Convergence to equilibrium for a second-order time semi-discretization ofthe Cahn-Hilliard equation

Paola F. Antonietti; Benoît Merlet; Morgan Pierre; Marco Verani

doi:10.3934/Math.2016.3.178

AIMS Mathematics, 2016, 1(3): 178-194. doi: 10.3934/Math.2016.3.178. Research article Special Issues

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Convergence to equilibrium for a second-order time semi-discretization ofthe Cahn-Hilliard equation

Paola F. Antonietti, Benoît Merlet, Morgan Pierre, Marco Verani

1 MOX– Laboratory for Modeling and Scientific Computing, Dipartimento di Matematica,Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
2 Laboratoire Paul Painlevé, U.M.R. CNRS 8524, Université Lille 1, Cité Scientifique, F-59655 Villeneuve d’Ascq Cedex, France
3 Team RAPSODI, Inria Lille - Nord Europe, 40 av. Halley, F-59650 Villeneuve d’Ascq, France
4 Laboratoire de Mathématiques et Applications, Université de Poitiers, CNRS, F-86962 Chasseneuil, France

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Special Issues: Nonlinear Evolution PDEs, Interfaces and Applications

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We consider a second-order two-step time semi-discretization of the Cahn-Hilliard equation with an analytic nonlinearity. The time-step is chosen small enough so that the pseudo-energy associated withdiscretization is nonincreasing at every time iteration. We prove that the sequence generated by the scheme converges to a steady state as time tends to infinity. We also obtain convergence rates in the energy norm. The proof is based on the Łojasiewicz-Simon inequality.

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Keywords: Łojasiewicz-Simon inequality; gradient flow; Cahn-Hilliard; backward differentiation formula

Citation: Paola F. Antonietti, Benoît Merlet, Morgan Pierre, Marco Verani. Convergence to equilibrium for a second-order time semi-discretization ofthe Cahn-Hilliard equation. AIMS Mathematics, 2016, 1(3): 178-194. doi: 10.3934/Math.2016.3.178

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