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Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions

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Trends in Nonlinear Analysis

Abstract

An overview will be given of some nonlinear parabolic-like evolution problems which are off the classical beaten track, but have increased in importance during the past decade. The emphasis is on problems which are nonlocal, pattern-forming (including exhibiting propagative phenomena), and/or lead in some singular limit to free boundary problems. In all cases they have been proposed as models for phenomena in the natural sciences. Also emphasized are the relationships among these various trends.

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References

  1. N. Alikakos, P. Bates, and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rat. Mech. Anal. 128,165–205 (1994). [The object of study is (3.17) in the scaled form ut + A[e/u — e 1 f (u)] = 0 in a bounded domain. Assume the Mullins-Sekerka problem (3.22) has a classical solution in [0, T].It divides the domain into two subdomains. Then with appropriate initial and boundary conditions, there is a family of solutions (depending on e) converging as e-+0 to +1 in the two subdomains. From Math. Reviews 97b:35174. In this paper, the authors carry out a program of rigorous justification of the connection between the Cahn-Hilliard equation and the Hele-Shaw problem in an appropriate asymptotic limit which corresponds to a sharpening of the transition layer between phase… The Cahn-Hilliard equation, a fourth-order nonlinear partial differential equation, serves as a well-regarded model for the process of phase separation and coarsening in a melted alloy. It has been the subject of much work over the past ten to fifteen years. In particular, formal asymptotic analysis [106] established the connection between the level sets of the solution to the Cahn-Hilliard equation and interfacial motion in the Hele-Shaw (or Mullins-Sekerka) problem. For the time interval in which the Hele-Shaw problem possesses a classical solution, the present paper now makes this formal connection rigorous. To set this result in context, one can compare it to the large body of work devoted to an asymptotic analysis of the (second-order) Allen-Cahn equation—which tends to dissipate the same energy as the Cahn-Hilliard equation but which, unlike the Cahn-Hilliard equation, does not respect conservation of mass. The connection between the level sets of solutions to the Allen-Cahn equation and the problem of motion by mean curvature was made rigorous by Evans, Soner and Souganidis [49], using a weak formulation of the curvature flow via viscosity theory. At roughly the same time, de Mottoni and Schatzman [91] used a detailed spectral analysis to justify the formal asymptotics for the Allen-Cahn equation for as long as the mean curvature flow retained a classical solution. The paper under review is in spirit the Cahn-Hilliard cousin of this last reference. (We should note that more recently Cheri [39] developed a weak formulation of the Hele-Shaw problem using varifolds and carried out the asymptotic connection in this setting as well.) The procedure combines a delicate and nonstandard application of the method of matched asympotic expansions with detailed spectral analysis for the Cahn-Hilliard equation, previously worked out in [4, 37].]

    Google Scholar 

  2. N. Alikakos, P. Bates, and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Diff. Equations 90,81–135 (1991). [See Section 3.4.1.]

    Google Scholar 

  3. N. Alikakos, L. Bronsard, and G. Fusco, Slow motion in the gradient theory of phase transitions via energy and spectrum, Calculus of Variations 6, 39–66 (1998).

    MathSciNet  MATH  Google Scholar 

  4. N. Alikakos, G. Fusco, The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions, Indiana Mathematics Journal 42,637674 (1993). [See Section 3.4.1.]

    Google Scholar 

  5. N. Alikakos, G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions. Part I: Spectral estimates, Comm. in P.D.E. 19, 1397–1447 (1994).

    Google Scholar 

  6. N. Alikakos, G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: the motion of bubbles, Arch. Rat. Mech. Anal. 141, 1–61 (1998). [See Section 3.4.1.]

    Google Scholar 

  7. N. Alikakos, G. Fusco, The equations of Ostwald ripening for dilute systems, J. Stat. Phys. 95,851–866 (1999). [See Section 3.4.3.]

    Google Scholar 

  8. N. Alikakos, G. Fusco, Ostwald ripening for dilute systems under quasistationary dynamics, preprint. [This is an expanded version, with proofs, of [7]. See Section 3.4.3.]

    Google Scholar 

  9. S. M. Allen, J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta. Mater. 27, 10841095 (1979).

    Google Scholar 

  10. H. W. Alt and I. Pawlow, Existence of solutions for non-isothermal phase separation. Adv. Math. Sci. Appl. 1,319–409 (1992). [From Math. Reviews 95h:80003: A model describing the dynamics of nonisothermal phase separation of a two-component system is studied. The coupled concentration and temperature fields are governed by thermodynamically unstable mechanisms. The model has the form of a nonlinear parabolic system consisting of a fourth-order equation for the concentration and a second-order equation for the temperature. Existence of solutions is established. The proof is based on implicit time discretization of the problem and appropriate energy and time compactness estimates for the time-discrete solutions.]

    Google Scholar 

  11. H. W. Alt and I. Pawlow, A mathematical model of dynamics of nonisothermal phase separation, Physica D 59,389–416, (1992). [From Math. Reviews 92f:35080: A Landau-Ginzburg free energy functional approach to the kinetics of a phase transition in a binary system is presented. Thermodynamic compatibility of the model is demonstrated by showing that the Clausius-Duhem inequality is satisfied, and the linear stability of stationary solutions of the kinetic equations, giving the rates of change of mass and energy, is investigated.]

    Google Scholar 

  12. H. W. Alt and I. Pawlow, A mathematical model and an existence theory for nonisothermal phase separation, Numerical methods for free boundary problems Internat. Schriftenreihe Numer. Math. 99,1–32, Birkhäuser, Basel, 1991. [Summary: “We present a mathematical model for nonisothermal phase separation in binary systems. The model is constructed within the Landau-Ginzburg theory of phase transitions and nonequilibrium thermodynamics. It consists of a system of nonlinear parabolic differential equations for the order parameter and the energy as conserved quantities. The model is conformable with the first principles of thermodynamics. Besides a description of the model we study the stability of stationary solutions, and give a survey of the existence theory for the system of governing differential equations.”]

    Google Scholar 

  13. M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Phys. Rev. A 41,6763–6771 (1990). [See Sec. 3.8.1.]

    Google Scholar 

  14. G. Barles, P. E. Souganidis, A new approach to front propagation problems: theory and applications. Arch. Rational Mech. Anal. 141 (1998), no. 3, 237296. [The authors’ summary: “In this paper we present a new definition for the global-in-time propagation (motion) of fronts (hypersurfaces, boundaries) with a prescribed normal velocity, past the first time they develop singularities. We show that if this propagation satisfies a geometric maximum principle (inclusion-avoidance-type property), then the normal velocity must depend only on the position of the front, its normal direction and principal curvatures. This new approach, which is more geometric and, as it turns out, equivalent to the level-set method, is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction-diffusion equations. We finally present a number of new asymptotic results. Among them are the asymptotics of (i) reaction-diffusion equations with rapidly oscillating coefficients, (ii) fully nonlinear nonlocal (integral differential) equations, and (iii) stochastic Ising models with long-range anisotropic interactions and general spin-flip dynamics.”]

    Google Scholar 

  15. P. Bates and A. Chmaj,A discrete convolution model for phase transitions, Arch. Rat. Mech. Anal. 150 281–305 (1999). [From Math. Reviews 2001c:82026: In this interesting paper, the authors consider an infinite system of coupled ODEs úsn = (J * u)sn — usn — A (usn),where n E Z or, in some cases, n E Z d with d = 1,2, 3, • • •; {J(i)},-11,12,… is a given sequence such that Ez#o J(i) = 1 and (J * u) n = Ego J(i)u.i À 0 is a parameter, and f,as they indicate, is bistable. First, they demonstrate a connection of this system with a model of statistical mechanics and with a semilinear heat equation. Next, under certain hypotheses, they show the existence of travelling waves and investigate their properties, including the stability and the uniqueness of a travelling wave with a nonzero speed. In the last part of the paper, the authors consider the existence of stationary solutions, present a characterization of all such solutions, give a criterion for their stability and show the admissibility, for sufficiently large A0, of a property that they call pinning.]

    Google Scholar 

  16. P. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions, J. Stat. Phys. 95, 1119–1139 (1999). [From Math. Reviews 2000j:82020 “A large variety of integrodifferential equations describing various physically interesting phenomena are considered intensively in the current literature. The non-local version of the Cahn-Hilliard equation for the phase segregation dynamics in certain particle systems was studied very recently in [G. Giacomin and J. L. Lebowitz, Phys. Rev. Lett. 76 (1996), no. 7, 1094–1097]; and in M. A. Katsoulakis and P. E. Souganidis [80]. A non-local, natural generalization of the familiar Landau-Ginzburg free energy functional is provided by S1 [u] (3.8). In particular, the L2 gradient of this functional leads to the so-called non-local version of the Allen-Cahn equation, which is the main topic of the present paper. Different behaviours of the kernel j model a variety of interesting physical phenomena, which is the motivation for considering particular versions of the non-local equations arising from minizations of the free energy functional… ” The main results in the paper relate to the possibility, under some conditions, of the existence of stationary solutions in higher space dimensions whose set of discontinuity is rather arbitrary.]

    Google Scholar 

  17. P. Bates and P. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations and time scales for coarsening, Physica D 43,335–348 (1990). [The Allen-Cahn, Cahn-Hilliard, and phase field models share certain features; for example they have similar stationary solutions and the last two exhibit phenomena like “spinodal decomposition” and coarsening. Insight into these phenomena can be gained by spectral analysis. This paper compares corresponding spectra for these models.]

    Google Scholar 

  18. P. Bates, P. Fife, X. Ren, and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rat. Mech. Anal. 138 (1997), 105–136. [This gives the basic theory for traveling waves of the “nonlocal Allen-Cahn” equation (3.10).]

    Google Scholar 

  19. P. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard equation: Part I, J. Differential Equations 111,421–457 (1995). [See Section 3.4.1.]

    Google Scholar 

  20. P. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard equation: Part II, J. Differential Equations 117, 165–216 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  21. D. Blömker, S. Maier-Paape, and T. Wanner, Spinodal decomposition for the Cahn-Hilliard-Cook equation, preprint.

    Google Scholar 

  22. A. Bonami, D. Hilhorst and E. Logak, Modified motion by mean curvature: local existence and uniqueness and qualitative properties, Differential Integral Equations 13, 1371–1392 (2000).

    MathSciNet  MATH  Google Scholar 

  23. D. Brandon and R. Rogers, Nonlocal superconductivity, Z. angew. Math. Phys. 45, 135–152 (1994).

    MathSciNet  MATH  Google Scholar 

  24. L. Bronsard and B. Stoth, Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J. Math. Anal. 28, 769–807 (1997). [Summary: “We study the asymptotic behavior of radially symmetric solutions of the nonlocal equation Eçst — e-1W’(q5) —) €(t) = 0 in a bounded spherically symmetric domain Q C Ri’, where Age(t) = E-1 f 5 QW’(¢)dx, with a Neumann boundary condition. The analysis is based on `energy methods’ combined with some a priori estimates, the latter being used to approximate the solution by the first two terms of an asymptotic expansion. We only need to assume that the initial data as well as their energy are bounded. We show that, in the limit as e - 0, the interfaces move by a nonlocal mean curvature flow, which preserves mass. As a by-product of our analysis, we obtain an L2 estimate on the `Lagrange multiplier’ a.E(t), which holds in the nonradial case as well. In addition, we show rigorously (in general geometry) that the nonlocal Ginzburg-Landau equation and the Cahn-Hilliard equation occur as special degenerate limits of a viscous Cahn-Hilliard equation.”]

    Google Scholar 

  25. L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Rational Mech. Anal. 124 355–379 (1993). [From Math. Reviews 94h:35122: The authors study the vector-valued Ginzburg-Landau equation ust = 2eL\u — OsuW (u) as a model for three-phase boundary motion, i.e. W has three minima. Formal asymptotic analysis is used to derive equations for the evolution of the phase boundaries, with one time scale used to analyze the evolution near the triple junction and the boundary of the domain of u, and another time scale used to analyze the motion on the rest of the boundary. Short term existence and uniqueness of solutions are shown by first proving existence for the linearized system, then constructing a contraction mapping whose fixed point is the desired solution. This method applies to a class of parabolic problems, as the authors illustrate by applying the method to an equation modeling eutectic solidification.]

    Google Scholar 

  26. L. Bronsard and R. Kohn, On the slowness of the phase boundary motion in one space dimension, Comm Pure Appl. Math. 43, 983–997 (1990).

    MathSciNet  MATH  Google Scholar 

  27. G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations, Phys. Rev. A 39, 5887— 5896 (1989).

    Google Scholar 

  28. G. Caginalp and P. Fife, Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math. 48,506–518 (1988). [A systematic formal layer reduction analysis for a simple phase-field model. It includes the reduction of (3.1) (with parameter e) to the motion-by-curvature law as a special case.]

    Google Scholar 

  29. J. W. Cahn, On spinodal decomposition, Acta Metall. 9,795–801 (1961). [The linearized Cahn-Hilliard evolution equation is proposed, based on the flux being proportional to the gradient of the chemical potential. The implications for stability are explored.]

    Google Scholar 

  30. J. Cahn, P. Fife and O. Penrose, A phase-field model for diffusion-induced grain boundary motion, Acta Materialia 45, 4397— 4413 (1997). [A gradient flow with Ginzburg-Landau energy plus elastic energy additions is proposed as a model for diffusion induced grain boundary motion.]

    Google Scholar 

  31. J. Cahn and J. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Physics 28,258— 267 (1958). [The free energy of a system with nonuniform concentration or density is proposed to be an integral with bulk free energy and quadratic gradient terms. On this basis, the character of a flat interface between phases, including the interfacial free energy and interface thickness, is investigated.]

    Google Scholar 

  32. J. Carr and R. Pego, Metastable patterns in solutions of ut = e2uxx — f (u), Comm. Pure Appl. Math. 42,523— 576 (1989). [This and [65] are the first papers to prove super slow motion.]

    Google Scholar 

  33. J. Carr and O. Penrose, Asymptotic behaviour in a simplified Lifschitz-Slyosov equation, Physica D 124, 166— 176 (1998).

    Google Scholar 

  34. C. Charach and P. Fife, Solidification fronts and solute trapping in a binary alloy, SIAM J. Appl. Math. 58,1826–1851 (1998). [Elaborate asymptotics applied to a phase field model for a solidifying alloy.]

    Google Scholar 

  35. C-K Chen and P. Fife, Nonlocal models of phase transitions in solids, Adv. Math. Sciences and Applications, to appear.

    Google Scholar 

  36. F. Chen, Almost periodic traveling waves of nonlocal evolution equations, preprint. [Existence, uniqueness, and stability of almost periodic traveling wave solutions of (3.14).]

    Google Scholar 

  37. X. Chen, Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces, Commun. Partial Differential Equations 19, 13711395 (1994).

    Google Scholar 

  38. X. Chen, Existence, uniqueness, and asymptotic stability of travelling waves in nonlocal evolution equations, Adv. Differential Equations 2,125–160 (1997). [Uniqueness and global asymptotic stability are addressed.]

    Google Scholar 

  39. X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation, J. Differential Geometry 44, 262–311 (1996). [Limit of solutions with bounded energy as E-0. There is a subsequence converging to a certain type of weak solution. This is a weak formulation of the Mullins-Sekerka problem using varifolds. See review of [1].]

    Google Scholar 

  40. X. Chen, D. Hilhorst and E. Logak, Asymptotic behavior of solutions of an Allen-Cahn equation with a nonlocal term, Nonlinear Analysis, TMA 28,1283–1298 (1997). [The limiting FBP for ut = L\u + - f (u, e f u) with Neumann BC’s and IC. This in turn is the shadow limit of the system u t = (Au + f (u, E ~ v), Tvt = Dv + u - v as a and T- 0. (The FitzHugh-Nagumo system with certain parameters fits in here.) The FBP is V = -K+co (1Q+1-1f2-1). The free boundary separates Q+ from Q-, where in the limit u = +1. Here co depends on f. An existence theory for the FBP is included.]

    Google Scholar 

  41. R. Choksi, Scaling laws in microphase separation of diblock copolymers, J. Nonlinear Science 11–3,223–236 (2001). [See Section 3.8.1.]

    Google Scholar 

  42. M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Modern Phys. 65(3), 851–1112 (1993). [Comprehensive review of patterning in physical models.]

    Google Scholar 

  43. E. N. Dancer, D. Hilhorst, M. Mimura, and L. A. Peletier, Spatial segregation limit of a competition-diffusion system, European J. Appl. Math. 10,97–115 (1999). [A system of reaction-diffusion equations for competing species. When the competition strength becomes large, the habitats of (regions dominated by) the two species become disjoint, with a single curve separating them. The limiting problem is similar to the Stefan problem of phase transformation, except that there is zero latent heat of transformation. The behavior of the limiting problem is discussed, particularly the large time evolution of the habitats.]

    Google Scholar 

  44. A. De Masi, T. Gobron, and E. Presutti, Travelling fronts in nonlocal evolution equations, Arch. Rat. Mech. Anal. 132,143–205 (1995). [This is the model from [46]. Existence of traveling waves for small h is proved, and well as their nonlinear global stability.]

    Google Scholar 

  45. A. De Masi, E. Orlandi, E. Presutti, and L. Triolo, Motion by curvature by scaling nonlocal evolution equations, J. Stat. Physics 73,543–570 (1993). [Model introduced in [46] is studied: t = -m+tank{ßJ * m}. Convergence to a motion by mean curvature is proved under proper scaling.]

    Google Scholar 

  46. A. De Masi, E. Orlandi, E. Presutti, and L. Triolo, Glauber evolution with Kac potentials: I. Mesoscopic and macroscopic limits, interface dynamics, Nonlinearity 7, 633–696 (1994). [Evolution of an Ising spin system with Kac potential J in the limit when the inverse range of the potential ‘y goes to 0.]

    Google Scholar 

  47. K. Elder, T. Rogers, and R. Desai, Early stages of spinodal decomposition for the Cahn-Hilliard-Cook model of phase separation. Phys. Rev. B 38, 47254739 (1988).

    Google Scholar 

  48. G. B. Ermentrout and J. B. McLeod, Existence and uniqueness of traveling waves for a neural network, Proc. Roy. Soc. Edin. 123A,461–478 (1993). [Summary: “A one-dimensional scalar neural network with two stable steady states is analysed. It is shown that there exists a unique monotone travelling wave front which joins the two stable states. Some additional properties of the wave such as the direction of its velocity are discussed.”]

    Google Scholar 

  49. L. C. Evans, H. M. Soner and P. E. Souganidis, Comm. Pure Appl. Math. 45, 1097–1123 (1992).

    MathSciNet  MATH  Google Scholar 

  50. D. Eyre, Coarsening dynamics for solutions of the Cahn-Hilliard equation in one dimension, preprint.

    Google Scholar 

  51. D. Eyre, Systems of Cahn-Hilliard equations, SIAM J. Appl. Math. 53,16861712 (1993). [Summary: “The phase separation of alloys with two or more components is studied, with emphasis on more than two components. Particular attention is given to differences between multicomponent and binary alloys. Specific topics of the paper include equilibrium theory, aspects of the dynamics, and numerical simulations. In the equilibrium theory, it is found that there is an enriched equilibrium structure that allows for multiple coexisting phases and the presence of triple points in the solution. Dynamic results include the characterization of the spinodal region and of the concentration variations that lead to spinodal decomposition. Unlike the binary theory, not all composition variations lead to separation, and the compositions are not restricted to the convex hull of the equilibrium concentrations. Linear analysis is used to predict that a pseudo-binary will initially result from spinodal decomposition. Numerical simulations of the dynamics for a ternary alloy verify this initially, but more than two phases often separate. A sequential application of the dominant growth mode in linearly independent directions of composition variations appears to explain these additional phases. Finally, intermediate products are found that have both separated and metastable phases. This is not seen in binary materials.”]

    Google Scholar 

  52. P. Fife, Dynamics of Internal Layers and Diffusive Interfaces, CBMS-NSF Regional Conference Series in Applied Mathematics 53, Soc. Ind. Appl. Math, Philadelphia (1988).

    Google Scholar 

  53. P. Fife, Models for phase separation and their mathematics, Electron. J. Diff. Eqns. Vol. 2000,No. 48, 1— 26 (2000). (Part of the proceedings of a workshop held in Katata, Japan in August, 1990 [Nonlinear Partial Differential Equations with Applications to Patterns, Waves, and Interfaces, M. Mimura and T. Nishida, eds.]).

    Google Scholar 

  54. P. Fife, Well-posedness issues in models for phase transition with weak interaction, Nonlinearity 14,221— 238 (2001). [See Section 3.2.5.]

    Google Scholar 

  55. P. Fife, Pattern formation in gradient systems, to appear as a chapter in Handbook for Dynamical Systems, Vol. 3, Applications, B. Fiedler and N. Kopell, eds.

    Google Scholar 

  56. P. Fife, J. Cahn, and C. Elliott, A free boundary model for diffusion induced grain boundary motion, Interfaces and Free Boundaries 3, 291— 336 (2001). [Extensive further development of the model in [30]. The difficulties of degenerate mobility are addressed.]

    Google Scholar 

  57. P. Fife and D. Hilhorst, The Nishiura-Ohnishi free boundary problem in the 1D case, SIAM J. Math. Anal. 33,589— 606 (2001). [See Section 3.8.1.]

    Google Scholar 

  58. P. Fife and M. Kowalczyk, A class of pattern-forming models, J. Nonlinear Science 9,641— 669 (1999). [Development of ideas in Section 3.5.1. See also [55].]

    Google Scholar 

  59. P. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal. 65, 335— 361 (1977).

    Google Scholar 

  60. P. Fife and O. Penrose, Interfacial dynamics for thermodynamically consistent phase-field models with non-conserved order parameter, Elect. J. Differential Equations 1995,1— 49 (1995). [Full asymptotics for a class of phase field models.]

    Google Scholar 

  61. P. Fife and X. Wang, A convolution model for interfacial motion: the generation and propagation of internal layers in higher space dimensions, Advances in Differential Equations 5, 85–110 (1998). [Reduction of (3.10) (with parameter e) to motion by curvature.]

    Google Scholar 

  62. N. A. Fleck and J. W. Hutchinson, Strain gradient plasticity, Adv. Appl. Mech. 3 296— 381 (1997). [Survey of phenomenological theories of plastic materials in which stress depends not only on the strain but also on the spatial gradient of the strain. In this sense it is slightly “nonlocal”.]

    Google Scholar 

  63. R. Fosdick and D. Mason, Single phase energy minimizers for materials with nonlocal spatial dependence, Quart. Appl. Math. 54, 161— 195 (1996).

    Google Scholar 

  64. R. Fosdick and D. Mason, On a model of nonlocal continuum mechanics, Part I: Existence and regularity, SIAM J. Appl. Math. 58,1278— 1306 (1998). [The energy field for a body is assumed to depend not only on the local strain field but also on the field’s average with a weighting kernel Minimizers are considered. They must be periodic and piecewise smooth. An existence theorem is given.]

    Google Scholar 

  65. G. Fusco and J. Hale, Slow motion manifolds, dormant instability and singular perturbation, in Dynamics and Differential Equations 1,1989. [See comment under [32].]

    Google Scholar 

  66. K. Glasner, A diffuse interface approach to Hele-Shaw flow, preprint. [Gradient nature of the problem is exploited.]

    Google Scholar 

  67. R. Goldstein, D. J. Muraki, and D. M. Petrich, Interface proliferation and the growth of labyrinths in a reaction-diffusion system, Phys. Rev. E 53,39333957 (1996). [The paper starts with the FitzHugh-Nagumo model and goes to a shadow system. This is the opposite limit from the usual excitable structures. Thus, we get a nonlocal gradient flow. A formal expansion can lead to a 4th order problem. If one assumes the fronts are narrow in 1D, one gets a 1D FBP. Formal asymptotics are provided. Heuristics and asymptotics are given for the 2D front case. Linear stability for fronts, stripes, and discs is investigated. Patterns are revealed by simulations. Similarities are brought out with models from superconductors, magnetic fluids, and monolayers.]

    Google Scholar 

  68. A. A. Golovin, S. H. Davis and A. A. Nepomnyashchy, A convective Cahn-Hilliard model for the formation of facts and corners in crystal growth, Physics D 122,202— 230 (1998). [In 1D, the derivative u t is replaced by the convective derivative ut — Duu„]

    Google Scholar 

  69. C. P. Grant, Spinodal decomposition for the Cahn-Hilliard equation, Commun. in Partial Differential Equations 18,453–490 (1993). [See Section 3.3.1.]

    Google Scholar 

  70. C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal. 26, 21–34 (1995).

    MathSciNet  MATH  Google Scholar 

  71. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Vol. 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1981.

    Google Scholar 

  72. M. Henry, Singular limit of a fourth order problem arising in the micro-phase separation of diblock copolymers, Advances in Differential Equations, to appear.

    Google Scholar 

  73. D. Hilhorst and J.F. Rodrigues, On a nonlocal diffusion equation with discontinuous reaction, Advances in Differential Equations 5,657— 680 (2000). [btu = a (f u).Au + f (u, f u). Start with Btu = a(y)Liu + f (u, v) + an eqn for y, and make the shadow system. Here f could be discontinuous; we get nonuniqueness then.]

    Google Scholar 

  74. T. Ikeda and M. Mimura, An interfacial approach to regional segregation of two competing species mediated by a predator, J. Math. Biol. 31,215–240 (1993). [The possibility of coexistence (with spatial segregation) is considered when the diffusion rate of the prey tends to zero. The dynamics of spatial segregation is discussed by means of an interfacial dynamics approach.]

    Google Scholar 

  75. D. Jackson, R. Goldstein and A. Cebers, Hydrodynamics of fingering instabilities in dipolar fluids, Phys. Rev. E 50,298–307 (1994). [The subject is ferrofluids under magnetic field. In the introduction, there is a discussion of general patterning from competition between surface tension and body forces. The present context could be generalized. Reference is made to [83] as the simplest gradient-flow model for this physics. In that one, viscous dissipation by Darcy’s law was neglected except at the boundary. Here it is not. Law of motion for the boundary: y = —VI7 (generalized Darcy’s law), V2H = 0, H = +I,where I an integral operator along the boundary C. This is a gradient flow for E = L — fc ds fc ds’t(s)•t(s’)0(R(s, s’)). Here PO is a nonlinear function, the contour C is given parametrically by r(s), R =1r(s) — r(s’)1, t is the unit tangent vector, and L is the length of C. Time-dependence of the fastest growing modes is studied.]

    Google Scholar 

  76. Y. Kan-on and M. Mimura, Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics, SIAM J. Math. Anal. 29, 1519–1536 (1998) (electronic). [The effect of a predator on the coexistence question for competing species is considered in this model. Singular perturbation analysis may reveal coexisting spatially segregated species.]

    Google Scholar 

  77. S. Kawaguchi, and M. Mimura, Collision of travelling waves in a reaction-diffusion system with global coupling effect, SIAM J. Appl. Math. 59,920–941 (1999). [The system is that proposed by Krischer and Mikhailov. The destabilization of radially symmetric equilibria, the formation of traveling waves, and the collision of the latter are investigated.]

    Google Scholar 

  78. K. Kawasaki, T. Ohta, and M. Kohrogui, Equilibrium morphology of block copolymer melts. 2, Macromolecules 21, 2972— 2980 (1988).

    Google Scholar 

  79. M. Katsoulakis and P. E. Souganidis, Interacting particle systems and generalized mean curvature evolution, Arch. Rat. Mech. Anal. 127,133— 157 (1994). [Authors’ summary: “We study the limiting behavior (the macroscopic limit) of an appropriately scaled spin system with Glauber-Kawasaki dynamics. We rigorously establish the existence in the limit of an interface evolving according to motion by mean curvature. This limit is valid for all positive times, past possible geometric singularities of the motion, which is interpreted in the viscosity sense.”]

    Google Scholar 

  80. 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. 169,61— 97 (1995). [Math. Reviews 96m:60238: The mesoscopic limit of a ferromagnetic stochastic Ising model under Glauber dynamics and Kac potential leads to the non-local mean-field equation am/at = tanh ß(J * m + h) — in in the limit -y —0. The above solution m(t, x), x E R d,is the limit of the expected variable located at lattice point i = x/ry in Z d,evolving under an interaction potential with couplings rydJ(71i — j1) between particles at i, j E Z d and external field h. The macroscopic limit considered here is the large time—large space asymptotics of the model. The authors prove that the resealed solution m(t/A2, x/À) tends as A 0 to an interface motion which propagates with normal velocity 07(mean curvature)+const, when the external field brings some speed-up: h = const•A. For A = A(y)-0 slowly enough, they prove that the stochastic density field itself, rescaled as before, converges to the mean curvature flow. The flow is viewed as a viscosity solution, and convergence holds past the occurrence of singularities. The transport coefficient O has an explicit form, coming from the averaging effect of small A’s.]

    Google Scholar 

  81. M. Katsoulakis and P. E. Souganidis, Stochastic Ising models and anisotropic front propagation, J. Statist. Phys. 87,63–89 (1997). [From Math. Reviews 98k:82118: The Ising models studied here have general spin-flip dynamics in Zd+ 1 obeying the detailed balance law with respect to Gibbs measures with long range interactions (Kac potentials [46]). Resealing time, space and the interactions together, the authors obtain interfaces moving with normal velocity depending anisotropically on their principal curvatures and direction. The mobility depends on the direction and is computed explicitly in terms of the microscopic dynamics.]

    Google Scholar 

  82. M. Kuwamura, S.-I. Ei, and M. Mimura, Very slow dynamics for some reaction-diffusion systems of the activator-inhibitor type, Japan J. Indust. Appl. Math. 9,35–77 (1992). [Slow dynamics is considered for the shadow system consisting of the following two equations: (1) ust = E 2 usxx + o-2[u(1 — u2) — (], (2) (gt = —rye + f l sfu(x, t)dx,where x E (0, 1) and0.]

    Google Scholar 

  83. S. Langer, R. Goldstein, and D. Jackson, Dynamics of labyrinthine pattern formation in magnetic fluids, Phys. Rev. A 46,4894–4904 (1992). [See commentary under [75].]

    Google Scholar 

  84. L. Leibler, Theory of microphase separation in block copolymers, Macromolecules 13, 1602— 1617 (1980).

    Google Scholar 

  85. K. Leung, Theory of morphological instability in driven systems, J. Stat. Phys. 61, 345 (1990).

    Article  Google Scholar 

  86. I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids 19,35–50 (1961). [This and [125] are the original papers presenting LSW theory.]

    Google Scholar 

  87. E. Logak, Singular limit of reaction-diffusion systems and modified motion by mean curvature, preprint.

    Google Scholar 

  88. S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part I: Probability and wavelength estimate, Comm. Math. Physics 195, 435–464 (1998). [See Section 3.3.2.]

    Google Scholar 

  89. S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part II: Nonlinear dynamics, Arch. Rat. Mech. Anal. 151, 187— 219 (2000).

    Google Scholar 

  90. E. Meron, Pattern formation in excitable media, Physics Reports 218, No. 1,1–66 (1992). [Survey.]

    Google Scholar 

  91. P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc. it 347, 1533–1589 (1995). [The first rigorous justification of the motion-by-curvature property of Allen-Cahn layers.]

    Google Scholar 

  92. B. Niethammer, Derivation of the LSW-theory for Ostwald ripening by homogenization methods, Arch. Rat. Mech. Anal. 147, 119— 178 (1999). [Rigorous derivation of LSW theory in the regime 8 « d and Scor « -Scr. See [93] and Sec. 3.4.3.]

    Google Scholar 

  93. B. Niethammer and F. Otto, Ostwald ripening: The screening length revisited, Calc. Var. Partial Differential Equations 13,33–68 (2001). [See Section 3.4.3.]

    Google Scholar 

  94. B. Niethammer and R. Pego, Non-self-similar behavior in the LSW theory of Ostwald ripening, J. Stat. Phys. 95,867— 902 (1999). [The supposedly long time behavior of the LSW model (convergence to a self-similar solution) is not universal. If the particle distribution has compact support, the asymptotics depend sensitively on the behavior of the initial distribution near its support boundary.]

    Google Scholar 

  95. B. Niethammer and R. Pego, The LSW model for domain coarsening: Asymptotic behavior for conserved total mass, preprint. [The variation this time is that a constraint is made that the total mass, including the mean-field mass, is conserved, rather than just the total volume fraction of one phase. Some results are similar to the classical case.]

    Google Scholar 

  96. B. Niethammer and R. Pego, On the initial-value problem in the LifshitzSlyozov-Wagner theory of Ostwald ripening, SIAM J. Math. Anal. 31,467485 (2000). [Global existence, uniqueness and continuous dependence on initial data for measurue-valued solutions with compact support in particle size. The topology of solutions used is the Wasserstein metric, a natural one for the space of size distributions.]

    Google Scholar 

  97. Y. Nishiura, Nonlinear Problems, Part I, Amer. Math. Soc. (2001).

    Google Scholar 

  98. Y. Nishiura and I. Ohnishi, Some mathematical aspects of the micro-phase separation in diblock copolymers, Physica D 84,31–39 (1995). [Functional introduced in [102, 13]. It involves parameters e and u. The H-1 gradient dynamics introduced. Space and time rescaled according to powers of e and u. Powers chosen so that the resealed problem involves only one parameter b. Then letting 5–40, one gets a meaningful free boundary problem with no parameter in it. This is formal. Stability considerations discussed, a la [17].]

    Google Scholar 

  99. I. Ohnishi and Y. Nishiura, Spectral comparison between the second and fourth order equations of conservative type with non-local terms, Japan Jour. Industrial and Applied Mathematics 15,253— 262 (1998). [A comparison result for the spectra of two nonlocal operators is given. The corresponding evolutions are (3.50) and a similar but second order equation. The two equations have the same steady states, and their stabilities coincide.]

    Google Scholar 

  100. I. Ohnishi, Y. Nishiura, M. Imai, and Y. Matsushita, Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term, Chaos 9,329— 341 (1999). [The free energy is given by (3.57). A rigorous asymptotic expansion in e of the global minimizer in 1D. Numerics on the dynamical problem reveal local minima.]

    Google Scholar 

  101. T. Ohta and M. Mimura, Pattern dynamics in excitable reaction-diffusion media. Formation, dynamics and statistics of patterns, Vol. 1, 55–112, World Sci. Publishing, River Edge, NJ, 1990. [Recent investigations of pattern formation and pattern evolution in excitable reaction-diffusion media by means of the interface dynamics are reviewed. This involves interfacial equations of motion and the associated stability considerations. An example is a phase field model of crystal growth.]

    Google Scholar 

  102. T. Ohta and K. Kawasaki, Equilibrium morphology of block copoloymer melts, Macromolecules 19,2621–2632 (1986). [Stat. mech. theory developed by Helfand and Wasserman; mean field in the weak segregation limit was given by Leibler. Here, such a theory is given in the strong segregation limit. They use a local order parameter and develop a free energy functional.]

    Google Scholar 

  103. F. Otto, Evolution of microstructure in unstable porous media flow: A relaxational approach, Comm. Pure Appl. Math. LII,0873–0915 (1999). [Uses the notion of mass transference plan in study of microstructure evolution in unstable porous medium flow.]

    Google Scholar 

  104. F. Otto, Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory, Arch. Rat. Mech. Anal. 111,63–103 (1998). [Hele-Shaw with magnetic fluid under magnetic field. Patterns have been observed in the laboratory. Reference is made to [75, 83]. The formulation of the dynamics is that given in [75]. This is shown to be a gradient flow. The author uses the Wasserstein metric. He gets an evolution law for the envelop of the labyrinth. It is a porous medium equation.]

    Google Scholar 

  105. E. Orlandi and L. Triolo, Travelling fronts in nonlocal models for phase separation in an external field, Proc. Roy. Soc. Edinburgh Sect. A 127,823–835 (1997). [Model nonlocal equation in [46]. Existence, uniqueness, and linear stability of traveling waves. The existence in [44] was for small h; that is removed. Other results are similar; some different methods.]

    Google Scholar 

  106. R. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. R. Soc. Lond. A 422,261— 278 (1989). [Various kinds of singular reductions of the Cahn-Hilliard equation are revealed when a parameter is small. They lead to free boundary problems. One of them corresponds to what would later be called the Mullins-Sekerka problem.]

    Google Scholar 

  107. L. A. Peletier and W. C. Troy, Spatial Patterns, Higher Order Models in Physics and Mechanics, Birkhäuser, Boston, 2001. [An extensive study of the existence and properties of solutions of + q d + f (u) = 0, which includes the Swift-Hohenberg equation. Special attention is given to the case when f is bistable.]

    Google Scholar 

  108. G. Peng, F. Qiu, V. Ginzburg, D. Jasnow and A. Balazs, Forming supramolecular networks from nanoscale rods in binary, phase-separating mixtures, Science 288,1802–1804 (2000). [Patterning in a mixture with two fluids and rods. Modeling is done with a phase-field system.]

    Google Scholar 

  109. X. Ren and J. Wei, On the multiplicity of solutions of two nonlocal variational problems, SIAM J. Math. Anal. 31,909–924 (2000). [Two mass-constrained variational problems are considered, associated with the energy functional (3.57) and a variation on it arising from a model by Truskinovsky and Ren. The F-limits of these problems as e—*0 are found. In 1D, all local minima are characterized, using perturbation from the F-limit.]

    Google Scholar 

  110. X. Ren and J. Wei, Analysis of global minima of a nonlocal variational problem, in preparation. [A different parameter range from that in [109].]

    Google Scholar 

  111. R. Rogers, Nonlocal variational problems in nonlinear electromagnetoelastostatics, SIAM J. Math. Anal. 19,1329–1347 (1988). [The setting is a nonlinear deformable magnetizable nonconducting body. It is unshielded, which makes the problem nonlocal. Reference is made to a previous nonlocal problem from astrophysics.]

    Google Scholar 

  112. E. Sander and T. Wanner, Monte Carlo simulations for spinodal decomposition, J. Stat. Phys. 95,925— 948 (1999). [Simulations on the Cahn-Hilliard equation showing phase separation.]

    Google Scholar 

  113. E. Sander and T. Wanner, Unexpectedly linear behavior for the Cahn-Hilliard equation, SIAM J. Appl. Math. 60,2182–2202 (2000). [See Section 3.3.2. Some results already conjectured in [112].]

    Google Scholar 

  114. M. Seul and D. Andelman, Domain shapes and patterns: the phenomenology of modulated phases, Science 267, 476–483 (1995).

    Article  Google Scholar 

  115. W. Shen, Traveling waves in time almost periodic structures governed by bistable nonlinearities I. Stability and uniqueness, J. Differential Equations 159, 1-54 (1999) II. Existence, op. cit. 159,55–101 (1999). [See Sec. 3.2.3.]

    Google Scholar 

  116. A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhäuser, Basel-Boston-Berlin, 1997.

    MATH  Google Scholar 

  117. H. M. Soner, Convergence of the phase-field equations to the Mullins-Sekerka problem with kinetic undercooling, Arch. Rat. Mech. Anal. 131, 139–197 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  118. P. E. Souganidis, Recent developments in the theory of interface dynamics. Differential equations and applications (Hangzhou, 1996), 286–300, Internat. Press, Cambridge, MA, 1997. [From Math. Reviews 2000d:35238: The author presents a survey of recent results in the theory of interface dynamics, and some applications. The related proofs are contained in [14]. More precisely, a new definition of motion is described, which holds for any time and coincides with the regular solution up to the first time of singularity. This evolution has a simple geometric definition based on a maximum principle type avoidance-inclusion property, and turns out to be equivalent to the viscosity-type solution obtained through the level-set method [see Y. G. Chen, Y. Giga and S. Goto, J. Differential Geom. 33 (1991), no. 3, 749–786; MR 93a:35093]. A similar definition of weak evolution has been proposed by E. De Giorgi [“Barriers, boundaries, motion of manifolds”, in Proceedings of Capri Workshop (1990)] and further developed by G. Bellettini and Novaga [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), no. 1, 97–131; MR 99i:35081]. The author uses similar techniques to introduce a general method in order to study the large time asymptotics of a class of fully nonlinear differential equations including, among others, reaction-diffusion equations and limits of stochastic Ising models with long-range interactions. The advantage of the method is that the rigorous justification of the appearance of an interface is reduced to proving a similar result only in the case where everything is smooth and for small time intervals.]

    Google Scholar 

  119. M. Suzuki, T. Ohta, M. Mimura, and H. Sakaguchi, Breathing and wiggling motions in three-species laterally inhibitory systems, Phys. Rev. E 52,36453655 (1995). [Authors’ summary: “We study the layer dynamics in a coupled set of reaction-diffusion systems describing the interaction of one activator and two inhibitors. In some special limits, this set of equations is reduced to a Bonhoeffer-van der Pol-type excitable system or a model for electric glow discharge. Dynamics of layers is investigated by an interfacial approach and complementarily by computer simulations. By changing the parameters, these layers undergo several sustained oscillations such as breathing, wiggling, and quasiperiodic motions, which are due to the competition of two inhibitors and one activator.”]

    Google Scholar 

  120. J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A 15,319–328 (1977). [Derivation of the Swift-Hohenberg equation in the context indicated.]

    Google Scholar 

  121. J. J. L. Velazquez, The Becker-Döring equations and the Lifshitz-Slyozov theory of coarsening, J. Stat. Phys. 92,195–236 (1998). [The LSW evolution law is approximated by a Fokker-Planck equation.]

    Google Scholar 

  122. P. W. Voorhees, The theory of Ostwald ripening, J. Stat. Phys. 38,231–252 (1985). [Review]

    Google Scholar 

  123. P. W. Voorhees, Ostwald ripening of two-phase mixtures, A. Rev. Mater. Sci. 22,197–215 (1992). [Review]

    Google Scholar 

  124. J. D. van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, Verh. Konink. Acad. Wetensch. Amsterdam 1 (1893). [First use of Ginzburg-Landau style energy.]

    Google Scholar 

  125. C. Wagner, Theorie der Alterung von Neiderschlägen durch Umlösen, Z. Elektrochem. 65,581— 594 (1961). [This and [86] are the original papers presenting LSW theory.]

    Google Scholar 

  126. X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, preprint. [Metastability for (3.10). The results depend on how fast J decays. Results analogous to local case. Persistence, annihilation, stability of patterns.]

    Google Scholar 

  127. J. A. Warren and W. J. Boettinger, Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method, Acta Metal. Mater. 43, 689— 703 (1995).

    Google Scholar 

  128. A. A. Wheeler, G. B. McFadden and W. J. Boettinger, Phase field model for solidification of a eutectic alloy, Proc. Royal Soc. London A 495, 525 (1996).

    Google Scholar 

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  1. Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT, 84112-0090, USA

    Paul Fife

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  1. Mathematics Department and Center for Scientific Computing, University of Warwick, CV4 7AL, Coventry, UK

    Markus Kirkilionis

  2. Institute of Scientific Computing, University of Heidelberg, Im Neuheimer Feld 368, 69120, Heidelberg, Germany

    Susanne Krömker

  3. Institute of Applied Mathematics, University of Heidelberg, Im Neuheimer Feld 293, 69120, Heidelberg, Germany

    Rolf Rannacher

  4. Department of Mathematics, University of Heidelberg, Im Neuheimer Feld 288, 69120, Heidelberg, Germany

    Friedrich Tomi

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Fife, P. (2003). Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions. In: Kirkilionis, M., Krömker, S., Rannacher, R., Tomi, F. (eds) Trends in Nonlinear Analysis. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05281-5_3

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