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Stability of Transition Front Solutions in Multidimensional Cahn–Hilliard Systems

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Abstract

We consider nonlinear stability for planar transition front solutions \(\bar{u} (x_1)\) arising in multidimensional (i.e., \(x\in {\mathbb {R}}^n\)) Cahn–Hilliard systems. In previous work, the author has established conditions under which such waves are spectrally and linearly stable, and in this analysis, it is shown that spectral stability implies nonlinear stability for such systems.

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References

  • Alikakos, N.D., Betelu, S.I., Chen, X.: Explicit stationary solutions in multiple well dynamics and non-uniqueness of interfacial energy densities. Eur. J. Appl. Math. 17, 525–556 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Adams, R.A., Fournier, J.F.: Sobolev Spaces, 2nd edn. Academic Press, Cambridge (2003)

    MATH  Google Scholar 

  • Alikakos, N.D., Fusco, G.: On the connection problem for potentials with several global minima. Indiana U. Math. J. 57, 1871–1906 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Bernau, S.J.: The square root of a positive self-adjoint operator. J. Aust. Math. Soc. 8, 17–36 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  • Cahn, J.W.: On spinodal decomposition. Acta Metall. 9, 795–801 (1961)

    Article  Google Scholar 

  • Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system I: interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)

    Article  Google Scholar 

  • de Fontaine, D.: A computer simulation of the evolution of coherent composition variations in solid solutions, Ph. D. thesis (1967), Northwestern University, Advisor: John Hilliard

  • de Fontaine, D.: An analysis of clustering and ordering in multicomponent solid solutions I. Stability criteria. J. Phys. Chem. Solids 33, 297–310 (1972)

    Article  Google Scholar 

  • de Fontaine, D.: Private communication (2009)

  • Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, AMS (1998)

  • Eyre, D.J.: Systems of Cahn–Hilliard equations. SIAM J. Appl. Math. 53, 1686–1712 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Friedman, A.: Partial Differential Equations of Parabolic Type, Dover Prentice-Hall 1964; Reprinted by Dover in 2008

  • Howard, P.: Pointwise estimates for the stability of scalar conservation laws, Thesis at Indiana University 1998; Adv. K. Zumbrun

  • Howard, P.: Pointwise estimates on the Green’s function for a scalar linear convection-diffusion equation. J. Differ. Equ. 155, 327–367 (1999)

    Article  MATH  Google Scholar 

  • Howard, P.: Asymptotic behavior near transition fronts for equations of generalized Cahn–Hilliard form. Commun. Math. Phys. 269, 765–808 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Howard, P.: Asymptotic behavior near planar transition fronts for equations of Cahn–Hilliard type. Phys. D 229, 123–165 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Howard, P.: Spectral analysis of planar transition fronts for the Cahn–Hilliard equation. J. Differ. Equ. 245, 594–615 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Howard, P.: Spectral analysis of stationary solutions of the Cahn–Hilliard equation. Adv. Differ. Equ. 14, 87–120 (2009)

    MathSciNet  MATH  Google Scholar 

  • Howard, P.: Short-time existence theory toward stability for nonlinear parabolic systems. J. Evol. Equ. 15, 403–456 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Howard, P.: Spectral analysis for transition front solutions in multidimensional Cahn–Hilliard systems. J. Differ. Equ. 257, 3448–3465 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Howard, P.: Linear stability for transition front solutions in multidimensional Cahn-Hilliard systems. J. Dyn. Differ. Equ. (2015). doi:10.1007/s10884-015-9490-6

    Google Scholar 

  • Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)

    MATH  Google Scholar 

  • Howard, P., Kwon, B.: Spectral analysis for transition front solutions in Cahn–Hilliard systems. Discrete Contin. Dyn. Syst. 32, 125–166 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Howard, P., Kwon, B.: Asymptotic stability analysis for transition wave solutions in Cahn–Hilliard systems. Phys. D 241, 1193–1222 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Howard, P., Kwon, B.: Asymptotic \(L^p\) stability for transition fronts in Cahn–Hilliard systems. J. Differ. Equ. 252, 5814–5831 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Howard, P., Hu, C.: Nonlinear stability for multidimensional fourth order shock fronts. Arch. Ration. Mech. Anal. 181, 201–260 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Howard, P., Zumbrun, K.: Stability of undercompressive shock profiles. J. Differ. Equ. 225, 308–360 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Hoff, D., Zumbrun, K.: Green’s function bounds for multidimensional scalar viscous shock fronts. J. Differ. Equ. 183, 368–408 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Hoff, D., Zumbrun, K.: Asymptotic behavior of multidimensional scalar viscous shock fronts. Indiana Univ. Math. J. 49, 427–474 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Korvola, T.: Stability of Cahn-Hilliard fronts in three dimensions, Doctoral dissertation, University of Helsinki (2003)

  • Korvola, T., Kupiainen, A., Taskinen, J.: Anomalous scaling for three-dimensional Cahn–Hilliard fronts. Comm. Pure Appl. Math. LVIII, 1–39 (2005)

    MathSciNet  MATH  Google Scholar 

  • Kohn, R.V., Yan, X.: Coarsening rates for models of multicomponent phase separation. Interfaces Free Bound. 6, 135–147 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Mascia, C., Zumbrun, K.: Pointwise Green’s function bounds and stability of relaxation shocks. Indiana U. Math. J. 51(4), 773–904 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Mascia, C., Zumbrun, K.: Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems. Comm. Pure Appl. Math. 57(7), 841–876 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Prigogine, I.: Bull. Soc. Chim. Belge. 8–9, 115 (1943)

    Google Scholar 

  • Reed, M., Simon, B.: Method of Modern Mathematical Physics IV: Analysis of Operators. Academic Press, Cambridge (1978)

    MATH  Google Scholar 

  • Stefanopoulos, V.: Heteroclinic connections for multiple-well potentials: the anisotropic case. Proc. Roy. Soc. Edinburgh 138A, 1313–1330 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Zumbrun, K.: Multidimensional stability of planar viscous shock waves, TMR Summer School Lectures: Kochel am See, May 1999. Progress in nonlinear differential equations and their applications, Birkhauser’s series (2001)

  • Zumbrun, K. : Planar stability criteria for viscous shock waves of systems with real viscosity. Hyperbolic systems of balance laws, 229–326, Lecture Notes in Math., 1911, Springer, Berlin, (2007)

  • Zumbrun, K., Howard, P.: Pointwise semigroup methods and stability of viscous shock waves, Indiana U. Math. J. 47 (1998) 741–871. See also the errata for this paper: Indiana U. Math. J. 51 (2002) 1017–1021

  • Zumbrun, K., Serre, D.: Viscous and inviscid stability of multidimensional planar shock fronts. Indiana U. Math. J. 48, 937–992 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant Number DMS-0906370. This is a continuation of work the author began with Bongsuk Kwon, and the author is indebted to Dr. Kwon for several enlightening discussions.

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Authors and Affiliations

  1. Texas A&M University, College Station, TX, USA

    Peter Howard

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  1. Peter Howard
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Correspondence to Peter Howard.

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Communicated by Arnd Scheel.

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Howard, P. Stability of Transition Front Solutions in Multidimensional Cahn–Hilliard Systems. J Nonlinear Sci 26, 619–661 (2016). https://doi.org/10.1007/s00332-016-9295-8

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