Analysis of a perturbed Cahn-Hilliard model for Langmuir-Blodgett films

TL;DR

This paper investigates a modified Cahn-Hilliard model with transport effects relevant to thin film formation, establishing well-posedness, energy properties, and asymptotic behavior as transport diminishes.

Contribution

It introduces a novel advective Cahn-Hilliard model, proving existence, uniqueness, and convergence properties despite the lack of a gradient flow structure.

Findings

Existence and uniqueness of solutions are established.

The model has a global attractor when transport is small.

Solutions converge to the classical Cahn-Hilliard equation as transport vanishes.

Abstract

An advective Cahn-Hilliard model motivated by thin film formation is studied in this paper. The one-dimensional evolution equation under consideration includes a transport term, whose presence prevents from identifying a gradient flow structure. Existence and uniqueness of solutions, together with continuous dependence on the initial data and an energy equality are proved by combining a minimizing movement scheme with a fixed point argument. Finally, it is shown that, when the contribution of the transport term is small, the equation possesses a global attractor and converges, as the transport term tends to zero, to a purely diffusive Cahn-Hilliard equation.