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2012, 32(1): 125-166. doi: 10.3934/dcds.2012.32.125

## Spectral analysis for transition front solutions in Cahn-Hilliard systems

 1 Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, United States, United States

Received  July 2010 Revised  March 2011 Published  September 2011

We consider the spectrum associated with the linear operator obtained when a Cahn--Hilliard system on $\mathbb{R}$ is linearized about a transition wave solution. In many cases it's possible to show that the only non-negative eigenvalue is $\lambda = 0$, and so stability depends entirely on the nature of this neutral eigenvalue. In such cases, we identify a stability condition based on an appropriate Evans function, and we verify this condition under strong structural conditions on our equations. More generally, we discuss and implement a straightforward numerical check of our condition, valid under mild structural conditions.
Citation: Peter Howard, Bongsuk Kwon. Spectral analysis for transition front solutions in Cahn-Hilliard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 125-166. doi: 10.3934/dcds.2012.32.125
##### References:
 [1] N. D. Alikakos, S. I. Betelu, and X. Chen, Explicit stationary solutions in multiple well dynamics and non-uniqueness of interfacial energy densities,, European J. of Applied Mathematics, 17 (2006), 525. doi: 10.1017/S095679250600667X. [2] N. D. Alikakos and G. Fusco, On the connection problem for potentials with several global minima,, Indiana Univ. Math. J., 57 (2008), 1871. doi: 10.1512/iumj.2008.57.3181. [3] J. Alexander, R. Gardner and C. K. R. T. Jones, A topological invariant arising in the stability analysis of traveling waves,, J. Reine Angew. Math., 410 (1990), 167. [4] J. Bricmont, A. Kupiainen and J. Taskinen, Stability of Cahn-Hilliard fronts,, Comm. Pure Appl. Math., 52 (1999), 839. doi: 10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I. [5] F. Boyer and C. Lapuerta, Study of a three component Cahn-Hilliard flow model,, Mathematical Modeling and Numerical Analysis, 40 (2006), 653. doi: 10.1051/m2an:2006028. [6] J. W. Cahn, On spinodal decomposition,, Acta Metall., 9 (1961), 795. doi: 10.1016/0001-6160(61)90182-1. [7] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I: Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102. [8] D. de Fontaine, "A Computer Simulation of the Evolution of Coherent Composition Variations in Solid Solutions,'', Ph. D. thesis, (1967). [9] D. de Fontaine, An analysis of clustering and ordering in multicomponent solid solutions I. Stability criteria,, J. Phys. Chem. Solids, 33 (1972), 297. doi: 10.1016/0022-3697(72)90011-X. [10] D. de Fontaine, An analysis of clustering and ordering in multicomponent solid solutions II. Fluctuations and kinetics,, J. Phys. Chem. Solids, 34 (1973), 1285. doi: 10.1016/S0022-3697(73)80026-5. [11] , D. de Fontaine,, Private communication 2009., (2009). [12] D. J. Eyre, Systems of Cahn-Hilliard equations,, SIAM J. Appl. Math., 53 (1993), 1686. doi: 10.1137/0153078. [13] M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,", Annals of Mathematics Studies, (1983). [14] M. Giaquinta and E. Giusti, Differentiability of minima of non-differentiable functionals,, Invent. Math., 72 (1983), 285. doi: 10.1007/BF01389324. [15] C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems,, SIAM J. Math. Anal., 26 (1995), 21. doi: 10.1137/S0036141092226053. [16] R. Gardner and K. Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles,, Comm. Pure Appl. Math., 51 (1998), 797. doi: 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1. [17] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture notes in mathematics, 840 (1981). [18] P. Howard, Pointwise estimates and stability for degenerate viscous shock waves,, J. Reine Angew. Math., 545 (2002), 19. doi: 10.1515/crll.2002.034. [19] P. Howard, Local tracking and stability for degenerate viscous shock waves,, J. Differential Eqns., 186 (2002), 440. [20] P. Howard, Asymptotic behavior near transition fronts for equations of generalized Cahn-Hilliard form,, Commun. Math. Phys., 269 (2007), 765. doi: 10.1007/s00220-006-0102-5. [21] P. Howard, Asymptotic behavior near planar transition fronts for the Cahn-Hilliard equation,, Phys. D, 229 (2007), 123. doi: 10.1016/j.physd.2007.03.018. [22] P. Howard, Spectral analysis of stationary solutions of the Cahn-Hilliard equation,, Advances in Differential Equations, 14 (2009), 87. [23] P. Howard and B. Kwon, Stability for transition front solutions in Cahn-Hilliard Systems,, in preparation., (). [24] P. Howard and K. Zumbrun, The Evans function and stability criteria for degenerate viscous shock waves,, Discrete and Continuous Dynamical Systems, 10 (2004), 837. doi: 10.3934/dcds.2004.10.837. [25] J. J. Hoyt, Spinodal decomposition in ternary alloys,, Acta Metall., 37 (1989), 2489. doi: 10.1016/0001-6160(89)90047-3. [26] J. Kim and K. Kang, A numerical method for the ternary Cahn-Hilliard system with a degenerate mobility,, Applied Numerical Mathematics, 59 (2009), 1029. doi: 10.1016/j.apnum.2008.04.004. [27] R. V. Kohn and X. Yan, Coarsening rates for models of multicomponent phase separation,, Interfaces Free Bound., 6 (2004), 135. doi: 10.4171/IFB/94. [28] J. E. Morral and J. W. Cahn, Spinodal decomposition in ternary systems,, Acta Metall., 19 (1971), 1037. doi: 10.1016/0001-6160(71)90036-8. [29] , I. Prigogine,, Bull. Soc. Chim. Belge., 8-9 (1943), 8. [30] M. Reed and B. Simon, "Methods of Modern Mathematical Physics IV: Analysis of Operators,", Academic Press, (1978). [31] V. Stefanopoulos, Heteroclinic connections for multiple-well potentials: The anisotropic case,, Proc. Royal Soc. Edinburgh Sect. A, 138 (2008), 1313. doi: 10.1017/S0308210507000145. [32] K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves,, Indiana Univ. Math. J., 51 (1998), 741. doi: 10.1512/iumj.2002.51.2410.

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##### References:
 [1] N. D. Alikakos, S. I. Betelu, and X. Chen, Explicit stationary solutions in multiple well dynamics and non-uniqueness of interfacial energy densities,, European J. of Applied Mathematics, 17 (2006), 525. doi: 10.1017/S095679250600667X. [2] N. D. Alikakos and G. Fusco, On the connection problem for potentials with several global minima,, Indiana Univ. Math. J., 57 (2008), 1871. doi: 10.1512/iumj.2008.57.3181. [3] J. Alexander, R. Gardner and C. K. R. T. Jones, A topological invariant arising in the stability analysis of traveling waves,, J. Reine Angew. Math., 410 (1990), 167. [4] J. Bricmont, A. Kupiainen and J. Taskinen, Stability of Cahn-Hilliard fronts,, Comm. Pure Appl. Math., 52 (1999), 839. doi: 10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I. [5] F. Boyer and C. Lapuerta, Study of a three component Cahn-Hilliard flow model,, Mathematical Modeling and Numerical Analysis, 40 (2006), 653. doi: 10.1051/m2an:2006028. [6] J. W. Cahn, On spinodal decomposition,, Acta Metall., 9 (1961), 795. doi: 10.1016/0001-6160(61)90182-1. [7] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I: Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102. [8] D. de Fontaine, "A Computer Simulation of the Evolution of Coherent Composition Variations in Solid Solutions,'', Ph. D. thesis, (1967). [9] D. de Fontaine, An analysis of clustering and ordering in multicomponent solid solutions I. Stability criteria,, J. Phys. Chem. Solids, 33 (1972), 297. doi: 10.1016/0022-3697(72)90011-X. [10] D. de Fontaine, An analysis of clustering and ordering in multicomponent solid solutions II. Fluctuations and kinetics,, J. Phys. Chem. Solids, 34 (1973), 1285. doi: 10.1016/S0022-3697(73)80026-5. [11] , D. de Fontaine,, Private communication 2009., (2009). [12] D. J. Eyre, Systems of Cahn-Hilliard equations,, SIAM J. Appl. Math., 53 (1993), 1686. doi: 10.1137/0153078. [13] M. Giaquinta, "Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,", Annals of Mathematics Studies, (1983). [14] M. Giaquinta and E. Giusti, Differentiability of minima of non-differentiable functionals,, Invent. Math., 72 (1983), 285. doi: 10.1007/BF01389324. [15] C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems,, SIAM J. Math. Anal., 26 (1995), 21. doi: 10.1137/S0036141092226053. [16] R. Gardner and K. Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles,, Comm. Pure Appl. Math., 51 (1998), 797. doi: 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1. [17] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture notes in mathematics, 840 (1981). [18] P. Howard, Pointwise estimates and stability for degenerate viscous shock waves,, J. Reine Angew. Math., 545 (2002), 19. doi: 10.1515/crll.2002.034. [19] P. Howard, Local tracking and stability for degenerate viscous shock waves,, J. Differential Eqns., 186 (2002), 440. [20] P. Howard, Asymptotic behavior near transition fronts for equations of generalized Cahn-Hilliard form,, Commun. Math. Phys., 269 (2007), 765. doi: 10.1007/s00220-006-0102-5. [21] P. Howard, Asymptotic behavior near planar transition fronts for the Cahn-Hilliard equation,, Phys. D, 229 (2007), 123. doi: 10.1016/j.physd.2007.03.018. [22] P. Howard, Spectral analysis of stationary solutions of the Cahn-Hilliard equation,, Advances in Differential Equations, 14 (2009), 87. [23] P. Howard and B. Kwon, Stability for transition front solutions in Cahn-Hilliard Systems,, in preparation., (). [24] P. Howard and K. Zumbrun, The Evans function and stability criteria for degenerate viscous shock waves,, Discrete and Continuous Dynamical Systems, 10 (2004), 837. doi: 10.3934/dcds.2004.10.837. [25] J. J. Hoyt, Spinodal decomposition in ternary alloys,, Acta Metall., 37 (1989), 2489. doi: 10.1016/0001-6160(89)90047-3. [26] J. Kim and K. Kang, A numerical method for the ternary Cahn-Hilliard system with a degenerate mobility,, Applied Numerical Mathematics, 59 (2009), 1029. doi: 10.1016/j.apnum.2008.04.004. [27] R. V. Kohn and X. Yan, Coarsening rates for models of multicomponent phase separation,, Interfaces Free Bound., 6 (2004), 135. doi: 10.4171/IFB/94. [28] J. E. Morral and J. W. Cahn, Spinodal decomposition in ternary systems,, Acta Metall., 19 (1971), 1037. doi: 10.1016/0001-6160(71)90036-8. [29] , I. Prigogine,, Bull. Soc. Chim. Belge., 8-9 (1943), 8. [30] M. Reed and B. Simon, "Methods of Modern Mathematical Physics IV: Analysis of Operators,", Academic Press, (1978). [31] V. Stefanopoulos, Heteroclinic connections for multiple-well potentials: The anisotropic case,, Proc. Royal Soc. Edinburgh Sect. A, 138 (2008), 1313. doi: 10.1017/S0308210507000145. [32] K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves,, Indiana Univ. Math. J., 51 (1998), 741. doi: 10.1512/iumj.2002.51.2410.
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