Stabilization of solutions to a model of Langmuir-Blodgett films

TL;DR

This paper demonstrates the stabilization of solutions to a one-dimensional advective Cahn-Hilliard equation modeling Langmuir-Blodgett films, using gradient flow structure and abstract mathematical results for small advective terms.

Contribution

It establishes the stabilization of solutions in a perturbed gradient flow setting, extending understanding of Langmuir-Blodgett film models with advection.

Findings

Solutions are stabilized for small advective parameter β.

The equation retains a gradient flow structure in a weak sense.

Finite steady states imply long-term solution stabilization.

Abstract

We show stabilisation of solutions to one-dimensional advective Cahn-Hilliard equation modeling the Langmuir-Blodgett thin films. This problem has the structure of a gradient flow perturbed by a linear term $\beta u_x$. Through application of an abstract result by Carvalho-Langa-Robinson, we show that for small $\beta$ the equation has the structure of gradient flow in a weak sense. Combining this with the finite number of steady states implies stabilization of solutions.