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- ItemOptimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: A deep quench approach(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JürgenIn this paper, we investigate a distributed optimal control problem for a convective viscous CahnHilliard system with dynamic boundary conditions. Such systems govern phase separation processes between two phases taking place in an incompressible fluid in a container and, at the same time, on the container boundary. The cost functional is of standard tracking type, while the control is exerted by the velocity of the fluid in the bulk. In this way, the coupling between the state (given by the associated order parameter and chemical potential) and control variables in the governing system of nonlinear partial differential equations is bilinear, which presents a difficulty for the analysis. In contrast to the previous paper Optimal velocity control of a viscous CahnHilliard system with convection and dynamic boundary conditions by the same authors, the bulk and surface free energies are of double obstacle type, which renders the state constraint nondifferentiable. It is well known that for such cases standard constraint qualifications are not satisfied so that standard methods do not apply to yield the existence of Lagrange multipliers. In this paper, we overcome this difficulty by taking advantage of results established in the quoted paper for logarithmic nonlinearities, using a so-called deep quench approximation. We derive results concerning the existence of optimal controls and the first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint system.
- ItemA degenerating Cahn-Hilliard system coupled with complete damage processes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Heinemann, Christian; Kraus, ChristianeComplete damage in elastic solids appears when the material looses all its integrity due to high exposure. In the case of alloys, the situation is quite involved since spinodal decomposition and coarsening also occur at sufficiently low temperatures which may lead locally to high stress peaks. Experimental observations on solder alloys reveal void and crack growth especially at phase boundaries. In this work, we investigate analytically a degenerating PDE system with a time-depending domain for phase separation and complete damage processes under time-varying Dirichlet boundary conditions. The evolution of the system is described by a degenerating parabolic differential equation of fourth order for the concentration, a doubly nonlinear differential inclusion for the damage process and a degenerating quasi-static balance equation for the displacement field. All these equations are strongly nonlinearly coupled....
- ItemExistence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Heinemann, Christian; Kraus, ChristianeIn this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first adopt the notion of weak solutions introduced in [HK11]. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques.
- ItemModeling and analysis of a phase field system for damage and phase separation processes in solids(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bonetti, Elena; Heinemann, Christian; Kraus, Christiane; Segatti, AntonioIn this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function. The main aim of this paper is to show existence of weak solutions for the introduced model, where, in contrast to existing damage models in the literature, different elastic properties of damaged and undamaged material are regarded. To prove existence of weak solutions for the introduced model, we start with an approximation system. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques.
- ItemAsymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Colli, Pierluigi; Gilardi, Gianni; Rocca, Elisabetta; Sprekels, JürgenThis paper is concerned with a phase field system of Cahn-Hilliard type that is related to a tumor growth model and consists of three equations in gianni terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent papers citeCGH and citeCGRS gianni from the viewpoint of well-posedness, long time bhv and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in citeCGRS by showing two independent sets of results as just one of the coefficents tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates