Stabilization of solutions to a model of Langmuir-Blodgett films

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.