong solutions of compressible systems to their incompressible systems as the Mach number tends to zero.This theoretical framework establishes an essential linkage between compressible and incompressible phase field models, demonstrating that both formulations exhibit consistent physical fidelity in capturing interfacial flow dynamics.Furthermore, we provide some convergence rate estimates of the solutions.

Paper Structure

This paper contains 15 sections, 14 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. The problem
  3. Literatures review
  4. Statements on the proof
  5. Preliminaries
  6. Navier-Stokes/Cahn-Hilliard system
  7. Statements of the theorems
  8. Local existence and uniform stability
  9. Proof of Theorem \ref{['th2']}
  10. Proof of Theorem \ref{['th3']}
  11. Navier-Stokes/Allen-Cahn system
  12. Statements of the theorems.
  13. Local existence and uniform stability.
  14. Proof of Theorem \ref{['th2-2']}.
  15. Proof of Theorem \ref{['th3-2']}

Key Result

Lemma 1.1

(see H-L-95K-ML-Z) Let $s>\frac{N}{2}$. For any functions $f, g$ (possibly vector-valued in $\mathbb{R}^n$) in Sobolev space $H^s(\mathbb{T}^N, \mathbb{R}^n)$, $\Psi \in C^s(\mathbb{R}^n)$, $\|\nabla^j\Psi\|_\infty<\infty~(j=1,2,...,s)$, and multi-index $\alpha$ satisfying $|\alpha|\le s$, we have t the estimates based on the chain rule: the estimate based on Leibniz's rule: where the constant $

Theorems & Definitions (20)

  • Lemma 1.1
  • Lemma 1.2
  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.2
  • Lemma 2.3
  • ...and 10 more