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Author: Sprekels, Jürgen × End date: 2022 × Start date: 2020 × End date: 2022 × Start date: 2020 × Type: Text × Subject: Cahn--Hilliard systems × Subject: Fractional operators × WGL Institute: WIAS ×

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    Longtime behavior for a generalized Cahn-Hilliard system with fractional operators
    (Messina : Accademia Peloritana dei Pericolanti, 2020) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the Cahn-Hilliard system, with possibly singular potentials, that we have recently investigated in the paper Well-posedness and regularity for a generalized fractional Cahn-Hilliard system. More precisely, we study the ω-limit of the phase parameter y and characterize it completely. Our characterization depends on the first eigenvalues λ1≥0 of one of the operators involved: if λ1>0, then the chemical potential μ vanishes at infinity and every element yω of the ω-limit is a stationary solution to the phase equation; if instead λ1=0, then every element yω of the ω-limit satisfies a problem containing a real function μ∞ related to the chemical potential μ. Such a function μ∞ is nonunique and time dependent, in general, as we show by an example. However, we give sufficient conditions for μ∞ to be uniquely determined and constant.
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    An asymptotic analysis for a generalized Cahn--Hilliard system with fractional operators
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In a recent paper the same authors have proved existence, uniqueness and regularity results for a class of viscous and nonviscous Cahn--Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers in the spectral sense of general linear operators, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space of square-integrable functions on a bounded and smooth three-dimensional domain, and have compact resolvents. Here, for the case of the viscous system, we analyze the asymptotic behavior of the solution as the fractional power coefficient of the second operator tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of the second operator appears.