Analysis and Approximation of a Fractional Cahn--Hilliard Equation

We derive a fractional Cahn--Hilliard equation (FCHE) by considering a gradient flow in the negative order Sobolev space $H^{-\alpha}$, $\alpha\in [0,1]$, where the choice $\alpha=1$ corresponds to the classical Cahn--Hilliard equation while the choice $\alpha=0$ recovers the Allen--Cahn equation. The existence of a unique solution is established and it is shown that the equation preserves mass for all positive values of fractional order $\alpha$ and that it indeed reduces the free energy. We then turn to the delicate question of the $L_\infty$ boundedness of the solution and establish an $L_\infty$ bound for the FCHE in the case where the nonlinearity is a quartic polynomial. As a consequence of the estimates, we are able to show that the Fourier--Galerkin method delivers a spectral rate of convergence for the FCHE in the case of a semidiscrete approximation scheme. Finally, we present results obtained using computational simulation of the FCHE for a variety of choices of fractional order $\alpha$. It is observed that the nature of the solution of the FCHE with a general $\alpha>0$ is qualitatively (and quantitatively) closer to the behavior of the classical Cahn--Hilliard equation than to the Allen--Cahn equation, regardless of how close to zero the value of $\alpha$ is. An examination of the coarsening rates of the FCHE reveals that the asymptotic rate is rather insensitive to the value of $\alpha$ and, as a consequence, is close to the well-established rate observed for the classical Cahn--Hilliard equation.