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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 194, Issue 4, pp 1071–1106 | Cite as

Cahn–Hilliard equation with nonlocal singular free energies

  • Helmut Abels
  • Stefano Bosia
  • Maurizio GrasselliEmail author
Article

Abstract

We consider a Cahn–Hilliard equation which is the conserved gradient flow of a nonlocal total free energy functional. This functional is characterized by a Helmholtz free energy density, which can be of logarithmic type. Moreover, the spatial interactions between the different phases are modeled by a singular kernel. As a consequence, the chemical potential \(\mu \) contains an integral operator acting on the concentration difference \(c\), instead of the usual Laplace operator. We analyze the equation on a bounded domain subject to no-flux boundary condition for \(\mu \) and by assuming constant mobility. We first establish the existence and uniqueness of a weak solution and some regularity properties. These results allow us to define a dissipative dynamical system on a suitable phase–space, and we prove that such a system has a (connected) global attractor. Finally, we show that a Neumann-like boundary condition can be recovered for \(c\), provided that it is supposed to be regular enough.

Keywords

Cahn–Hilliard equation Nonlocal free energy Regional fractional Laplacian Logarithmic potential Monotone operators Global attractors 

Mathematical Subject Classification (2010)

35B41 37L99 45K05 47H05 47J35 80A22 

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Helmut Abels
    • 1
  • Stefano Bosia
    • 2
  • Maurizio Grasselli
    • 2
    Email author
  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  2. 2.Politecnico di MilanoDipartimento di MatematicaMilanItaly

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