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Global regular solvability of a nonuniformly nonlinear sixth order Cahn–Hilliard system

Volume 48 / 2021

Irena Pawłow, Wojciech M. Zajączkowski Applicationes Mathematicae 48 (2021), 1-35 MSC: Primary 35G31, 35A01, 35D35, 35K35; Secondary 35Q56. DOI: 10.4064/am2419-4-2021 Published online: 2 August 2021

Abstract

The aim of this paper is to prove existence, uniqueness and continuous dependence on the initial datum of a global regular solution (for arbitrarily large time) for a nonuniformly nonlinear sixth order Cahn–Hilliard system. The nonuniform nonlinearity means that the unknown enters nonlinearly the highest 6th order operator. Such nonlinearity results from the second order Ginzburg–Landau (GL) functional with the coefficients of the first and second gradient depending on the solution. The physical motivation of our system comes from the microemulsion model proposed by Gompper and Schick (1989) and Schmid and Schick (1993). However, other applications are expected to be relevant as well. We note that in a particular case our system is the same as the isotropic version of a modified phase-field crystal model. The mathematical treatment of the problem requires elaborate techniques extending our (2011) existence result on a sixth order Cahn–Hilliard system based on a GL functional with constant coefficient of the second order gradient term.

Authors

  • Irena PawłowInstitute of Mathematics and Cryptology
    Cybernetics Faculty
    Military University of Technology
    Kaliskiego 2
    00-908 Warszawa, Poland
  • Wojciech M. ZajączkowskiInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-656 Warszawa, Poland
    and
    Institute of Mathematics and Cryptology
    Cybernetics Faculty
    Military University of Technology
    Kaliskiego 2
    00-908 Warszawa, Poland
    ORCID: 0000-0003-1229-2162
    e-mail

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