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

Piotr Rybka; Glen Wheeler

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arXiv · 2205.05316

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

Piotr Rybka·Glen Wheeler

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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.

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Piotr Rybka, Glen Wheeler. 2022-05-11. Convergence of solutions to a convective Cahn-Hilliard type equation of the sixth order in case of small deposition rates. https://arxiv.org/abs/2205.05316

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