A continuous dependence result for a nonstandard system of phase field equations

Abstract

The present note deals with a nonstandard system of differential equations describing a two‐species phase segregation. This system naturally arises in the asymptotic analysis carried out recently by the same authors, as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. In particular, an existence result has been proved for the limit system in a very general framework. On the contrary, uniqueness was shown by assuming a constant mobility coefficient. Here, we generalize this result and prove a continuous dependence property in the case that the mobility coefficient suitably depends on the chemical potential. Copyright © 2013 John Wiley & Sons, Ltd.

Limiting Problems for a Nonstandard Viscous Cahn–Hilliard System with Dynamic Boundary Conditions

“…This estimate requires more effort. At first, observe that (2.14) implies that 11) where, thanks to (3.1), 1/(ε + 2g(ρ ε )) ≤ 1/(2g * ) for all ε > 0. Now, using (3.11), we find that…”

Global Existence for a Nonstandard Viscous Cahn--Hilliard System with Dynamic Boundary Condition

“…If one puts, without loss of generality, ε = δ = 1, then one obtains the more general system 1 + 2g(ρ) ∂ t µ + µ g ′ (ρ) ∂ t ρ − ∆µ = 0, (1.17) ∂ t ρ − ∆ρ + W ′ (ρ) = µ g ′ (ρ), (1.18) which was investigated in the contributions [11,15,17,19], still for no-flux boundary conditions, also from the side of the numerical approximation. The related phase relaxation system (in which the diffusive term −∆ρ disappears from (1.18)), has been dealt with in [12,13,21]. We also mention the recent article [25], where a nonlocal version of (1.17)-(1.18) -based on the replacement of the diffusive term of (1.18) with a nonlocal operator acting on ρ -has been largely investigated.…”

On an application of Tikhonov's fixed point theorem to a nonlocal Cahn–Hilliard type system modeling phase separation

“…Later, the local free energy density (1.1) was generalized to the form ψ = ψ(ρ, ∇ρ, µ) = −µ g(ρ) + F (ρ) + σ 2 |∇ρ| 2 (1.9) with a function g having suitable (see below) properties. If one puts, without loss of generality, ε = δ = 1, then one obtains the more general system 1 + 2g(ρ) ∂ t µ + µ g ′ (ρ) ∂ t ρ − ∆µ = 0 (1.10) ∂ t ρ − σ ∆ρ + F ′ (ρ) = µ g ′ (ρ), (1.11) which was investigated in the papers [12,13,15,17,19].…”