Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation

Matthieu Brachet (1) , Philippe Parnaudeau (2) , Morgan Pierre (1)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 7348]
2 PPrime [Poitiers] - Institut Pprime [UPR 3346]

We review space and time discretizations of the Cahn-Hilliard equation which are energy stable. In many cases, we prove that a solution converges to a steady state as time goes to infinity. The proof is based on Lyapunov theory and on a Lojasiewicz type inequality. In a few cases, the convergence result is only partial and this raises some interesting questions. Numerical simulations in two and three space dimensions illustrate the theoretical results. Several perspectives are discussed.

Mots clés: Cahn-Hilliard equation; gradient flow; energy stable scheme; Lyapunov stability; Lasalle's invariance principle; Lojasiewicz-Simon inequality; Kurdyka-Lojasiewicz inequality. Contents

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Matthieu Brachet, Philippe Parnaudeau, Morgan Pierre. Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation. 2021. ⟨hal-03466113⟩