A vanishing diffusion limit in a nonstandard system of phase field equations
- Colli, Pierluigi
- Gilardi, Gianni
- Krejčí, Pavel
- Sprekels, Jürgen
2010 Mathematics Subject Classification
- 35K61 35A05 35B40 74A15
- nonstandard phase field system, nonlinear partial differential equations, asymptotic limit, convergence of solutions
We are concerned with a nonstandard phase field model of Cahn--Hilliard type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated by Colli, Gilardi, Podio-Guidugli, and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011, and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treat the very general case in which the diffusivity coefficient of the parabolic PDE is allowed to depend nonlinearly on both variables. In the same framework, this paper investigates the asymptotic limit of the solutions to the initial-boundary value problems as the diffusion coefficient $sigma$ in the equation governing the evolution of the order parameter tends to zero. We prove that such a limit actually exists and solves the limit problem, which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics. In the case of a constant diffusivity, we are able to show uniqueness and to improve the regularity of the solution.
- Evol. Equ. Control Theory, 3 (2014) pp. 257--275.