Fractional Cahn-Hilliard Equation(s): Analysis, Properties and Approximation
Mark Ainsworth
Brown University
The classical Cahn-Hilliard equation [1] is a nonlinear, fourth order in space, parabolic partial differential equation which is often used as a diffuse interface model for the phase separation of a binary alloy. Despite the widespread adoption of the model, there are good reasons for preferring models in which fractional spatial derivatives appear [2,3].
We consider two such Fractional Cahn-Hilliard equations (FCHE). The first [4] corresponds to considering a gradient flow of the free energy functional in a negative order Sobolev space \(H^{-\alpha}\), \(\alpha\in[0,1]\) where the choice \(\alpha=1\) corresponds to the classical Cahn-Hilliard equation whilst the choice \(\alpha=0\) recovers the Allen-Cahn equation.
It is shown that the equation preserves mass for all positive values of fractional order and that it indeed reduces the free energy. The well-posedness of the problem is established in the sense that the \(H^1\)-norm of the solution remains uniformly bounded.
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 non-linearity 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 semi-discrete ap- proximation scheme.
Finally, we present results obtained using computational simulation of the FCHE for a variety of choices of fractional order \(\alpha\). We then consider an alternative FCHE [3,5] in which the free energy functional involves a fractional order derivative.
(joint work with Zhiping Mao)
References:
[1] J.W. Cahn and J.E. Hilliard, Free energy of a non-uniform system. I. Interfacial Free Energy, J.
Chem. Phys, 28, 258-267 (1958)
[2] L. Caffarelli and E. Valdinoci, A Priori Bounds for solutions of non-local evoluation PDE, Springer,
Milan 2013.
[3] G. Palatucci and O. Savin, Local and global minimisers for a variational energy involving a
fractional norm, Ann. Mat. Pura Appl., 4, 673-718 (2014).
[4] M. Ainsworth and Z. Mao, Analysis and Approximation of a Fractional Cahn-Hilliard Equation,
(In review, 2016).
[5] M. Ainsworth and Z. Mao, Well-posedness of the Cahn-Hilliard Equation with Fractional Free
Energy and Its Fourier-Galerkin Discretization, (In review, 2017).