We derive a new, effective macroscopic Cahn–Hilliard equation whose homogeneous free energy is represented by fourth-order polynomials, which form the frequently applied double-well potential. This upscaling is done for perforated/strongly heterogeneous domains. To the best knowledge of the authors, this seems to be the first attempt of upscaling the Cahn–Hilliard equation in such domains. The ...

We consider a diffuse interface model for tumour growth consisting of a Cahn– Hilliard equation with source terms coupled to a reaction-diffusion equation. The coupled system of partial differential equations models a tumour growing in the presence of a nutrient species and surrounded by healthy tissue. The model also takes into account transport mechanisms such as chemotaxis and active transpo...

The Cahn-Hilliard equation is a fourth-order parabolic partial differential equation that is one of the leading models for the study of phase separation in isothermal, isotropic, binary mixtures, such as molten alloys. When a spatially homogeneous alloy is rapidly quenched in a physical experiment, a fine-grained decomposition into two distinct phases is frequently observed; this phenomenon is ...

Large deviation for Markov processes can be studied by Hamilton– Jacobi equation techniques. The method of proof involves three steps: First, we apply a nonlinear transform to generators of the Markov processes, and verify that limit of the transformed generators exists. Such limit induces a Hamilton–Jacobi equation. Second, we show that a strong form of uniqueness (the comparison principle) ho...

We study initial boundary value problems for the convective Cahn-Hilliard equation ∂tu+ ∂ 4 xu+ u∂xu+ ∂ 2 x(|u| u) = 0. It is well-known that without the convective term, the solutions of this equation may blow up in finite time for any p > 0. In contrast to that, we show that the presence of the convective term u∂xu in the Cahn-Hilliard equation prevents blow up at least for 0 < p < 4 9 . We a...