Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations

2015/02/23 by Goro Akagi, Akagi, Goro, Giulio Schimperna +3 · 3 citations
Computer Science · Mathematics · #35B25 #35B40 #35K20 #35R11 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP #msc:35B25 #msc:35B40 #msc:35K20 #msc:35R11

paper · pdf · doi:10.48550/arxiv.1502.06383

33 pages

openalex publication_date 2015/02/23 · arxiv created 2015/03/05 · arxiv updated 2015/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain Ω of R^N and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the entire complement of Ω). After setting a proper functional framework, we prove existence and uniqueness of weak solutions to the related initial-boundary value problem. Then, we investigate some significant singular limits obtained as the order of either of the fractional Laplacians appearing in the equation is let tend to 0. In particular, we can rigorously prove that the fractional Allen-Cahn, fractional porous medium, and fractional fast-diffusion equations can be obtained in the limit. Finally, in the last part of the paper, we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits.

Citations

• The Pohozaev Identity for the Fractional Laplacian
• The random walk's guide to anomalous diffusion: a fractional dynamics approach
• Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

Cited by

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• High-order nonuniform time-stepping and MBP-preserving linear schemes for the time-fractional Allen-Cahn equation
• Weak solutions and incompressible limit of a quasi-incompressible Navier--Stokes/Cahn--Hilliard model for viscous two-phase flows

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