Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics
By S. Lisini, Daniel Matthes, Giuseppe Savar'e. 2012
95 citations recorded by the source corpus.
Abstract In this paper, we establish a novel approach to proving existence of non-negative weak solutions for degenerate parabolic equations of fourth order, like the Cahn–Hilliard and certain thin film equations. The considered evolution equations are in the form of a gradient flow for a perturbed Dirichlet energy with respect to a Wasserstein-like transport metric, and weak solutions are obtained as curves of maximal slope. Our main assumption is that the mobility of the particles is a concave function of their spatial density. A qualitative difference of our approach to previous ones is that essential properties of the solution – non-negativity, conservation of the total mass and dissipation of the energy – are automatically guaranteed by the construction from minimizing movements in the energy landscape.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/539e9ab4a37d82d4a0337bae3db949a21a50424c
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In Applied Mathematics
In Computational Theory and Mathematics
Citations in the corpus
- A Review on the Cahn-Hilliard Equation: Classical Results and Recent Advances in Dynamic Boundary Conditions (2021, background): “Moreover, in [30], the authors considered the Cahn–Hilliard type equation with some general nonlinear mobilities (e.g., nonnegative concave functions) as a gradient flow in certain weighted-Wasserstein metric spaces and proved the existence of weak solutions by the variational minimizing movement…”