Abstract
We study front propagation problems for forced mean curvature flows and their phase field variants that take place in stratified media, that is, heterogeneous media whose characteristics do not vary in one direction. We consider phase change fronts in infinite cylinders whose axis coincides with the symmetry axis of the medium. Using the recently developed variational approaches, we provide a convergence result relating asymptotic in time front propagation in the diffuse interface case to that in the sharp interface case, for suitably balanced nonlinearities of Allen-Cahn type. The result is established by using arguments in the spirit of Γ-convergence, to obtain a correspondence between the minimizers of an exponentially weighted Ginzburg-Landau type functional and the minimizers of an exponentially weighted area type functional. These minimizers yield the fastest traveling waves invading a given stable equilibrium in the respective models and determine the asymptotic propagation speeds for front-like initial data. We further show that generically these fronts are the exponentially stable global attractors for this kind of initial data and give sufficient conditions under which complete phase change occurs via the formation of the considered fronts.
Similar content being viewed by others
References
Alfaro M., Hilhorst D., Matano H.: The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system. J. Differ. Equ. 245, 505–565 (2008)
Alfaro M., Matano H.: On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete Cont. Dyn. Syst. B 17, 1639–1649 (2012)
Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metal. Mater. 27, 1085–1095 (1979)
Amar, M., De Cicco, V., Fusco, N.: Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands. ESAIM Control Opt. Calc. Var. 14, 456–477 (2008)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, New York, 2000
Aronson, D.G., Weinberger, H.F.: Multidimensional diffusion arising in population genetics. Adv. Math. 30, 33–58 (1978)
Barles, G., Cesaroni, A., Novaga, M.: Homogenization of fronts in highly heterogeneous media. SIAM J. Math. Anal. 43, 212–227 (2011)
Barles, G., Soner, H.M., Souganidis, P.E.: Front propagation and phase field theory. SIAM J. Control Optim. 31, 439–469 (1993)
Barles, G., Souganidis, P.E.: A new approach to front propagation problems: theory and applications. Arch. Rational Mech. Anal. 141, 237–296 (1998)
Berestycki, H., Nirenberg, L.: Travelling fronts in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéaire 9, 497–572 (1992)
Braides, A.: Γ-convergence for beginners. In: Oxford Lecture Series in Mathematics and its Applications, Oxford, 2002
Bronsard, L., Kohn, R.V.: Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics. J. Differ. Equ. 90, 211–237 (1991)
Caffarelli, L.A., Córdoba, A.: Uniform convergence of a singular perturbation problem. Comm. Pure Appl. Math. 48, 1–12 (1995)
Caginalp, G.: The role of microscopic anisotropy in the macroscopic behavior of a phase boundary. Ann. Phys. 172, 136–155 (1986)
Caginalp, G., Fife, P.C.: Phase-field methods for interfacial boundaries. Phys. Rev. B 33, 7792–7794 (1986)
Cardaliaguet, P., Lions, P.L., Souganidis, P.E.: A discussion about the homogenization of moving interfaces. J. Math. Pures Appl. 91, 339–363 (2009)
Cesaroni, A., Muratov, C.B., Novaga, M.: Asymptotic behavior of attractors for inhomogeneous Allen-Cahn equations. RIMS Kokyuroku, to appear
Cesaroni, A., Novaga, M.: Long-time behavior of the mean curvature flow with periodic forcing. Comm. Partial Differ. Equ. 38, 780–801 (2013)
Chen, X.: Generation and propagation of interfaces in reaction-diffusion equations: J. Differ. Equ. 96, 116–141 (1992)
De Mottoni, P., Schatzman, M.: Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347, 1533–1589 (1995)
Evans, L.C., Soner, H.M., Souganidis, P.E.: Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math. 45, 1097–1123 (1992)
Farina, A., Valdinoci, E.: Geometry of quasiminimal phase transitions. Calc. Var. Partial Differ. Equ. 33, 1–35 (2008)
Fife, P.C.: Pattern formation in reacting and diffusing systems. J. Chem. Phys. 64, 554–563 (1976)
Fife, P.C.: Dynamics of Internal Layers and Diffusive Interfaces. Society for Industrial and Applied Mathematics, Philadelphia, 1988
Fife, P.C., McLeod, J.B.: The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65, 335–361 (1977)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin 1983
Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser Verlag, Basel, 1984
Grüter, M.: Boundary regularity for solutions of a partitioning problem. Arch. Rational Mech. Anal. 97(3), 261–270 (1987)
Grüter, M.: Optimal regularity for codimension one minimal surfaces with a free boundary. Manuscripta Math. 58(3), 295–343 (1987)
Grüter, M.: Regularity results for minimizing currents with a free boundary. J. Reine Angew. Math. 375/376, 307–325 (1987)
Ilmanen T.: Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38, 417–461 (1993)
Jones, C.K.R.T.: Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions. Rocky Mountain J. Math. 13, 355–364 (1983)
Lucia, M., Muratov, C.B., Novaga, M.: Existence of traveling wave solutions for Ginzburg-Landau-type problems in infinite cylinders. Arch. Rat. Mech. Anal. 188, 475–508 (2008)
Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems. Cambridge Studies in Advanced Mathematics, Cambridge, 2012
Modica, L., Mortola, S.: Un esempio di Γ -convergenza. Bollettino U.M.I. 14-B, 285–299 (1977)
Mugnai, L., Röger, M.: Convergence of perturbed Allen-Cahn equations to forced mean curvature flow. Indiana Univ. Math. J. 60, 41–75 (2011)
Muratov, C.B.: A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type. Discrete Cont. Dyn. Syst. B 4, 867–892 (2004)
Muratov, C.B., Novaga, M.: Front propagation in infinite cylinders I. A variational approach. Comm. Math. Sci. 6, 799–826 (2008)
Muratov, C.B., Novaga, M.: Front propagation in infinite cylinders. II. The sharp reaction zone limit. Calc. Var. Partial Differ. Equ. 31, 521–547 (2008)
Muratov, C.B., Novaga, M.: Global exponential convergence to variational traveling waves in cylinders. SIAM J. Math. Anal. 44, 293–315 (2012)
Novaga, M., Valdinoci, E.: The geometry of mesoscopic phase transition interfaces. Discrete Cont. Dyn. Syst. A 19, 777–798 (2008)
Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer, New York 1984
Rubinstein, J., Sternberg, P., Keller, J.B.: Fast reaction, slow diffusion, and curve shortening. SIAM J. Appl. Math. 49, 116–133 (1989)
Taylor, J.E.: Boundary regularity for solutions to various capillarity and free boundary problems. Comm. Partial Differ. Equ. 2, 323–357 (1977)
Vega J.M.: The asymptotic behavior of the solutions of some semilinear elliptic equations in cylindrical domains. J. Differ. Equ. 102, 119–152 (1993)
Xin J.: Front propagation in heterogeneous media. SIAM Rev. 42, 161–230 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. Otto
Rights and permissions
About this article
Cite this article
Cesaroni, A., Muratov, C.B. & Novaga, M. Front Propagation in Geometric and Phase Field Models of Stratified Media. Arch Rational Mech Anal 216, 153–191 (2015). https://doi.org/10.1007/s00205-014-0804-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-014-0804-3