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

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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

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

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.

Keywords: Two-phase flow, Navier-Stokes equation, diffuse interface model, mixtures of viscous fluids, Cahn-Hilliard equation, non-local operators.

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

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.

show all references

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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