Abstract
We introduce and analyze the nonlocal variants of two Cahn-Hilliard type equations with reaction terms. The first one is the so-called Cahn-Hilliard-Oono equation which models, for instance, pattern formation in diblock-copolymers as well as in binary alloys with induced reaction and type-I superconductors. The second one is the Cahn-Hilliard type equation introduced by Bertozzi et al. to describe image inpainting. Here we take a free energy functional which accounts for nonlocal interactions. Our choice is motivated by the work of Giacomin and Lebowitz who showed that the rigorous physical derivation of the Cahn-Hilliard equation leads to consider nonlocal functionals.The equations also have a transport term with a given velocity field and are subject to a homogenous Neumann boundary condition for the chemical potential, i.e., the first variation of the free energy functional. We first establish the well-posedness of the corresponding initial and boundary value problems in a weak setting. Then we consider such problems as dynamical systems and we show that they have bounded absorbing sets and global attractors.
Highlights
Consider an incompressible isothermal and isotropic binary solution of constant molar volume which is initially homogeneous
Given a bounded domain Ω ⊂ Rd with d ≤ 3 occupied by a binary mixture of components A and B with a mass fraction of φA(x) and φB(x), respectively, setting φ := φA − φB, the free energy functional is given by
Some works have been devoted to study the nonlocal Cahn-Hilliard equation coupled with the Navier-Stokes system
Summary
Consider an incompressible isothermal and isotropic binary solution of constant molar volume which is initially homogeneous. Some works have been devoted to study the nonlocal Cahn-Hilliard equation coupled with the Navier-Stokes system (see [18, 24, 25]) This is a nonlocal variant of a well-known diffuse interface model for phase separation in two-phase incompressible and isothermal fluids (model H). By defining φ as the difference between the concentrations, the following Cahn-Hilliard type equation is proposed (see [31]) This equation is known as the CHO equation and was introduced in [2] (cf [48], ) for phase separation in diblock copolymers (see [47]). We adapt the strategy used in [18]
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