ANALYSIS AND APPROXIMATION OF A FRACTIONAL CAHN-HILLIARD EQUATION

摘要

We derive a fractional Cahn-Hilliard equation (FCHE) by considering a gradient flow in the negative order Sobolev space H-alpha, a is an element of [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 a and that it indeed reduces the free energy. We then turn to the delicate question of the L infinity, boundedness of the solution and establish an L infinity, 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 a. 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 a is. An examination of the coarsening rates of the FCHE reveals that the asymptotic rate is rather insensitive to the value of a and, as a consequence, is close to the well-established rate observed for the classical Cahn-Hilliard equation.