Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions,part I: Convergence result

Abstract

We consider the sharp interface limit for the scalar-valued and vector-valued Allen–Cahn equation with homogeneous Neumann boundary condition in a bounded smooth domain Ω of arbitrary dimension $N ⩾ 2$ in the situation when a two-phase diffuse interface has developed and intersects the boundary $\partial Ω$ . The limit problem is mean curvature flow with 90°-contact angle and we show convergence in strong norms for well-prepared initial data as long as a smooth solution to the limit problem exists. To this end we assume that the limit problem has a smooth solution on $[0, T]$ for some time $T > 0$ . Based on the latter we construct suitable curvilinear coordinates and set up an asymptotic expansion for the scalar-valued and the vector-valued Allen–Cahn equation. In order to estimate the difference of the exact and approximate solutions with a Gronwall-type argument, a spectral estimate for the linearized Allen–Cahn operator in both cases is required. The latter will be shown in a separate paper, cf. (Moser (2021)).

Keywords

Sharp interface limit mean curvature flow contact angle Allen–Cahn equation vector-valued Allen–Cahn equation

Get full access to this article

View all access options for this article.

References

Abels and

Fei, Sharp interface limit for a Navier–Stokes/Allen–Cahn system with different viscosities, Preprint, 2022. arXiv:2201.09343.

Abels and

Liu, Sharp interface limit for a Stokes/Allen–Cahn system, Arch. Rational Mech. Anal. 229(1) (2018), 417–502. doi:10.1007/s00205-018-1220-x.

Abels and

Marquardt, Sharp interface limit of a Stokes/Cahn–Hilliard system, part I: Convergence result, Preprint, 2020. arXiv:2003.03139.

Abels and

Marquardt, Sharp interface limit of a Stokes/Cahn–Hilliard system, part II: Approximate solutions, J. Math. Fluid Mech. 23(38) (2021), 1–48.

Abels and

Moser, Convergence of the Allen–Cahn equation to the mean curvature flow with 90°-contact angle in 2D, Interfaces and Free Boundaries 21(3) (2019), 313–365. doi:10.4171/IFB/425.

Abels and

Moser, Convergence of the Allen–Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to 90°, SIAM J. Math. Anal. 54(1) (2022), 114–172. doi:10.1137/21M1424925.

Abels and

Schaubeck, Sharp interface limit for the Cahn–Larché system, Asymptotic Analysis 91 (2015), 283–340. doi:10.3233/ASY-141268.

R.A.

Adams and

J.J.F.

Fournier, Sobolev Spaces, 2nd edn, Elsevier Ltd., 2003.

N.D.

Alikakos,

P.W.

Bates and

Chen, Convergence of the Cahn–Hilliard equation to the Hele–Shaw model, Arch. Rational Mech. Anal. 128 (1994), 165–205. doi:10.1007/BF00375025.

10.

N.D.

Alikakos and

Fusco, Entire solutions to equivariant elliptic systems with variational structure, Arch. Rational Mech. Anal. 202 (2011), 567–597. doi:10.1007/s00205-011-0441-z.

11.

S.M.

Allen and

J.W.

Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica 27 (1979), 1085–1095. doi:10.1016/0001-6160(79)90196-2.

12.

H.W.

Alt, Linear Functional Analysis. An Application-Oriented Introduction, Springer-Verlag, London, 2016.

13.

Amann and

Escher, Analysis III, Birkhäuser, Basel, 2009.

14.

D.M.

Anderson,

G.B.

McFadden and

A.A.

Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech. 30 (1998), 139–165. doi:10.1146/annurev.fluid.30.1.139.

15.

Baiocchi,

Comincioli,

Magenes and

Pozzi, Free boundary problems in the theory of fluid flow through porous media: Existence and uniqueness theorems, Annali di Matematica Pura ed Applicata 97(1) (1973), 1–82. doi:10.1007/BF02414909.

16.

Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn–Hilliard fluids, Ann. Inst. Henri Poincaré 7(2) (1990), 67–90. doi:10.1016/s0294-1449(16)30304-3.

17.

Barles and

Da Lio, A geometrical approach to front propagation problems in bounded domains with Neumann-type boundary conditions, Interfaces and Free Boundaries 5 (2003), 239–274. doi:10.4171/IFB/79.

18.

Barles and

P.E.

Souganidis, A new approach to front propagation problems: Theory and applications, Arch. Rational Mech. Anal. 141 (1998), 237–296. doi:10.1007/s002050050077.

19.

Barrett,

Garcke and

Nürnberg, Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation, SMAI Journal of Computational Mathematics 4 (2018), 151–195. doi:10.5802/smai-jcm.32.

20.

Bartels, Numerical Methods for Nonlinear Partial Differential Equations, Springer Series in Computational Mathematics, Springer, 2015.

21.

Bellettini, Lecture Notes on Mean Curvature Flow, Barriers and Singular Perturbations, Scuola Normale Superiore, Pisa, 2013.

22.

Bronsard,

Gui and

Schatzman, A three-layered minimizer in

R^{2}

for a variational problem with a symmetric three-well potential, Comm. Pure Appl. Math. 49(7) (1996), 677–715. doi:10.1002/(SICI)1097-0312(199607)49:7<677::AID-CPA2>3.0.CO;2-6.

23.

Bronsard and

Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg–Landau equation, Arch. Rational Mech. Anal. 124(4) (1993), 355–379. doi:10.1007/BF00375607.

24.

Caginalp and

Chen, Convergence of the phase field model to its sharp interface limits, Euro. Jnl. of Applied Mathematics 9 (1998), 417–445. doi:10.1017/S0956792598003520.

25.

Caginalp,

Chen and

Eck, A rapidly converging phase field model, Discrete and continuous dynamical systems 15(4) (2006), 1017–1034. doi:10.3934/dcds.2006.15.1017.

26.

Carr and

R.L.

Pego, Metastable patterns in solutions of

u_{t} = ε^{2} u_{x x} - f (u)

, Comm. Pure Appl. Math. 42 (1989), 523–576. doi:10.1002/cpa.3160420502.

27.

Chen, Generation and propagation of interfaces for reaction–diffusion equations, Journal of Differential Equations 96 (1992), 116–141. doi:10.1016/0022-0396(92)90146-E.

28.

Chen, Spectrum for the Allen–Cahn, Cahn–Hilliard, and phase-field equations for generic interfaces, Commun. in Partial Differential Equations 19(7&8) (1994), 1371–1395. doi:10.1080/03605309408821057.

29.

Chen,

Hilhorst and

Logak, Mass conserving Allen–Cahn equation and volume preserving mean curvature flow, Interfaces and Free Boundaries 12 (2010), 527–549. doi:10.4171/IFB/244.

30.

de Mottoni and

Schatzman, Geometrical evolution of developed interfaces, Transactions of the Americal Mathematical Society 347(5) (1995), 207–220.

31.

Denk,

Hieber and

Prüss, Optimal

L^{p}

L^{q}

-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z. 257 (2007), 193–224. doi:10.1007/s00209-007-0120-9.

32.

Depner, Stability analysis of geometric evolution equations with triple lines and boundary contact, PhD thesis, Univ. of Regensburg, 2010. urn:nbn:de:bvb:355-epub-160479.

33.

Ebenbeck, Cahn–Hilliard–Brinkman models for tumour growth: Modelling, analysis and optimal control, PhD thesis, Univ. of Regensburg, 2019. urn:nbn:de:bvb:355-epub-433763.

34.

Eck,

Garcke and

Knabner, Mathematical Modeling, Springer International Publishing AG, 2017.

35.

L.C.

Evans, Partial Differential Equations, 2nd edn, American Mathematical Society, Providence, Rhode Island, 2010.

36.

L.C.

Evans and

R.F.

Gariepy, Measure Theory and Fine Properties of Functions, revised edn, Taylor & Francis Group, LLC, 2015.

37.

L.C.

Evans,

H.M.

Soner and

P.E.

Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45 (1992), 1097–1123. doi:10.1002/cpa.3160450903.

38.

Fei and

Liu, Phase-field approximation of the Willmore flow, Preprint, 2020. arXiv:1904.11139.

39.

Fischer and

Hensel, Weak-strong uniqueness for the Navier–Stokes equation for two fluids with surface tension, Arch. Ration. Mech. Anal. 236 (2020), 967–1087. doi:10.1007/s00205-019-01486-2.

40.

Fischer,

Laux and

T.M.

Simon, Convergence rates of the Allen–Cahn equation to mean curvature flow: A short proof based on relative entropies, SIAM J. Math. Anal. 52(6) (2020), 6222–6233. doi:10.1137/20M1322182.

41.

Fischer and

Marveggio, Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow, Preprint, 2022. arXiv:2203.17143.

42.

Friedman, Free boundary problems in science and technology, Notices of the AMS 47(8) (2000), 854–861.

43.

Garcke and

Haas, Modelling of non-isothermal multi-component, multi-phase systems with convection, in: Phase Transformations in Multicomponent Melts, Wiley-VCH Verlag, Weinheim, 2008, pp. 325–338. doi:10.1002/9783527624041.ch20.

44.

Gilbarg and

N.S.

Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn, Springer-Verlag, Berlin, Heidelberg, 2001.

45.

Haas, Modeling and analysis for general non-isothermal convective phase field systems, PhD thesis, Univ. of Regensburg, 2007. urn:nbn:de:bvb:355-opus-7834.

46.

Hebey, Sobolev Spaces on Riemannian Manifolds, Springer, Berlin, Heidelberg, 1996.

47.

Hensel and

Moser, Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime, Preprint, 2021. arXiv:2112.11173.

48.

Huisken, Non-parametric mean curvature evolution with boundary conditions, Journal of Differential Equations 77 (1989), 369–378. doi:10.1016/0022-0396(89)90149-6.

49.

Ilmanen, Convergence of the Allen–Cahn equation to Brakke’s motion by mean curvature, J. Differential Geom. 38 (1993), 417–461. doi:10.4310/jdg/1214454300.

50.

Kagaya, Convergence of the Allen–Cahn equation with a zero Neumann boundary condition on non-convex domains, Math. Ann. 373 (2019), 1485–1528. doi:10.1007/s00208-018-1720-x.

51.

Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg, 1995.

52.

Katsoulakis,

G.T.

Kossioris and

Reitich, Generalized motion by mean curvature with Neumann conditions and the Allen–Cahn model for phase transitions, The Journal of Geometric Analysis 5(2) (1995), 255–279. doi:10.1007/BF02921677.

53.

Kreuter, Spaces of vector-valued functions, Master Thesis, Univ. of Ulm, 2015.

54.

Kusche, Spectral analysis for linearizations of the Allen–Cahn equation around rescaled stationary solutions with triple junction, PhD thesis, Univ. of Regensburg, 2006. urn:nbn:de:bvb:355-opus-6471.

55.

Laux and

Liu, Nematic–isotropic phase transition in liquid crystals: A variational derivation of effective geometric motions, Arch. Ration. Mech. Anal. 241 (2021), 1785–1814. doi:10.1007/s00205-021-01681-0.

56.

Laux and

T.M.

Simon, Convergence of the Allen–Cahn equation to multiphase mean curvature flow, Comm. Pure Appl. Math. 71(8) (2018), 1597–1647. doi:10.1002/cpa.21747.

57.

D.S.

Lee and

J.S.

Kim, Mean curvature flow by the Allen–Cahn equation, Euro. Jnl. of Applied Mathematics 26 (2015), 535–559. doi:10.1017/S0956792515000200.

58.

Leoni, A First Course in Sobolev Spaces, 2nd edn, American Mathematical Society, Providence, Rhode Island, 2017.

59.

Liu, Sharp interface limit of an anisotropic Ginzburg–Landau equation, Preprint, 2021. arXiv:2111.15061.

60.

Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Springer, Basel, 1995.

61.

Lunardi,

Sinestrari and

Von Wahl, A semigroup approach to the time-dependent parabolic initial boundary value problem, Diff. Int. Eqns. 5 (1992), 1275–1306.

62.

Marquardt, Sharp interface limit for a Stokes/Cahn–Hilliard system, PhD thesis, Univ. of Regensburg, 2019. urn:nbn:de:bvb:355-epub-384308.

63.

Miranville, The Cahn–Hilliard Equation: Recent Advances and Applications, SIAM, 2019, xiv+216.

64.

Mizuno and

Tonegawa, Convergence of the Allen–Cahn equation with Neumann boundary conditions, SIAM J. Math. Anal. 47(3) (2015), 1906–1932. doi:10.1137/140987808.

65.

Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal. 98 (1987), 123–142. doi:10.1007/BF00251230.

66.

Moser, Sharp interface limits for diffuse interface models with contact angle, PhD thesis, Univ. of Regensburg, 2020. urn:nbn:de:bvb:355-epub-443894.

67.

Moser, Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions, part II: Spectral estimates, Preprint, 2021.

68.

Novick-Coen, The Cahn–Hilliard equation, in: Handbook of Differential Equations: Evolutionary Equations, Vol. 4, Elsevier, 2008, pp. 201–228. doi:10.1016/S1874-5717(08)00004-2.

69.

O’Neill, Semi Riemannian Geometry. With Applications to Relativity, Academic Press, 1983.

70.

Prüss and

Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Birkhäuser, Basel, 2016.

71.

Renardy and

R.C.

Rogers, An Introduction to Partial Differential Equations, 2nd edn, Springer-Verlag, New York, 2004.

72.

Rubinstein,

Sternberg and

Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math. 49(1) (1989), 116–133. doi:10.1137/0149007.

73.

Sáez Trumper, Relaxation of the curve shortening flow via the parabolic Ginzburg–Landau equation, Calc. Var. 31(3) (2008), 359–386. doi:10.1007/s00526-007-0118-5.

74.

Sáez Trumper, Relaxation of the flow of triods by curve shortening flow via the vector-valued parabolic Allen–Cahn equation, J. Reine Angew. Math. 634 (2009), 143–168.

75.

Schaubeck, Sharp interface limits for diffuse interface models, PhD thesis, Univ. of Regensburg, 2014. urn:nbn:de:bvb:355-epub-294622.

76.

Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal. 101 (1988), 209–260. doi:10.1007/BF00253122.

77.

Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Company, Amsterdam, New York, Oxford, 1978.

78.

Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.

79.

Vogel, Sufficient conditions for capillary surfaces to be energy minima, Pac. J. Math. 194(2) (2000), 469–489. doi:10.2140/pjm.2000.194.469.