Convergence of the Allen-Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to \(90^\circ\). (English) Zbl 1485.35020
In this paper, the authors study a parabolic Allen-Cahn equation with a nonlinear Robin condition in a bounded smooth domain in \(\mathbb{R}^2\). The limiting problem is the mean curvature flow with a contact angle \(\alpha\) on the boundary. When \(\alpha\) is close to \(\pi/2\), by assuming the limiting problem has a local smooth solution, they prove that solutions to the Allen-Cahn equation converges to the limit in a smooth way. This is done by taking an expansion of the solution in \(\varepsilon\) and then estimating the difference. For this purpose, some linearized estimates are developed, and the restriction on \(\alpha\) arises from the linearization problem at the boundary contact point.
Reviewer: Kelei Wang (Wuhan)
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35B36 | Pattern formations in context of PDEs |
| 35K57 | Reaction-diffusion equations |
| 35K61 | Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations |
| 35R37 | Moving boundary problems for PDEs |
| 53E10 | Flows related to mean curvature |
Keywords:
sharp interface limit; Allen-Cahn equation; Robin boundary condition; mean curvature flow; contact angleReferences:
| [1] | H. Abels and Y. Liu, Sharp interface limit for a Stokes/Allen-Cahn system, Arch. Ration. Mech. Anal., 229 (2018), pp. 417-502. · Zbl 1394.35343 |
| [2] | H. Abels and M. Moser, Convergence of the Allen-Cahn equation to the mean curvature flow with 90°-contact angle in 2D, Interfaces Free Bound., 21 (2019), pp. 313-365. · Zbl 1430.35127 |
| [3] | R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed., Elsevier, Amsterdam, 2003. · Zbl 1098.46001 |
| [4] | N. D. Alikakos, P. W. Bates, and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Ration. Mech. Anal., 128 (1994), pp. 165-205. · Zbl 0828.35105 |
| [5] | S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), pp. 1085-1095. |
| [6] | H. W. Alt, Linear Functional Analysis, An Application-Oriented Introduction, Springer, London (2016). · Zbl 1358.46002 |
| [7] | H. Amann and J. Escher, Analysis III. Birkhäuser, Basel, 2009. · Zbl 1187.28001 |
| [8] | L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Ration. Mech. Anal., 124 (1993), pp. 355-379. · Zbl 0785.76085 |
| [9] | G. Caginalp, X. Chen, and C. Eck, A rapidly converging phase field model, Discrete Contin. Dyn. Syst., 15 (2006), pp. 1017-1034. · Zbl 1119.80009 |
| [10] | X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), pp. 116-141. · Zbl 0765.35024 |
| [11] | X. Chen, Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces, Comm. Partial Differential Equations, 19 (1994), pp. 1371-1395. · Zbl 0811.35098 |
| [12] | X. Chen, D. Hilhorst, and E. Logak, Mass conserving Allen-Cahn equation and volume preserving mean curvature flow, Interfaces Free Bound., 12 (2010), pp. 527-549. · Zbl 1219.35018 |
| [13] | P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), pp. 207-220. · Zbl 0840.35010 |
| [14] | D. Depner, Stability Analysis of Geometric Evolution Equations with Triple Lines and Boundary Contact, Ph.D thesis, University of Regensburg, Regensburg, Germany, 2010. |
| [15] | L. C. Evans, Partial Differential Equations, 2nd ed., American Mathematical Society, Providence, RI, 2010. · Zbl 1194.35001 |
| [16] | L. C. Evans, H. M. Soner, and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), pp. 1097-1123. · Zbl 0801.35045 |
| [17] | J. Fischer, T. 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 (2020), pp. 6222-6233. · Zbl 1454.53079 |
| [18] | T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, J. Differential Geom., 38 (1993), pp. 417-461. · Zbl 0784.53035 |
| [19] | T. Kagaya, Convergence of the Allen-Cahn equation with a zero Neumann boundary condition on non-convex domains, Math. Ann., 373 (2019), pp. 1485-1528. · Zbl 1421.35192 |
| [20] | T. Kagaya and Y. Tonegawa, A singular perturbation limit of diffused interface energy with a fixed contact angle condition, Indiana Univ. Math. J., 67 (2018), pp. 1425-1437. · Zbl 1403.35027 |
| [21] | M. Katsoulakis, G. T. Kossioris, and F. Reitich, Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn Model for phase transitions, J. Geom. Anal., 5 (1995), pp. 255-279. · Zbl 0827.35003 |
| [22] | T. Laux and T. M. Simon, Convergence of the Allen-Cahn equation to multiphase mean curvature flow, Comm. Pure Appl. Math., 71 (2018), pp. 1597-1647. · Zbl 1393.35122 |
| [23] | G. Leoni, A First Course in Sobolev Spaces, 2nd ed., American Mathematical Society, Providence, RI, 2017. · Zbl 1382.46001 |
| [24] | A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Springer, Basel, 1995. · Zbl 0816.35001 |
| [25] | A. Lunardi, E. Sinestrari, and W. Von Wahl, A semigroup approach to the time-dependent parabolic initial boundary value problem, Differential Integral Equations, 5 (1992), pp. 1275-1306. · Zbl 0758.34048 |
| [26] | M. Mizuno and Y. Tonegawa, Convergence of the Allen-Cahn equation with Neumann boundary conditions, SIAM J. Math. Anal., 47 (2015), pp. 1906-1932. · Zbl 1330.35196 |
| [27] | L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal., 98 (1987), pp. 123-142. · Zbl 0616.76004 |
| [28] | L. Modica, Gradient theory of phase transitions with boundary contact energy, Ann. Inst. H. Poincaré Non-Linéaire, 4 (1987), pp. 487-512. · Zbl 0642.49009 |
| [29] | M. Moser, Sharp Interface Limits for Diffuse Interface Models with Contact Angle, Ph.D. thesis, University of Regensburg, Regensburg, Germany, 2020. |
| [30] | M. Moser, Convergence of the Scalar- and Vector-Valued Allen-Cahn Equation to Mean Curvature Flow with 90°-Contact Angle in Higher Dimensions, preprint, https://arxiv.org/abs/2105.07100 (2021). |
| [31] | N. C. Owen and P. Sternberg, Gradient flow and front propagation with boundary contact energy, Proc. A, 437 (1992), pp. 715-728. |
| [32] | J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Birkhäuser, Basel, 2016. · Zbl 1435.35004 |
| [33] | S. Schaubeck, Sharp interface limits for diffuse interface models, Ph.D thesis, University of Regensburg, Regensburg, Germany, 2014. |
| [34] | H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. · Zbl 0387.46033 |
| [35] | H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983. · Zbl 0546.46028 |
| [36] | T. Vogel, Sufficient conditions for capillary surfaces to be energy minima, Pacific J. Math., 194 (2000), pp. 469-489. · Zbl 1021.58014 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
