Abels, Helmut and Bosia, Stefano and Grasselli, Maurizio (2015) Cahn-Hilliard equation with nonlocal singular free energies. ANNALI DI MATEMATICA PURA ED APPLICATA, 194 (4). pp. 1071-1106. ISSN 0373-3114, 1618-1891
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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.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | PHASE SEGREGATION DYNAMICS; LONG-RANGE INTERACTIONS; LOGARITHMIC FREE-ENERGY; GLOBAL ATTRACTORS; PARTICLE-SYSTEMS; BOUNDARY-PROBLEM; MODELS; Cahn-Hilliard equation; Nonlocal free energy; Regional fractional Laplacian; Logarithmic potential; Monotone operators; Global attractors |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics > Prof. Dr. Helmut Abels |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 19 Jun 2019 09:16 |
| Last Modified: | 19 Jun 2019 09:16 |
| URI: | https://pred.uni-regensburg.de/id/eprint/5094 |
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