Global solutions to a generalized Cahn-Hilliard equation with viscosity. (English) Zbl 1032.35075

Bonetti, Elena; Dreyer, Wolfgang; Schimperna, Giulio

Adv. Differ. Equ. 8, No. 2, 231-256 (2003).

The authors of this paper study a viscous Cahn-Hilliard equation modeling the phase separation in a binary metallic alloy. The major advantage of this model is that it takes into account the viscous effects in the diffusion process and the external mechanical forces acting on the alloy. Mathematically, the equation governing the scaled concentration \(\chi\) is given by \(\partial_t \chi -\nabla \cdot(M(\chi) \nabla w) =0\), where \(M(\chi)\) is the mobility matrix and \(w\) is the chemical potential. To incorporate the viscous effects and external forces into the model, \(w\) is assumed to be of the form \[ w =\varepsilon^\lambda \partial_t \chi- \varepsilon^\nu a(\chi) \Delta \chi + g(\chi) + \partial I (\chi) + f, \] where \(\varepsilon\), \(\lambda\) and \(\nu\) are parameters, \(a(\chi)\) characterizes the surface tension, \(g+f\) accounts for the mechanical contributions, and \(\partial I\) denotes the subdifferential of the indicator function \(I\) over \([-1,1]\). It is worth mentioning that the first term in \(w\) represents the viscous effects. Assuming the no-flux boundary condition and that the initial datum \(\chi_0\) is in \(H^1\), the authors established the existence of solutions for the viscous Cahn-Hilliard equation. The major ingredient of the proof is the Galerkin approximation. In addition, the authors have also investigated the asymptotic behavior of the solution as \(\varepsilon\to 0\). In case of \(\lambda =0\) and \(\nu=1\), the weak limit was shown to satisfy a sharp interface model. When \(\lambda=1\) and \(\mu=0\), the weak limit solves a variant of the Cahn-Hilliard equation with a nonconstant mobility.

Reviewer: Jiahong Wu (Stillwater)

MSC:

35K35	Initial-boundary value problems for higher-order parabolic equations
35R35	Free boundary problems for PDEs
35K55	Nonlinear parabolic equations
74N25	Transformations involving diffusion in solids
82B26	Phase transitions (general) in equilibrium statistical mechanics

Keywords:

viscous Cahn-Hilliard equation; existence of solutions; asymptotic limit; no-flux boundary condition; Galerkin approximation