Convergence of solutions for the fractional Cahn-Hilliard system

TL;DR

This paper establishes the global existence, smoothing effects, and convergence to equilibrium of solutions for the fractional Cahn-Hilliard system, using a novel ojasiewicz-Simon inequality adapted for fractional operators.

Contribution

It introduces a new convergence proof for the fractional Cahn-Hilliard equation based on a ojasiewicz-Simon inequality tailored for fractional Laplacians and non-analytic nonlinearities.

Findings

Proves global existence of weak solutions.

Demonstrates parabolic smoothing effects.

Shows solutions converge to equilibrium.

Abstract

This paper deals with the Cauchy-Dirichlet problem for the fractional Cahn-Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called \L ojasiewicz-Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.