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estimates for the Cahn–Hilliard equation with obstacle free energy

Ľubomír BaňasRobert Nürnberg — 2009

ESAIM: Mathematical Modelling and Numerical Analysis

We derive estimates for a discretization in space of the standard Cahn–Hilliard equation with a double obstacle free energy. The derived estimates are robust and efficient, and in practice are combined with a heuristic time step adaptation. We present numerical experiments in two and three space dimensions and compare our method with an existing heuristic spatial mesh adaptation algorithm.

Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere

Ľubomír BaňasZdzisław BrzeźniakMikhail NeklyudovMartin OndrejátAndreas Prohl — 2015

Czechoslovak Mathematical Journal

We study ergodic properties of stochastic geometric wave equations on a particular model with the target being the 2D sphere while considering only solutions which are independent of the space variable. This simplification leads to a degenerate stochastic equation in the tangent bundle of the 2D sphere. Studying this equation, we prove existence and non-uniqueness of invariant probability measures for the original problem and obtain also results on attractivity towards an invariant measure. We also...

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