Abstract
We describe a new theory for studying the geometric motion of interfaces past the first times singularities develop. We also show how this theory allows one to justify and study the development of such interfaces in a wide range of applications, from the asymptotics of recation-diffusion equations and systems and hydrodynamic limits of particles systems in phase transitions to turbulent combustion.
Partially supported by the NSF.
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References
S. M. Allen and J. W. Cahn, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallica 27 (1979), 1085–1095.
L. Alvarez, F. Guichard, P.-L. Lions, and J.-M. Morel, Axioms and fundamental equations of image processing, Arch. Rational Mech. Anal. 123 (1992), 199–257.
S. Angenent, D. L. Chopp, and T. Ilmanen, A computed example of nonuniqueness of mean curvature flow is ℝ3, Comm. Partial Differential Equations 20 (1995), 1937–1958.
S. Angenent, T. Ilmanen, and J. L. Velazquez, Nonuniqueness in geometric heat flows, preprint.
D. G. Aronson and H. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math. 30 (1978), 33–76.
G. Barles, Remark on a flame propagation model, Rapport INRIA 464 (1985).
G. Barles, L. Bronsard, and P. E. Souganidis, Front propagation for reaction-diffusion equations of bistable type, Anal. Nonlin. 9 (1992), 479–506.
G. Barles, L. C. Evans, and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE, Duke Math. J. 61 (1990), 835–858.
G. Barles, H. M. Soner, and P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim. 31 (1993), 439–469.
G. Barles and P. E. Souganidis, Convergence of approximation schemes for full nonlinear equations, Asymptotic Anal. 4 (1989), 271–283.
G. Barles and P. E. Souganidis, A new approach to front propagation: Theory and applications, Arch. Rational Mech. Anal. 141 (1998), 237–296.
G. Barles and P. E. Souganidis, A remark on the asymptotic behavior of the solution of the KPP equation, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), 679–684.
E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi Equations with convex Hamiltonians, to appear in Comm. Partial Differential Equations.
P. Bates, P. Fife, X. Ren, and X. Wang, Traveling waves in a convolution model for phase transitions, preprint.
G. Bellettini and M. Paolini, Two examples of fattening for the mean curvature flow with a driving force, Math. Appl. 5 (1994), 229–236.
G. Bellettini and M. Paolini, Some results on minimal barriers in the sense of DeGiorgi to driven motion by mean curvature, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 19 (1995), 43–67.
G. Bellettini and M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J. 25 (1996), 537–566.
L. Bonaventura, Motion by curvature in an interacting spin system, preprint.
K. A. Brakke, The Motion of a Surface by Its Mean Curvature, Princeton University Press, Princeton, NJ, 1978.
L. Bronsard and R. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau model, J. Differential Equations 90 (1991), 211–237.
G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. 92 (1986), 205–245.
G. Caginalp, Mathematical models of phase boundaries, in: Material Instabilities in Continuum Mechanics and Related Mathematical Problems (J. Ball, ed.), Clarendon Press, Oxford, 1988, 35–52.
X.-Y. Chen, Generation and propagation of the interface for reaction-diffusion equation, J. Differential Equations 96 (1992), 116–141.
Y.-G. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), 749–786.
M. G. Crandall, Viscosity solutions: a primer, in: Viscosity Solutions and Applications (I. Capuzzo Dolcetta and P.-L. Lions, eds.), Lecture Notes in Math. 1660, Springer-Verlag, Berlin, 1997, 1–43.
M. G. Crandall, H. Ishii, and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 1–67.
M. G. Crandall, P.-L. Lions, and P. E. Souganidis, Universal bounds and maximal solutions for certain evolution equations, Arch. Rational Mech. Anal. 105 (1989), 163–190.
E. DeGiorgi, Some conjectures on flow by mean curvature, Proc. Capri Workshop 1990 (Benevento, Bruno, and Sbardone, eds.).
A. DeMasi, P. Ferrari, and J. Lebowitz, Reaction-diffusion equations for interacting particle systems, J. Statist. Phys. 44 (1986), 589–644.
A. DeMasi, E. Orlandi, E. Presutti, and L. Triolo, Glauber evolution with Kač potentials: I. Mesoscopic and macroscopic limits, interface dynamics, Nonlinearity 7 (1994), 633–696.
A. DeMasi, E. Orlandi, E. Presutti, and L. Triolo, Motion by curvature by scaling non-local evolution equations, J. Statist Phys. 73 (1993), 543–570.
A. DeMasi, E. Orlandi, E. Presutti, and L. Triolo, Stability of the interface in a model of phase separation, Proc. Royal Soc. Edinburgh Sect. A 124 (1994), 1013–1022.
A. De Masi and E. Presutti, Mathematical Methods for Hydrodynamic Limits,Lecture Notes in Math. 1501, Springer-Verlag, Berlin, 1991.
P. DeMottoni and M. Schatzman, Development of interfaces in ℝN, Proc. Royal Soc. Edinburgh Sect. A 116 (1990), 207–220.
P. F. Embid, A. Majda, and P. E. Souganidis, Effective geometric front dynamics for premixed turbulent combustion with separated velocity scales, Combustion Sci. Techn. 103 (1994), 85–115.
P. F. Embid, A. Majda and P. E. Souganidis, Comparison of turbulent flame speeds from complete averaging and the G-equation, Phys. Fluids 7 (1995), 2052–2060.
P. F. Embid, A. Majda, and P. E. Souganidis, Examples and counterexamples for Huygens Principle in premixed combustion, Combustion Sci. Techn. 120 (1996), 273–303.
L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Royal Soc. Edinburgh Sect. A 111 (1989), 359–375.
L. C. Evans, Regularity for fully nonlinear elliptic equations and motion by mean curvature, in: Viscosity Solutions and Applications (I. Capuzzo Dolcetta and P.-L. Lions, eds.), Lecture Notes in Math. 1660, Springer-Verlag, Berlin, 1997, 98–133.
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.
L. C. Evans and P. E. Souganidis, A PDE approach to geometric optics for certain reaction-diffusion equations, Indiana Univ. Math. J. 38 (1989), 141–172.
L. C. Evans and P. E. Souganidis, A PDE approach to certain large deviation problems for systems of parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (Suppl.) (1994), 229–258.
L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom. 33 (1991), 635–681.
P. C. Fife, Nonlinear diffusive waves, CMBS Conf., Utah 1987, CMBS Conf. Series (1989).
P. C. Fife and B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling solutions, Arch. Rational Mech. Anal 65 (1977), 335–361.
M. Freidlin, Functional Integration and Partial Differential Equations, Ann. of Math. Stud. 109, Princeton University Press, Princeton, NJ, 1985.
M. Freidlin, Limit theorems for large deviations of reaction-diffusion equations,Ann. Probab. 13 (1985), 639–675.
M. Freidlin and Y. T. Lee, Wave front propagation for a class of space non-homogeneous reaction-diffusion systems, preprint.
J. Gärtner, Bistable reaction-diffusion equations and excitable media, Math.Nachr. 112 (1983), 125–152.
Y. Giga and M.-H. Sato, Generalized interface evolution with Neumann boundary condition, Proc. Japan Acad. Ser. A Math. Sci. 67 (1991), 263–266.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. 224, Springer-Verlag, New York, 1983.
S. Goto, Generalized motion of hypersurfaces whose growth speed depends super-linearly on curvature tensor, J. Differential Integral Equations 7 (1994), 323–343.
W. Grabowski, Cumulus entrainment, fine scale mixing, and bouyancy reversal,Quart. J. Roy. Meteorol. Soc. 119 (1993), 935–956.
W. Grabowski and P. Smolarkiewicz, Two-time level semi-Lagrangian modeling of precipitating clouds, Monthly Weather Rev. 124 (1996), 487–497.
M. E. Gurtin, Multiphase thermodynamics with interfacial structure. 1. Heat conduction and the capillary balance law, Arch. Rational Mech. Anal. 104 (1988), 185–221.
M. E. Gurtin, H. M. Soner, and P. E. Souganidis, Anisotropic motion of an interface relaxed by the formation of infinitesimal wrinkles, J. Differential Equations 119 (1995), 54–108.
T. Ilmanen, The level-set flow on a manifold, in: Differential Geometry: Partial Differential Equations on Manifolds (R. Greene and S.-T. Yau, eds.), Proc. Sympos. Pure Math. 54, Amer. Math. Soc, Providence, RI, 1993, 193–204.
T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, J. Differential Geom. 38 (1993), 417–461.
H. Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Engrg. Chuo Univ. Ser. I Math. 26 (1985), 5–24.
H. Ishii, Parabolic pde with discontinuities and evolution of interfaces, Adv. Differential Equations 1 (1996), 51–72.0
H. Ishii, G. Pires, and P. E. Souganidis, Threshold dynamics type approximation schemes for propagating fronts, J. Math. Soc. Japan 51 (1999), 267–308.
H. Ishii and P. E. Souganidis, Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor, Tohoku Math. J. 47 (1995), 227–250.
R. Jensen, P.-L. Lions, and P. E. Souganidis, A uniqueness result for viscosity solutions of second-order fully nonlinear pde’s, Proc. Amer. Math. Soc. 102 (1988), 975–978.
R. Jerrard, Fully nonlinear phase transitions and generalized mean curvature motions,Comm. Partial Differential Equations 20 (1995), 223–265.
M. Katsoulakis, G. Kossioris, and F. Reitich, Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions, to appear in J. Geom. Anal..
M. Katsoulakis and P. E. Souganidis, Interacting particle systems and generalized mean curvature evolution, Arch. Rational Mech. Anal 127 (1994), 133–157.
M. Katsoulakis and P. E. Souganidis, Generalized motion by mean curvature as a macroscopic limit of stochastic Ising models with long range interactions and Glauber dynamics, Comm. Math. Phys. 169 (1995), 61–97.
M. Katsoulakis and P. E. Souganidis, Stochastic Ising models and anisotropic front propagation, to appear in J. Statist. Phys..
I. C. Kim, Singular limits of chemotaxis — growth model, Nonlinear Anal. 46 (2001), 817–834.
A. Linan and F. Williams, Fundamental Aspects of Combustion, Oxford University Press, 1993, Chapters 3 and 5.
J. Lebowitz and O. Penrose, Rigorous treatment of the Van der Waals Maxwell theory of the liquid vapour transition, J. Math. Phys. 98 (1966), 98–113.
T. Liggett, Interacting Particle Systems, Grundlehren Math. Wiss. 276, Springer-Verlag,New York, 1985.
P.-L. Lions, G. Papanicolaou, and S. R. S. Varadhan, Homogenization of the Hamilton-Jacobi equations, preprint.
P.-L. Lions and P. E. Souganidis, Viscosity solutions of fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 1085–1092.
P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic partial differential equations: Nonsmooth equations and applications, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 735–741.
P.-L. Lions and P. E. Souganidis, Uniqueness of weak solutions for fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 783–790.
P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic partial differential equations with semilinear stochastic dependence, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 617–624.
A. Majda and P. E. Souganidis, Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity 7 (1994), 1–30.
A. Majda and P. E. Souganidis, Bounds on enhanced turbulent flame speeds for combustion with fractal velocity fields, J. Statist. Phys. 83 (1996), 933–954.
A. Majda and P. E. Souganidis, Flame fronts in a turbulent combustion model with fractal velocity fields, Comm. Pure Appl. Math. 51 (1998), 1337–1348.
A. Majda and P. E. Souganidis, The effect of turbulence on mixing in prototype reaction diffusion systems, Comm. Pure Appl. Math. 53 (2000), 1284–1304.
R. H. Nochetto, M. Paolini, and C. Verdi, Optimal interface error estimates for the mean curvature flow, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), 193–212.
T. Ohta, D. Jasnow, and K. Kawasaki, Universal scaling in the motion of random intervaces, Phys. Rev. Lett. 49 (1982), 1223–1226.
S. Osher and J. Sethian, Fronts moving with curvature dependent speed: Algorithms based on Hamilton-Jacobi equations, J. Comput. Phys. 79 (1988), 12–49.
J. Rubinstein, P. Sternberg, and J. B. Keller, Fast reaction, slow diffusion and curve shortening, SIAM J. Appl. Math. 49 (1989), 116–133.
H. M. Soner, Motion of a set by the curvature of its boundary, J. Differential Equations 101 (1993), 313–372.
H. M. Soner, Ginzburg-Landau equation and motion by mean curvature, I: Convergence, J. Geom. Anal. 7 (1997), 437–475, II to appear ibid.
H. M. Soner and P. E. Souganidis, Uniqueness and singularities of rotationally symmetric domains moving by mean curvature, Comm. Partial Differential Equations 18 (1993), 859–894.
P. Soravia, Generalized motion of a front along its normal direction: A differential games approach, Nonlinear Anal. 22 (1994), 1242–1262.
P. Soravia and P. E. Souganidis, Phase field theory for FitzHugh-Nagumo type systems, SIAM J. Math. Anal. 42 (1996), 1341–1359.
P. E. Souganidis, Front propagation: theory and application, in: Viscosity Solutions and Applications (I. Capuzzo Dolcetta and P.-L. Lions, eds.), Lecture Notes in Math. 1660, Springer, Berlin, 1997, 186–242.
P. E. Souganidis, Stochastic Homogenization for Hamilton-Jacobi equations and applications, Asymptotic Anal 20 (1999), 1–11.
H. Spohn, Large Scale Dynamics of Interacting Particles, Springer-Verlag, New York, 1991.
H. Spohn, Interface motion in models with stochastic dynamics, J. Statist Phys.71 (1993), 1081–1132.
F. Williams, Combustion Theory, 2nd ed., Addison-Wesley.
J. X. Xin, Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media, J. Statist Phys. 73 (1993), 893–926.
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Souganidis, P.E. (2002). Recent developments in the theory of front propagation and its applications. In: Bourlioux, A., Gander, M.J., Sabidussi, G. (eds) Modern Methods in Scientific Computing and Applications. NATO Science Series, vol 75. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0510-4_11
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