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Nonlocal-to-local convergence rates for strong solutions to a Navier-Stokes-Cahn-Hilliard system with singular potential

Published 16 Mar 2024 in math.AP, math-ph, and math.MP | (2403.10947v1)

Abstract: The main goal of this paper is to establish the nonlocal-to-local convergence of strong solutions to a Navier--Stokes--Cahn--Hilliard model with singular potential describing immiscible, viscous two-phase flows with matched densities, which is referred to as the Model H. This means that we show that the strong solutions to the nonlocal Model H converge to the strong solution to the local Model H as the weight function in the nonlocal interaction kernel approaches the delta distribution. Compared to previous results in the literature, our main novelty is to further establish corresponding convergence rates. Before investigating the nonlocal-to-local convergence, we first need to ensure the strong well-posedness of the nonlocal Model H. In two dimensions, this result can already be found in the literature, whereas in three dimensions, it will be shown in the present paper. Moreover, in both two and three dimensions, we establish suitable uniform bounds on the strong solutions of the nonlocal Model H, which are essential to prove the nonlocal-to-local convergence results.

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References (48)
  1. H. Abels. On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Ration. Mech. Anal., 194(2):463–506, 2009.
  2. Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities. J. Math. Fluid Mech., 15(3):453–480, 2013.
  3. On an incompressible Navier–Stokes/Cahn–Hilliard system with degenerate mobility. Annales de l’Institut Henri Poincaré C, 30(6):1175–1190, 2013.
  4. H. Abels and H. Garcke. Weak solutions and diffuse interface models for incompressible two-phase flows. In Handbook of mathematical analysis in mechanics of viscous fluids, pages 1267–1327. Springer, Cham, 2018.
  5. Global regularity and asymptotic stabilization for the incompressible Navier–Stokes-Cahn–Hilliard model with unmatched densities. Math. Ann., 2023. DOI: 10.1007/s00208-023-02670-2.
  6. Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci., 22(3):1150013, 40, 2012.
  7. Mathematical Analysis of a Diffuse Interface Model for Multi-Phase Flows of Incompressible Viscous Fluids with Different Densities. To appear in J. Math. Fluid Mech., arXiv preprint: arXiv:2308.11813[math.AP], 2023.
  8. H. Abels and C. Hurm. Strong Nonlocal-to-Local Convergence of the Cahn-Hilliard Equation and its Operator. arXiv preprint: arXiv:2307.02264 [math.AP], 2023.
  9. H. Abels and Y. Terasawa. Convergence of a nonlocal to a local diffuse interface model for two-phase flow with unmatched densities. Discrete Contin. Dyn. Syst. Ser. S, 15(8):1871–1881, 2022.
  10. H. Abels and J. Weber. Local well-posedness of a quasi-incompressible two-phase flow. J. Evol. Equ., 21(3):3477–3502, 2021.
  11. F. Boyer. Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal., 20(2):175–212, 1999.
  12. Global existence of weak solutions to a nonlocal Cahn-Hilliard-Navier-Stokes system. J. Math. Anal. Appl., 386(1):428–444, 2012.
  13. Degenerate nonlocal Cahn-Hilliard equations: well-posedness, regularity and local asymptotics. Ann. Inst. H. Poincaré C Anal. Non Linéaire, 37(3):627–651, 2020.
  14. Local asymptotics for nonlocal convective Cahn-Hilliard equations with W1,1superscript𝑊11W^{1,1}italic_W start_POSTSUPERSCRIPT 1 , 1 end_POSTSUPERSCRIPT kernel and singular potential. J. Differential Equations, 289:35–58, 2021.
  15. Nonlocal-to-local convergence of Cahn-Hilliard equations: Neumann boundary conditions and viscosity terms. Arch. Ration. Mech. Anal., 239(1):117–149, 2021.
  16. C. Elbar and J. Skrzeczkowski. Degenerate Cahn-Hilliard equation: from nonlocal to local. J. Differential Equations, 364:576–611, 2023.
  17. 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(3):647–687, 2021.
  18. On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions. J. Nonlinear Sci., 26(4):847–893, 2016.
  19. S. Frigeri and M. Grasselli. Global and trajectory attractors for a nonlocal Cahn-Hilliard-Navier-Stokes system. J. Dynam. Differential Equations, 24(4):827–856, 2012.
  20. S. Frigeri and M. Grasselli. Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials. Dyn. Partial Differ. Equ., 9(4):273–304, 2012.
  21. Strong solutions for two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems. J. Differential Equations, 255(9):2587–2614, 2013.
  22. A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility. Nonlinearity, 28(5):1257–1293, 2015.
  23. The nonlocal Cahn-Hilliard equation with singular potential: well-posedness, regularity and strict separation property. J. Differential Equations, 263(9):5253–5297, 2017.
  24. The separation property for 2D Cahn-Hilliard equations: local, nonlocal and fractional energy cases. Discrete Contin. Dyn. Syst., 43(6):2270–2304, 2023.
  25. Global well-posedness and convergence to equilibrium for the Abels-Garcke-Grün model with nonlocal free energy. J. Math. Pures Appl. (9), 178:46–109, 2023.
  26. C. G. Gal and M. Grasselli. Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D. Ann. Inst. H. Poincaré C Anal. Non Linéaire, 27(1):401–436, 2010.
  27. Cahn–Hilliard–Navier–Stokes systems with moving contact lines. Calc. Var. Partial Differential Equations, 55(3):Art. 50, 47, 2016.
  28. Allen-Cahn–Navier-Stokes-Voigt systems with moving contact lines. J. Math. Fluid Mech., 25(4):Paper No. 89, 57, 2023.
  29. 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(1):1–56, 2019.
  30. C. G. Gal and A. Poiatti. Unified framework for the separation property in binary phase-segregation processes with singular entropy densities. To appear in European J. Appl. Math., ResearchGate preprint, DOI: 10.13140/RG.2.2.35972.30089/1, 2023.
  31. The anisotropic Cahn-Hilliard equation: regularity theory and strict separation properties. Discrete Contin. Dyn. Syst. Ser. S, 16(12):3622–3660, 2023.
  32. 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(3):Paper No. 100, 40, 2021.
  33. A. Giorgini. Existence and stability of strong solutions to the Abels-Garcke-Grün model in three dimensions. Interfaces Free Bound., 24(4):565–608, 2022.
  34. The Cahn-Hilliard-Oono equation with singular potential. Math. Models Methods Appl. Sci., 27(13):2485–2510, 2017.
  35. A. Giorgini and P. Knopf. Two-phase flows with bulk-surface interaction: thermodynamically consistent Navier-Stokes-Cahn-Hilliard models with dynamic boundary conditions. J. Math. Fluid Mech., 25(3):Paper No. 65, 44, 2023.
  36. Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system. SIAM J. Math. Anal., 51(3):2535–2574, 2019.
  37. Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci., 6(6):815–831, 1996.
  38. Theory of dynamic critical phenomena. Rev. Mod. Phys., 49:435–479, 1977.
  39. From nonlocal to local Cahn-Hilliard equation. Adv. Math. Sci. Appl., 28(2):197–211, 2019.
  40. A. Miranville. The Cahn-Hilliard equation, volume 95 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019. Recent advances and applications.
  41. A. Miranville and S. Zelik. Robust exponential attractors for Cahn-Hilliard type equations with singular potentials. Math. Methods Appl. Sci., 27(5):545–582, 2004.
  42. A. Poiatti. The 3D strict separation property for the nonlocal Cahn–Hilliard equation with singular potential. To appear in Anal. PDE, arXiv preprint: arXiv:2303.07745 [math.AP], 2022.
  43. A. Poiatti and A. Signori. Regularity results and optimal velocity control of the convective nonlocal Cahn-Hilliard equation in 3D. ESAIM Control Optim. Calc. Var., 2024. DOI: 10.1051/cocv/2024007.
  44. A. C. Ponce. An estimate in the spirit of Poincaré’s inequality. J. Eur. Math. Soc. (JEMS), 6(1):1–15, 2004.
  45. A. C. Ponce. A new approach to Sobolev spaces and connections to ΓΓ\Gammaroman_Γ-convergence. Calc. Var. Partial Differential Equations, 19(3):229–255, 2004.
  46. W. A. Strauss. On continuity of functions with values in various Banach spaces. Pacific J. Math., 19:543–551, 1966.
  47. R. Temam. Navier-Stokes equations and nonlinear functional analysis, volume 66 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 1995.
  48. N. S. Trudinger. On imbeddings into Orlicz spaces and some applications. J. Math. Mech., 17:473–483, 1967.
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