# American Institute of Mathematical Sciences

August  2022, 15(8): 1871-1881. doi: 10.3934/dcdss.2022117

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

 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

Received  January 2022 Revised  April 2022 Published  August 2022 Early access  May 2022

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

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.

Citation: Helmut Abels, Yutaka Terasawa. Convergence of a nonlocal to a local diffuse interface model for two-phase flow with unmatched densities. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 1871-1881. doi: 10.3934/dcdss.2022117
##### References:
 [1] H. Abels, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Comm. Math. Phys., 289 (2009), 45-73.  doi: 10.1007/s00220-009-0806-4. [2] H. Abels, S. Bosia and M. Grasselli, Cahn-Hilliard equation with nonlocal singular free energies, Ann. Mat. Pura Appl., 194 (2015), 1071-1106.  doi: 10.1007/s10231-014-0411-9. [3] H. Abels, D. Depner and H. Garcke, Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities, J. Math. Fluid Mech., 15 (2013), 453-480.  doi: 10.1007/s00021-012-0118-x. [4] H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22 (2012), 1150013, 40 pp. doi: 10.1142/S0218202511500138. [5] H. Abels and Y. Terasawa, Weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities and nonlocal free energies, Math. Methods Appl. Sci., 43 (2020), 3200-3219.  doi: 10.1002/mma.6111. [6] H. Abels and J. Weber, Local well-posedness of a quasi-incompressible two-phase flow, J. Evol. Equ., 21 (2021), 3477-3502.  doi: 10.1007/s00028-020-00646-2. [7] E. Davoli, H. Ranetbauer, L. Scarpa and L. Trussardi, Degenerate nonlocal Cahn-Hilliard equations: Well-posedness, regularity and local asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 627-651.  doi: 10.1016/j.anihpc.2019.10.002. [8] E. Davoli, L. Scarpa and L. Trussardi, Local asymptotics for nonlocal convective Cahn-Hilliard equations with $W^{1, 1}$ kernel and singular potential, J. Differential Equations, 289 (2021), 35-58.  doi: 10.1016/j.jde.2021.04.016. [9] E. Davoli, L. Scarpa and L. Trussardi, Nonlocal-to-local convergence of Cahn-Hilliard equations: Neumann boundary conditions and viscosity terms, Arch. Ration. Mech. Anal., 239 (2021), 117-149.  doi: 10.1007/s00205-020-01573-9. [10] S. Frigeri, Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities, Math. Models Methods Appl. Sci., 26 (2016), 1955-1993.  doi: 10.1142/S0218202516500494. [11] S. Frigeri, On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 38 (2021), 647-687.  doi: 10.1016/j.anihpc.2020.08.005. [12] S. Frigeri, C. G. Gal and and M. Grasselli, Regularity results for the nonlocal Cahn-Hilliard equation with singular potential and degenerate mobility, J. Differential Equations, 287 (2021), 295-328.  doi: 10.1016/j.jde.2021.03.052. [13] C. G. Gal, M. Grasselli and H. Wu, Global weak solutions to a diffuse interface model for incompressible two-phase flows with moving contact lines and different densities, Arch. Ration. Mech. Anal., 234 (2019), 1-56.  doi: 10.1007/s00205-019-01383-8. [14] A. Giorgini, Well-posedness of the two-dimensional Abels-Garcke-Grün model for two-phase flows with unmatched densities, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 100, 40 pp. doi: 10.1007/s00526-021-01962-2. [15] Z. Liang, On the existence of weak solutions to non-local Cahn-Hilliard/Navier-Stokes equations and its local asymptotics, Commun. Math. Sci., 18 (2020), 2121-2147.  doi: 10.4310/CMS.2020.v18.n8.a2. [16] S. Melchionna, H. Ranetbauer, L. Scarpa and L. Trussardi, From nonlocal to local Cahn-Hilliard equation, Adv. Math. Sci. Appl., 28 (2019), 197-211. [17] A. C. Ponce, An estimate in the spirit of Poincaré's inequality, J. Eur. Math. Soc. (JEMS), 6 (2004), 1-15. [18] A. C. Ponce, A new approach to Sobolev spaces and connections to $\Gamma$-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.  doi: 10.1007/s00526-003-0195-z. [19] J. Weber, Analysis of Diffuse Interface Models for Two-Phase Flows with and without Surfactants, Ph.D thesis, University Regensburg, urn: nbn: de: bvb: 355-epub-342471, 2016.

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##### References:
 [1] H. Abels, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Comm. Math. Phys., 289 (2009), 45-73.  doi: 10.1007/s00220-009-0806-4. [2] H. Abels, S. Bosia and M. Grasselli, Cahn-Hilliard equation with nonlocal singular free energies, Ann. Mat. Pura Appl., 194 (2015), 1071-1106.  doi: 10.1007/s10231-014-0411-9. [3] H. Abels, D. Depner and H. Garcke, Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities, J. Math. Fluid Mech., 15 (2013), 453-480.  doi: 10.1007/s00021-012-0118-x. [4] H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22 (2012), 1150013, 40 pp. doi: 10.1142/S0218202511500138. [5] H. Abels and Y. Terasawa, Weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities and nonlocal free energies, Math. Methods Appl. Sci., 43 (2020), 3200-3219.  doi: 10.1002/mma.6111. [6] H. Abels and J. Weber, Local well-posedness of a quasi-incompressible two-phase flow, J. Evol. Equ., 21 (2021), 3477-3502.  doi: 10.1007/s00028-020-00646-2. [7] E. Davoli, H. Ranetbauer, L. Scarpa and L. Trussardi, Degenerate nonlocal Cahn-Hilliard equations: Well-posedness, regularity and local asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 627-651.  doi: 10.1016/j.anihpc.2019.10.002. [8] E. Davoli, L. Scarpa and L. Trussardi, Local asymptotics for nonlocal convective Cahn-Hilliard equations with $W^{1, 1}$ kernel and singular potential, J. Differential Equations, 289 (2021), 35-58.  doi: 10.1016/j.jde.2021.04.016. [9] E. Davoli, L. Scarpa and L. Trussardi, Nonlocal-to-local convergence of Cahn-Hilliard equations: Neumann boundary conditions and viscosity terms, Arch. Ration. Mech. Anal., 239 (2021), 117-149.  doi: 10.1007/s00205-020-01573-9. [10] S. Frigeri, Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities, Math. Models Methods Appl. Sci., 26 (2016), 1955-1993.  doi: 10.1142/S0218202516500494. [11] S. Frigeri, On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 38 (2021), 647-687.  doi: 10.1016/j.anihpc.2020.08.005. [12] S. Frigeri, C. G. Gal and and M. Grasselli, Regularity results for the nonlocal Cahn-Hilliard equation with singular potential and degenerate mobility, J. Differential Equations, 287 (2021), 295-328.  doi: 10.1016/j.jde.2021.03.052. [13] C. G. Gal, M. Grasselli and H. Wu, Global weak solutions to a diffuse interface model for incompressible two-phase flows with moving contact lines and different densities, Arch. Ration. Mech. Anal., 234 (2019), 1-56.  doi: 10.1007/s00205-019-01383-8. [14] A. Giorgini, Well-posedness of the two-dimensional Abels-Garcke-Grün model for two-phase flows with unmatched densities, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 100, 40 pp. doi: 10.1007/s00526-021-01962-2. [15] Z. Liang, On the existence of weak solutions to non-local Cahn-Hilliard/Navier-Stokes equations and its local asymptotics, Commun. Math. Sci., 18 (2020), 2121-2147.  doi: 10.4310/CMS.2020.v18.n8.a2. [16] S. Melchionna, H. Ranetbauer, L. Scarpa and L. Trussardi, From nonlocal to local Cahn-Hilliard equation, Adv. Math. Sci. Appl., 28 (2019), 197-211. [17] A. C. Ponce, An estimate in the spirit of Poincaré's inequality, J. Eur. Math. Soc. (JEMS), 6 (2004), 1-15. [18] A. C. Ponce, A new approach to Sobolev spaces and connections to $\Gamma$-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.  doi: 10.1007/s00526-003-0195-z. [19] J. Weber, Analysis of Diffuse Interface Models for Two-Phase Flows with and without Surfactants, Ph.D thesis, University Regensburg, urn: nbn: de: bvb: 355-epub-342471, 2016.
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