Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations
By Goro Akagi, Giulio Schimperna, Antonio Segatti. 2015
119 citations recorded by the source corpus.
We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain $\Omega$ of $R^N$ and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the entire complement of $\Omega$). 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.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/323cea24cf62a7c47d222962da7968a090334d43
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In Applied Mathematics
In Computational Theory and Mathematics
In Numerical Analysis
Citations in the corpus
- Well-posedness of the Cahn–Hilliard equation with fractional free energy and its Fourier Galerkin approximation (2017, background): “To appear [3] Akagi G, Schimperna G, Segatti A.”
- The phase field method for geometric moving interfaces and their numerical approximations (2019, unknown): “Such models have attracted the attention of community in recent years [8, 239, 327, 405, 408].”
- What is the fractional Laplacian? A comparative review with new results (2018, unknown): “ds for discretizing the fractional Laplacians. Fractional Equation Domain Diusion-Reaction [3] @ tu+ ( ) =2u+ c(t;x)u= 0 (0;+1) Rd Quasi-geostrophic [4] @ t+ ur+ ( ) =2= f [0;T] R2 Cahn-Hilliard [5, 6] @ tu+ ( ) =2( "2 u+ f(u)) = 0 (0;T] (0;2ˇ)2 Porous Medium [5, 7] @ tu+ ( ) =2(jujm 1signu) = ”