TY - JOUR

T1 - Time-fractional Cahn–Hilliard equation

T2 - Well-posedness, degeneracy, and numerical solutions

AU - Fritz, Marvin

AU - Rajendran, Mabel L.

AU - Wohlmuth, Barbara

N1 - Publisher Copyright:
© 2022 Elsevier Ltd

PY - 2022/2/15

Y1 - 2022/2/15

N2 - In this paper, we derive the time-fractional Cahn–Hilliard equation from continuum mixture theory with a modification of Fick's law of diffusion. This model describes the process of phase separation with nonlocal memory effects. We analyze the existence, uniqueness, and regularity of weak solutions of the time-fractional Cahn–Hilliard equation. In this regard, we consider degenerating mobility functions and free energies of Landau, Flory–Huggins and double-obstacle type. We apply the Faedo–Galerkin method to the system, derive energy estimates, and use compactness theorems to pass to the limit in the discrete form. In order to compensate for the missing chain rule of fractional derivatives, we prove a fractional chain inequality for semiconvex functions. The work concludes with numerical simulations and a sensitivity analysis showing the influence of the fractional power. Here, we consider a convolution quadrature scheme for the time-fractional component, and use a mixed finite element method for the space discretization.

AB - In this paper, we derive the time-fractional Cahn–Hilliard equation from continuum mixture theory with a modification of Fick's law of diffusion. This model describes the process of phase separation with nonlocal memory effects. We analyze the existence, uniqueness, and regularity of weak solutions of the time-fractional Cahn–Hilliard equation. In this regard, we consider degenerating mobility functions and free energies of Landau, Flory–Huggins and double-obstacle type. We apply the Faedo–Galerkin method to the system, derive energy estimates, and use compactness theorems to pass to the limit in the discrete form. In order to compensate for the missing chain rule of fractional derivatives, we prove a fractional chain inequality for semiconvex functions. The work concludes with numerical simulations and a sensitivity analysis showing the influence of the fractional power. Here, we consider a convolution quadrature scheme for the time-fractional component, and use a mixed finite element method for the space discretization.

KW - Cahn–Hilliard equation

KW - Degenerate mobility

KW - Fractional chain inequality

KW - Time-fractional PDE

KW - Weak solutions

KW - Well-posedness

UR - http://www.scopus.com/inward/record.url?scp=85122644227&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2022.01.002

DO - 10.1016/j.camwa.2022.01.002

M3 - Article

AN - SCOPUS:85122644227

SN - 0898-1221

VL - 108

SP - 66

EP - 87

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

ER -