Well-posedness and Long-time Behavior for a Nonstandard Viscous Cahn–Hilliard System

Analysis and Optimal Boundary Control of a Nonstandard System of Phase Field Equations

“…, N} the problem (2.12)-(2.15) has a unique solution satisfying (2.16)-(2.18), where the index n is replaced by n − 1 . Then it follows with exactly the same argument as in the proof of Theorem 2.1 in [5] that the initial-boundary value problem (2.14), (2.15) has a unique solution ρ n that satisfies (2.16) and the first inequality in (2.18). Substituting ρ n in (2.12), we infer that the linear initial-boundary value problem (2.12), (2.13) has a unique solution µ n satisfying (2.17).…”

“…We differentiate Eq. (2.9) formally with respect to t and test the resulting equation with ρ t (this argument can be made rigorous, see [5]). Since, owing to the convexity of f 1 , f ′′ 1 (ρ) is nonnegative almost everywhere, we find the estimate…”

“…Exactly as in the proof of Theorem 2.1 in [5], the domain integral in the second and third lines of (2.23) can be estimated from above by an expression of the form…”

“…It has been introduced recently in [16] and [5]; for the general physical background, we refer the reader to [16]. The unknown variables are the order parameter ρ , interpreted as a volumetric density, and the chemical potential µ .…”

“…The model, which was introduced in [16], describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been studied recently in [5], [6] for the case of homogeneous Neumann boundary conditions. In this paper, we investigate the case that the boundary condition for one of the unknowns of the system is of third kind and nonhomogeneous.…”