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Stabilization of solutions to a model of Langmuir-Blodgett films

Marco Morandotti, Piotr Rybka, Glen Wheeler

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 $βu_x$. Through application of an abstract result by Carvalho-Langa-Robinson, we show that for small $β$ 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.

Stabilization of solutions to a model of Langmuir-Blodgett films

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 . Through application of an abstract result by Carvalho-Langa-Robinson, we show that for small 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.
Paper Structure (10 sections, 23 theorems, 125 equations)

This paper contains 10 sections, 23 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. The case $\beta = 0$
  3. Continuation to small beta
  4. Semigroup background
  5. Existence of weak solutions to \ref{['eq2']}
  6. Basics of the analytic semigroup theory
  7. Estimates uniform in $\beta$
  8. Stabilization of solutions
  9. Tools of the dynamical systems
  10. Convergence of solutions

Key Result

Theorem 1.1

There exist a positive number $\Lambda_0$ and an at most countable set $E\subset [\Lambda_0,\infty)$ such that, if $L\in (0,\infty)\setminus E$, then there is $\beta^*=\beta^*(L)>0$ with the following properties. If $u_0\in \{u\in H^1(0, L)\colon u(0)=0\}$ and $u$ is the corresponding solution to eq

Theorems & Definitions (47)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • proof
  • ...and 37 more