Abstract
In this paper we study the asymptotic behavior (∈→0) of the Ginzburg-Landau equation:
. where the unknownu ∈ is a real-valued function of [0. ∞)×Rd, and the given nonlinear functionf(u) = 2u(u 2−1) is the derivative of a potential W(u) = (u 2−l)2/2 with two minima of equal depth. We prove that there are a subsequence ∈n and two disjoint, open subsetsP, N of (0, ∞) ×R d satisfying
uniformly inP andN (here 1 A is the indicator of the setA). Furthermore, the Hausdorff dimension of the interface Γ = complement of (P∪N) ⊂ (0, ∞)×R d is equal tod and it is a weak solution of the mean curvature flow as defined in [13,92]. If this weak solution is unique, or equivalently if the level-set solution of the mean curvature flow is “thin,” then the convergence is on the whole sequence. We also show thatu ∈n has an expansion of the form
whereq(r) = tanh(r) is the traveling wave associated to the cubic nonlinearityf, O(∈) → 0 as ∈ → 0, andd(t, x) is the signed distance ofx to thet-section of Γ. We prove these results under fairly general assumptions on the initial data,u 0. In particular we donot assume thatu ∈(0.x) = q(d(0,x)/∈), nor that we assume that the initial energy, ε∈(u ∈(0, .)), is uniformly bounded in ∈. Main tools of our analysis are viscosity solutions of parabolic equations, weak viscosity limit of Barles and Perthame, weak solutions of mean curvature flow and their properties obtained in [13] and Ilmanen’s generalization of Huisken’s monotonicity formula.
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References
Allen, S., and Cahn, J. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening.Acta Metall. 27, 1084–1095 (1979).
Alikakos, N. D., Bates, P. W., and Chen, X. Convergence of the Cahn-Hilliard equation to the Hele-Shaw model.Arch. Rat. Mech. Anal. 128, 165–205 (1994).
Alikakos, N. D., Bates, P. W., and Fusco, G. Slow motion for the Cahn-Hilliard equation in one space dimension.J. Diff. Equations 90, 81–135 (1991).
Almgren, F., and Taylor, J. E. Flat flow is motion by crystalline curvature for curves with crystalline energies.J. Differential Geom. 42, 1–22 (1995).
Almgren, F., Taylor, J. E., and Wang, L. Curvature driven flows: a variational approach.SIAM J. Cont. and Opt. 31, 387–438 (March 1993). Issue dedicated to W. H. Fleming.
Almgren, F., and Wang, L. Mathematical existence of crystal growth with Gibbs-thompson curvature effects. Forthcoming.
Angenent, S. Parabolic equations for curves on surfaces. I. Curves with p-integrable curvature.Ann. Math. 132, 451–483(1990).
Angenent, S. Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions.Ann. Math. 133, 171–217(1991).
Barles, G. Remark on a flame propogation model. Rapport INRIA # 464 (1985).
Barles, G., Bronsard, L., and Souganidis, P. E. Front propogation for reaction-diffusion equations of bi-stable type.Ann. I.H.P. Anal. Nonlin. 9, 479–496 (1992).
Barles, G., and Georgelin, C. A simple proof of convergence for an approximation scheme for computing motions by mean curvature.SIAM J. Num. Anal. 32, 484–500(1995).
Barles, G., and Perthame, B. Discontinuous solutions of deterministic optimal stopping problems.Math. Modelling Numerical Analysis 21, 557–579 (1987).
Barles, G., Soner, H. M., and Souganidis, P. E. Front propagation and phase field theory.SIAM. J. Cont. Opt. (March 1993). Issue dedicated to W. H. Fleming.
Bence, J. Merriman, B., and Osher, S. Diffusion generated motion by mean curvature. Preprint (1992).
Blowey, J. F., and Elliot, C. M. Curvature dependent phase boundary motion and parabolic double obstacle problems.IMA 47, 19–60. Springer, New York, 1993.
Bonaventura, L. Motion by curvature in an interacting spin system. Preprint (1992).
Brakke, K.A.The Motion of a Surface by Its Mean Curvature. Princeton University Press, Princeton, NJ, 1978.
Bronsard, L., and Kohn, R. On the slowness of the phase boundary motion in one space dimension.Comm. Pure Appl. Math. 43, 983–998 (1990).
Bronsard, L., and Kohn, R. Motion by mean curvature as the singular limit of Ginzburg-Landau model.Jour. Diff. Equations 90, 211–237 (1991).
Bronsard, L., and Reitich, F. On the three-phase boundary motion and the singular limit of a vector-valued Ginzburgh-Landau equation.Arch. Rat. Mech. An. 124, 355–379 (1993).
Caginalp, G. Surface tension and supercooling in solidification theory.Lecture Notes in Physics 216, 216–226 (1984).
Caginalp, G. An analysis of a phase field model of a free boundary.Arch. Rat. Mech. An. 92, 205–245 (1986).
Caginalp, G. Stefan and Hele Shaw type models as asymptotic limits of the phase-field equations.Physical Review A 39/11, 5887–5896(1989).
Caginalp, G., and Chen, X. Phase field equations in the singular limit of sharp interface equations. Preprint (1992).
Caginalp, G., and Fife, P. Dynamics of layered interfaces arising from phase boundaries.SIAM J. Appl. Math. 48, 506–518 (1988).
Caginalp, G., and Socolovsky, E. A. Efficient computation of a sharp interface by spreading via phase field methods.Appl. Math. Let. 2/2, 117–120 (1989).
Chen, X. Generation and propagation of the interface for reaction-diffusion equations.Jour. Diff. Equations 96, 116–141 (1992).
Chen, X. Spectrums for the Allen-Cahn, Cahn-Hilliard and phase-field equations for generic interfaces. Preprint (1993).
Chen, X., and Elliot, C. M. Asymptotics for a parabolic double obstacle problem. Preprint (1991).
Chen, Y.-G., Giga, Y., and Goto, S. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations.J. Differential Geometry 33, 749–786 (1991). (Announcement:Proc. Japan Academy Ser. A 67/10, 323–328 (1991).)
Chen, Y.-G., Giga, Y., and Goto, S. Analysis toward snow crystal growth.Proc. Func. Analysis and Rel. Topics (S. Koshi, ed.) Sapporo, 1990. To appear.
Crandall, M. G., Evans, L. C., and Lions, P.-L. Some properties of viscosity solutions of Hamilton-Jacobi equations.Trans. AMS 282, 487–502 (1984).
Crandall, M. G., Ishii, H., and Lions, P.-L. User’s guide to viscosity solutions of second order partial diffrential equations.Bull. AMS 27/1, 1–67 (1992).
Crandall, M. G., and Lions, P.-L. Viscosity solutions of Hamilton-Jacobi equations.Trans. AMS 277, 1–43 (1983).
Collins, J. B., and Levine, H. Diffuse interface model of diffusion-limited crystal growth.Phys. Rev. B. 31, 6119–6122 (1985).
Dang, H., Fife, P. C., and Peletier, L. A. Saddle solutions of the bi-stable diffusion equation. Preprint (1992).
DeGiorgi, E. Some conjectures on flow by mean curvature.Proceedings of Capri Workshop (1990).
Evans, L. C. Convergence of an algorithm for mean curvature motion. Preprint (1993).
Evans, L. C., and Spruck, J. Motion of level sets by mean curvature.J. Differential Geometry 33, 635–681 (1991).
Evans, L. C., and Spruck, J. Motion of level sets by mean curvature II.Trans. AMS 330, 635–681 (1992).
Evans, L. C., and Spruck, J. Motion of level sets by mean curvature III.J. Geom. Analysis 2, 121–150 (1992).
Evans, L. C., and Spruck, J. Motion of level sets by mean curvature IV.J. Geom. Analysis 5, 77–114 (1995).
Evans, L. C., Soner, H. M., and Souganidis, P. E. Phase transitions and generalized motion by mean curvature.Comm. Pure Appl. Math. 45, 1097–1123 (1992).
Fife, P.C. Dynamics of internal layers and diffusive interfaces.CBMS-NSF Regional Conference Series in Applied Math. 53, (1988), SIAM, Philadelphia.
Fife, P. C., and McLeod, B. The appoarch of solutions of nonlinear diffusion equation to travelling front solutions.Arc. Rat. Mech. An. 65, 335–361 (1977).
Fix, G. Phase field methods for free boundary problems. InFree Boundary Problems: Theory and Applications (B. Fasano and M. Primicerio, eds.). Pitman, London 1983, pp. 580–589.
Fleming, W. H., and Soner, H. M.Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, New York, 1993.
Fonseca, I., and Tartar, L. The gradient theory of phase transitions for systems with two potential wells.Proc. Royal Soc. Edinburgh Sect. A 111, 89–102 (1989).
Fried, E., and Gurtin, M. Continuum theory of thermally induced phase transitions based on an order parameter.Physica D 68, 326–343 (1993).
Gärtner, J. Bistable reaction-diffusion equations and excitable media.Math. Nachr. 112, 125–152 (1983).
Gage, M., and Hamilton, R. S. The heat equation shrinking convex plane curves.J. Differential Geometry 23, 69–95 (1986).
Giga, Y., and Goto, S. Motion of hypersurfaces and geometric equations.J. Math. Soc. Japan 44/1, 99–111 (1992).
Giga, Y., Goto, S., Ishii, H., and Sato, M. H. Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains.Indiana Math. J. 40/2, 443–470 (1991).
Giga, Y., and Sato, M. H. Generalized interface condition with the Neumann boundary condition.Proc. Japan Acad. Ser. A Math 67, 263–266 (1991).
Grayson, M. A. The heat equation shrinks embedded plane curves to round points.J. Differential Geometry 26, 285–314 (1987).
Grayson, M. A. A short note on the evolution of surfaces via mean curvature.Duke Math. J. 58, 555–558 (1989).
Gurtin, M. E. Multiphase thermomechanics with interfacial structure 1. Heat conduction and the capillary balance law.Arch. Rat. Mech. An. 104, 195–221 (1988).
Gurtin, M. E. Multiphase thermomechanics with interfacial structure. Towards a nonequilibrium thermomechanics of two phase materials.Arch. Rat. Mech. An. 104, 275–312 (1988).
Gurtin, M. E.Thermodynamics of Evolving Phase Boundaries in the Plane. Oxford University Press, 1993.
Gurtin, M. E., Soner, H. M., and Souganidis, P. E. Anisotropic motion of an interface relaxed by the formation of infinitesimal wrinkles.Journal of Differential Equations 119, 54–108 (1995).
Huisken, G. Asymptotic behavior for singularities of the mean curvature flow.J. Differential Geometry 31, 285–299 (1990).
Ilmanen, T. Convergence of the Allen-Cahn equation to the Brakke’s motion by mean curvature.J. Differential Geom. 38, 417–461(1993).
Ilmanen, T. Elliptic regularization and partial regularity for motion by mean curvature.Mem. AMS 108, (1994).
Ilmanen, T. Generalized motion of sets by mean curvature on a manifold.Univ. Indiana Math. J. 41, 671–705 (1992).
Ishii, H. A simple direct proof for uniqueness for solutions of the Hamilton-Jacobi equations of the Eikonial type.Proc. AMS. 100, 247–251 (1987).
Ishii, H., and Souganidis, P. E. Forthcoming.
Jensen, R. The maximum principle for viscosity solutions of second order fully nonlinear partial differential equations.Arch. Rat. Mech. An. 101, 1–27 (1988).
Katsoulakis, M., Kossioris, G., and Reitrich, F. Generalized motion by mean curvature with Neumann condition and the Allen-Cahn model for phase transitions. Preprint (1992).
Katsoulakis, M., and Souganidis, P. E. Interacting particle systems and generalized mean curvature evolution. Preprint (1992).
Langer, J. S. Unpublished notes (1978). 475
Lasry, J. M., and Lions, P.-L. A remark on regularization in Hilbert spaces.Israel J. Math. 551, 257–266 (1988).
Luckhaus, S. Solutions of the two phase Stefan problem with the Gibbs-Thompson law for the melting temperature.European J. Appl. Math. 1, 101–111 (1990).
DeMasi, A., Orlandi, E., Presutti, E., and Triolo, L. Motion by curvature by scaling non local evolution equations. Preprint (1993).
Modica, L. Gradient theory of phase transitions and the minimal interface criteria.Arch. Rat. Mech. An. 98, 123–142 (1987).
Modica, L., and Mortola, S. Il limite nella Γ-convergenza di una famiglia di funzionali ellittici.Boll. Un. Math. Ital. A14(1977).
deMottoni, P., and Schatzman, M. Geometrical evolution of developed interfaces. Trans. AMS. To appear. (Announcement: Evolution géométric d’interfaces.C.R. Acad. Sci. Sér. I. Math. 309, 453–458 (1989).
deMottoni, P., and Schatzman, M. Development of surfaces inR d.Proc. Royal Edinburgh Sect. A 116, 207–220 (1990).
Mullins, W. Two dimensional motion of idealized grain boundaries.J. Applied Physics 27, 900–904 (1956).
Nochetto, R. H., Paolini, M., and Verdi, C. Optimal interface error estimates for the mean curvature flow.Ann. Scuola Nor. Pisa 21, 193–212 (1994).
Ohnuma, M., and Sato, M. Singular degenerate parabolic equations with applications to geometric evolutions. Preprint (1992).
Osher, S., and Sethian, J. Fronts propogating with curvature dependent speed.J. Comp. Phys. 79, 12–49 (1988).
Otha, T., Jasnow, D.,and Kawasaki, K. Universal scaling in the motion of a random interface.Physics Review Letters 49, 1223–1226 (1982).
Owen, N., Rubinstein, J., and Sternberg, P. Minimizers and gradient flow for singularly perturbed bi-stable potentials with a Dirichlet condition.Proc. Royal Soc. London, A,429, 505–532 (1990).
Pego, R. L. Front migration in the nonlinear Cahn-Hilliard equation.Proc. Roy. Soc. London A 422, 261–278 (1989).
Penrose, O., and Fife, P. Theormodynamically consistent models for the kinetics of phase transitions.Physica D 43, 44–62 (1990).
Rubinstein, J., Sternberg, P., and Keller, J. B. Fast reaction, slow diffusion and curve shortening.SIAM J. Appl. Math. 49, 116–133 (1989).
Rubinstein, J., Sternberg, P., and Keller, J. B. Reaction diffusion processes and evolution to harmonic maps.SIAM J. Appl. Math. 49, 1722–1733 (1989).
Rubinstein, J., and Sternberg, P. Nonlocal reaction diffusion equations and nucleation.J. IMA. To appear.
Schatzman, M. On the stability of the saddle solution of Allen-Cahn’s equation. Preprint (1993).
Sethian, J. Curvature and evolution of fronts.Comm. Math. Physics 101, 487–495 (1985).
Sethian, J., and Strain, J. Crystal growth and dentritic solidification.Jour. Comp. Physics 2, 231–253 (1992).
Soner, H. M. Motion of a set by the curvature of its boundary.Jour. Diff. Equations 101/2, 313–372 (1993).
Soner, H. M. Convergence of the phase-field equations to the Mullins-Sekerka problem with kinetic undercooling.131, 139–197 (1995).
Soner, H. M. Ginzburg-Landau equation and motion by mean curvature, II: Development of the initial interface.J. Geom. Anal. 7, 000–000 (1997).
Sternberg, P. The effect of a singular perturbation on nonconvex variational problems.Arch. Rat. Mech. An. 101, 209–260 (1988).
Sternberg, P., and Ziemer, W. Generalized motion by curvature with a Dirichlet condition. Preprint (1992).
Sternberg, P., and Ziemer, W. Local minimizers of a three phase partition problem with triple junctions. Preprint (1993).
Stoth, B. A model with sharp interface as limit of phase-field equations in one space dimension.European J. Appl. Math.(1992). To appear.
Stoth, B. The Stefan problem with the Gibbs-Thompson law as singular limit of phase-field equations in the radial case.European J. Appl. Math. (1992). To appear.
Strain, J. A boundary integral approach to unstable solidification.Jour. Comp. Physics 85, 342–389 (1989).
Taylor, J. E. Motion of curves by crystalline curvature, including triple junctions and boundary points.Proc. Symp. Pure Math. To appear.
Taylor, J. E. The motion of multiple phase junctions under prescribed phase-boundary velocities. Preprint (1992).
Taylor, J. E., Cahn, J. W., and Handwerker, A. C. Geometric models of crystal growth.Acta Met. 40, 1443–1474 (1992).
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Communicated by David Kinderlehrer
Partially supported by the NSF grant DMS-9200801 and by the Army Research Office through the Center for Nonlinear Analysis.
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Saner, H.M. Ginzburg-Landau equation and motion by mean curvature, I: Convergence. J Geom Anal 7, 437–475 (1997). https://doi.org/10.1007/BF02921628
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DOI: https://doi.org/10.1007/BF02921628