Convergence of a nonlocal to a local diffuse interface model for two-phase flow with unmatched densities

Helmut Abels; Yutaka Terasawa

doi:10.3934/dcdss.2022117

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2022, Volume 15, Issue 8: 1871-1881. Doi: 10.3934/dcdss.2022117

This issue Previous Article Preface Next Article A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs

Convergence of a nonlocal to a local diffuse interface model for two-phase flow with unmatched densities

Helmut Abels^1, , and
Yutaka Terasawa^2,

1.
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
2.
Graduate School of Mathematics, Nagoya University, Furocho Chikusaku, Nagoya, 464-8602, Japan

^* Corresponding author: Helmut Abels
^* Corresponding author: Helmut Abels

Received: December 31, 2021

Revised: March 31, 2022

Published: August 2022

The second author has been supported by JSPS KAKENHI number 17K17804

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Abstract

We prove convergence of suitable subsequences of weak solutions of a diffuse interface model for the two-phase flow of incompressible fluids with different densities with a nonlocal Cahn-Hilliard equation to weak solutions of the corresponding system with a standard "local" Cahn-Hilliard equation. The analysis is done in the case of a sufficiently smooth bounded domain with no-slip boundary condition for the velocity and Neumann boundary conditions for the Cahn-Hilliard equation. The proof is based on the corresponding result in the case of a single Cahn-Hilliard equation and compactness arguments used in the proof of existence of weak solutions for the diffuse interface model.

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Mathematics Subject Classification: Primary: 76T99; Secondary: 35Q30, 35Q35, 76D03, 76D05, 76D27, 76D45.

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