Convergence of solutions to a convective Cahn-Hilliard type equation of the sixth order in case of small deposition rates

TL;DR

This paper proves the stabilization of solutions to a sixth-order convective Cahn-Hilliard equation with small deposition rates, using abstract gradient flow theory and Liouville theorems for eternal solutions.

Contribution

It demonstrates the stabilization of solutions for a complex sixth-order PDE under small perturbations, extending gradient flow analysis to this context.

Findings

Solutions stabilize for small deposition rates.

The equation exhibits gradient flow structure in a weak sense.

A Liouville theorem for eternal solutions is established.

Abstract

We show stabilisation of solutions to the sixth-order convective Cahn-Hilliard equation. {The problem} has the structure of a gradient flow perturbed by a quadratic destabilising term with coefficient $\delta>0$. Through application of an abstract result by Carvalho-Langa-Robinson we show that for small $\delta$ the equation has the structure of gradient flow in a weak sense. On the way we prove a kind of Liouville theorem for eternal solutions to parabolic problems. Finally, the desired stabilisation follows from a powerful theorem due to Hale-Raugel.