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- ItemDistributed optimal control of a nonstandard system of phase field equations : dedicated to Prof. Dr. Ingo Müller on the occasion of his 75th birthday(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Colli, Pierluigi; Gilardi, Gianni; Podio-Guidugli, Paolo; Sprekels, Jürgen; Müller, IngoWe investigate a distributed optimal control problem for a phase field model of Cahn-Hilliard type. The model describes two-species phase segregation on an atomic lattice under the presence of diffusion; it has been introduced recently in [4], on the basis of the theory developed in [15], and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.
- ItemAn asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Colli, Pierluigi; Gilardi, Gianni; Podio-Guidugli, Paola; Sprekels, JürgenThis paper is concerned with a diffusion model of phase-field type, consisting of a parabolic system of two partial differential equations, interpreted as balances of microforces and microenergy, for two unknowns: the problem's order parameter $rho$ and the chemical potential $mu$; each equation includes a viscosity term -- respectively, $varepsilon,partial_tmu$ and $delta,partial_trho$ -- with $varepsilon$ and $delta$ two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper [5], we proved that this problem is well-posed and investigated the long-time behavior of its $(varepsilon,delta)-$solutions. Here we discuss the asymptotic limit of the system as $eps$ tends to 0. We prove convergence of $(varepsilon,delta)-$solutions to the corresponding solutions for the case $eps$ =0, whose long-time behavior we characterize; in the proofs, we employ compactness and monotonicity arguments.
- ItemGlobal existence for a strongly coupled Cahn-Hilliard system with viscosity : in memory of Enrico Magenes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Colli, Pierluigi; Gilardi, Gianni; Podio-Guidugli, Paolo; Sprekels, Jürgen; Magenes, EnricoAn existence result is proved for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system is meant to model two-species phase segregation on an atomic lattice under the presence of diffusion. A similar system has been recently introduced and analyzed in [CGPS11]. Both systems conform to the general theory developed in [Pod06]: two parabolic PDEs, interpreted as balances of microforces and microenergy, are to be solved for the order parameter $rho$ and the chemical potential $mu$. In the system studied in this note, a phase-field equation in $rho$ fairly more general than in [CGPS11] is coupled with a highly nonlinear diffusion equation for $mu$, in which the conductivity coefficient is allowed to depend nonlinearly on both variables.
- ItemA vanishing diffusion limit in a nonstandard system of phase field equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Colli, Pierluigi; Gilardi, Gianni; Krejci, Pavel; Sprekels, JürgenWe are concerned with a nonstandard phase field model of CahnHilliard 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 ơ 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.
- ItemContinuous dependence for a nonstandard Cahn-Hilliard system with nonlinear atom mobility(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Colli, Pierluigi; Gilardi, Gianni; Podio-Guidugli, Paolo; Sprekels, JürgenThis note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice [Podio-Guidugli 2006]; it consists of the balance equations of microforces and microenergy; the two unknowns are the order parameter $rho$ and the chemical potential $mu$. Some recent results obtained for this class of problems is reviewed and, in the case of a nonconstant and nonlinear atom mobility, uniqueness and continuous dependence on the initial data are shown with the help of a new line of argumentation developed in Colli/Gilardi/Podio-Guidugli/Sprekels 2012.
- ItemGlobal existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Colli, Pierluigi; Gilardi, Gianni; Podio-Guidugli, Paolo; Sprekels, JürgenExistence and uniqueness are investigated for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model two-species phase segregation on an atomic [19]; in the balance equations of microforces and microenergy, the two unknowns are the order parameter $rho$ and the chemical potential $mu$. A simpler version of the same system has recently been discussed in [8]. In this paper, a fairly more general phase-field equation for $rho$ is coupled with a genuinely nonlinear diffusion equation for $mu$. The existence of a global-in-time solution is proved with the help of suitable a priori estimates. In the case of costant atom mobility, a new and rather unusual uniqueness proof is given, based on a suitable combination of variables.