Well-Posedness and Global Attractors for Viscous Fractional Cahn-Hilliard Equations with Memory

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ÖZTÜRK E., Shomberg J. L.

FRACTAL AND FRACTIONAL, vol.6, no.9, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 6 Issue: 9
  • Publication Date: 2022
  • Doi Number: 10.3390/fractalfract6090505
  • Journal Name: FRACTAL AND FRACTIONAL
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Social Sciences Citation Index (SSCI), Scopus, INSPEC, Directory of Open Access Journals
  • Keywords: Cahn-Hilliard equation, fractional Laplacian, memory, PHASE-FIELD SYSTEM, ROBUST EXPONENTIAL ATTRACTORS, UNIFORM ATTRACTORS, LONGTIME BEHAVIOR, SINGULAR LIMIT, RELAXATION, DYNAMICS, MODEL
  • Hacettepe University Affiliated: Yes

Abstract

We examine a viscous Cahn-Hilliard phase-separation model with memory and where the chemical potential possesses a nonlocal fractional Laplacian operator. The existence of global weak solutions is proven using a Galerkin approximation scheme. A continuous dependence estimate provides uniqueness of the weak solutions and also serves to define a precompact pseudometric. This, in addition to the existence of a bounded absorbing set, shows that the associated semigroup of solution operators admits a compact connected global attractor in the weak energy phase space. The minimal assumptions on the nonlinear potential allow for arbitrary polynomial growth.