Overview

A diffuse phase field model is a mathematical model for describing microstructural phenomena and for predicting morphological evolution on the mesoscale. It is applied to a wide variety of material processes such as solidification, coarsening in alloys, crack propagation and martensitic transformations.
The phase field models are usually based on a free energy functional depending on an order parameter (the phase field). The interfacial dynamics is modeled by a partial evolution equation for the order parameter. For a system of two components, the values of the solution are close to two different values, that represent the two phases, and varies smoothly between these values in the zone around the interface. The evolution equation for the order parameter is typically coupled with an additional evolution equation, which comprises the balance of momentum or energy. This often leads to highly nonlinear PDE-systems of elliptic-parabolic type with constraints. Such PDE-systems are investigated at WIAS concerning their analytically properties such as existence, uniqueness and regularity of solutions. In a further step, numerical simulations of the PDE-systems are carried out at WIAS.

Since the interfaces in a diffuse phase field model have a small transition area no boundary conditions are needed. In particular, no assumptions on the shape of the interfaces or mutual distributions are required. For this reason phase field models have also become a useful tool for numerical simulations, predicting complex morphological evolutions.


Phase separation and simultaneously cooling on the left edge for a binary alloy
(Markus Radszuweit)

Typical applications where phase field models are utilized are for instance for phase separation, coarsening and damage processes. The picture displays coarsening processes and crack propagtion in binary alloys.
These processes can be described by Cahn-Hilliard equations where the order parameter models the chemical concentration in the binary alloy coupled with elasticity and a variational inequality for the inner damage variable.


Snapshots from a numerical simulation of phase separation and damage with material-dependent elasticity tensor and time-varying loading. From top left to bottom right: Phase field, elastic energy, damage field (with grid), and temperature field (Markus Radszuweit)


Highlights

Modeling, analysis, and numerics of a temperature-dependent phase field model for phase separation and damage

A novel model for phase separation and damage in thermoviscoelastic materials is derived and studied in Preprint 2164. The main novelty of the paper consists in the fact that, in contrast with previous works in the literature, thermal processeses are encompassed in the model, nonlinearly coupled with the damage, concentration and displacement evolutions. More in particular, existence of "entropic weak solutions" are proven, resorting to a solvability for the study of PDE systems for phase transition and damage. The global-in-time existence result is obtained by passing to the limit in a carefully devised time-discretization scheme.

Publications

  Monographs

  • P. Colli, A. Damlamian, N. Kenmochi, M. Mimura, J. Sprekels, eds., Proceedings of International Conference on: Nonlinear Phenomena with Energy Dissipation: Mathematical Analysis, Modeling and Simulation, 29 of Gakuto International Series Mathematical Sciences and Applications, Gakkōtosho, Tokyo, 2008, 475 pages, (Collection Published).

  • P. Colli, N. Kenmochi, J. Sprekels, eds., Dissipative Phase Transitions, 71 of Series on Advances in Mathematics for Applied Sciences, World Scientific, Singapore, 2006, xii+300 pages, (Collection Published).

  Articles in Refereed Journals

  • P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Asymptotic analyses and error estimates for a Cahn--Hilliard type phase field system modelling tumor growth, Discrete and Continuous Dynamical Systems, 10 (2017) pp. 37--54.
    Abstract
    This 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

  • S.P. Frigeri, Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities, Mathematical Models & Methods in Applied Sciences, 26 (2016) pp. 1957--1993.
    Abstract
    We consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one recently derived by Abels, Garcke and Grün and consists of a Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. The density of the mixture depends on an order parameter. For this nonlocal system we prove existence of global dissipative weak solutions for the case of singular double-well potentials and non degenerate mobilities. To this goal we devise an approach which is completely independent of the one employed by Abels, Depner and Garcke to establish existence of weak solutions for the local Abels et al. model.

  • M. Dai, E. Feireisl, E. Rocca, G. Schimperna, M.E. Schonbek, On asymptotic isotropy for a hydrodynamic model of liquid crystals, Asymptotic Analysis, 97 (2016) pp. 189--210.
    Abstract
    We study a PDE system describing the motion of liquid crystals by means of the Q?tensor description for the crystals coupled with the incompressible Navier-Stokes system. Using the method of Fourier splitting, we show that solutions of the system tend to the isotropic state at the rate (1 + t)?? as t ? ? 1 for a certain ? > 2 .

  • P. Colli, G. Gilardi, J. Sprekels, A boundary control problem for the viscous Cahn--Hilliard equation with dynamic boundary conditions, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 73 (2016) pp. 195--225.
    Abstract
    A boundary control problem for the viscous Cahn-Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved.

  • P. Colli, G. Gilardi, J. Sprekels, On an application of Tikhonov's fixed point theorem to a nonlocal Cahn--Hilliard type system modeling phase separation, Journal of Differential Equations, 260 (2016) pp. 7940--7964.
    Abstract
    This paper investigates a nonlocal version of a model for phase separation on an atomic lattice that was introduced by P. Podio-Guidugli in Ric. Mat. 55 (2006) 105-118. The model consists of an initial-boundary value problem for a nonlinearly coupled system of two partial differential equations governing the evolution of an order parameter ρ and the chemical potential μ. Singular contributions to the local free energy in the form of logarithmic or double-obstacle potentials are admitted. In contrast to the local model, which was studied by P. Podio-Guidugli and the present authors in a series of recent publications, in the nonlocal case the equation governing the evolution of the order parameter contains in place of the Laplacian a nonlocal expression that originates from nonlocal contributions to the free energy and accounts for possible long-range interactions between the atoms. It is shown that just as in the local case the model equations are well posed, where the technique of proving existence is entirely different: it is based on an application of Tikhonov's fixed point theorem in a rather unusual separable and reflexive Banach space.

  • M. Korzec, A. Münch, E. Süli, B. Wagner, Anisotropy in wavelet based phase field models, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 21 (2016) pp. 1167--1187.
    Abstract
    Anisotropy is an essential feature of phase-field models, in particular when describing the evolution of microstructures in solids. The symmetries of the crystalline phases are reflected in the interfacial energy by introducing corresponding directional dependencies in the gradient energy coefficients, which multiply the highest order derivative in the phase-field model. This paper instead considers an alternative approach, where the anisotropic gradient energy terms are replaced by a wavelet analogue that is intrinsically anisotropic and linear. In our studies we focus on the classical coupled temperature - Ginzburg-Landau type phase-field model for dendritic growth. For the resulting derivative-free wavelet analogue existence, uniqueness and continuous dependence on initial data for weak solutions is proved. The ability to capture dendritic growth similar to the results obtained from classical models is investigated numerically.

  • M.H. Farshbaf Shaker, C. Hecht, Optimal control of elastic vector-valued Allen--Cahn variational inequalities, SIAM Journal on Control and Optimization, 54 (2016) pp. 129--152.
    Abstract
    In this paper we consider a elastic vector-valued Allen--Cahn MPCC (Mathematical Programs with Complementarity Constraints) problem. We use a regularization approach to get the optimality system for the subproblems. By passing to the limit in the optimality conditions for the regularized subproblems, we derive certain generalized first-order necessary optimality conditions for the original problem.

  • S.P. Frigeri, E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal Cahn--Hilliard/Navier--Stokes system in two dimensions, SIAM Journal on Control and Optimization, 54 (2016) pp. 221 -- 250.
    Abstract
    We study a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids coupling the Navier-Stokes system with a convective nonlocal Cahn-Hilliard equation in two dimensions of space. We apply recently proved well-posedness and regularity results in order to establish existence of optimal controls as well as first-order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid flow.

  • E. Rocca, R. Rossi, ``Entropic'' solutions to a thermodynamically consistent PDE system for phase transitions and damage, SIAM Journal on Mathematical Analysis, 74 (2015) pp. 2519--2586.
    Abstract
    In this paper we analyze a PDE system modelling (non-isothermal) phase transitions and dam- age phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L^1. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as ``entropic'', where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its ``entropic'' formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.

  • E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal convective Cahn--Hilliard equation by the velocity in three dimensions, SIAM Journal on Control and Optimization, 53 (2015) pp. 1654--1680.
    Abstract
    In this paper we study a distributed optimal control problem for a nonlocal convective Cahn-Hilliard equation with degenerate mobility and singular potential in three dimensions of space. While the cost functional is of standard tracking type, the control problem under investigation cannot easily be treated via standard techniques for two reasons: the state system is a highly nonlinear system of PDEs containing singular and degenerating terms, and the control variable, which is given by the velocity of the motion occurring in the convective term, is nonlinearly coupled to the state variable. The latter fact makes it necessary to state rather special regularity assumptions for the admissible controls, which, while looking a bit nonstandard, are however quite natural in the corresponding analytical framework. In fact, they are indispensable prerequisites to guarantee the well-posedness of the associated state system. In this contribution, we employ recently proved existence, uniqueness and regularity results for the solution to the associated state system in order to establish the existence of optimal controls and appropriate first-order necessary optimality conditions for the optimal control problem.

  • S.P. Frigeri, M. Grasselli, E. Rocca, A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility, Nonlinearity, 28 (2015) pp. 1257--1293.
    Abstract
    We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier-Stokes system coupled with a convective nonlocal Cahn-Hilliard equation with non-constant mobility. We first prove the existence of a global weak solution in the case of non-degenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of the global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective nonlocal Cahn-Hilliard equation with degenerate mobility and singular potential in dimension three.

  • M. Heida, Existence of solutions for two types of generalized versions of the Cahn--Hilliard equation, Applications of Mathematics, 60 (2015) pp. 51--90.
    Abstract
    We show existence of solutions to two types of generalized anisotropic Cahn-Hilliard problems: In the first case, we assume the mobility to be dependent on the concentration and its gradient, where the system is supplied with dynamic boundary conditions. In the second case, we deal with classical no-flux boundary conditions where the mobility depends on concentration u, gradient of concentration ?u and the chemical potential ?u?s?(u). The existence is shown using a newly developed generalization of gradient flows by the author and the theory of Young measures.

  • M. Heida, On systems of Cahn--Hilliard and Allen--Cahn equations considered as gradient flows in Hilbert spaces, Journal of Mathematical Analysis and Applications, 423 (2015) pp. 410--455.

  • E. Bonetti, Ch. Heinemann, Ch. Kraus, A. Segatti, Modeling and analysis of a phase field system for damage and phase separation processes in solids, Journal of Partial Differential Equations, 258 (2015) pp. 3928--3959.
    Abstract
    In 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 system, we start with a regularized version. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques.

  • P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Vanishing viscosities and error estimate for a Cahn--Hilliard type phase field system related to tumor growth, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 26 (2015) pp. 93--108.
    Abstract
    In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn--Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in [Colli-Gilardi-Hilhorst 2015], letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.

  • P. Colli, G. Gilardi, J. Sprekels, A boundary control problem for the pure Cahn--Hilliard equation with dynamic boundary conditions, Advances in Nonlinear Analysis, 4 (2015) pp. 311--325.
    Abstract
    A boundary control problem for the pure Cahn--Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first-order necessary conditions for optimality are proved.

  • P. Colli, M.H. Farshbaf Shaker, G. Gilardi, J. Sprekels, Optimal boundary control of a viscous Cahn--Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM Journal on Control and Optimization, 53 (2015) pp. 2696--2721.
    Abstract
    In this paper, we investigate optimal boundary control problems for Cahn--Hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace--Beltrami operator. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy, which follows the lines of the recent approach by Colli, Farshbaf-Shaker, Sprekels (see Appl. Math. Optim., 2014) to the (simpler) Allen--Cahn case, is the following: we use the results that were recently established by Colli, Gilardi, Sprekels in the preprint arXiv:1407.3916 [math.AP] for the case of (differentiable) logarithmic potentials and perform a so-called ``deep quench limit''. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (non-differentiable) double obstacle potentials.

  • P. Colli, M.H. Farshbaf Shaker, G. Gilardi, J. Sprekels, Second-order analysis of a boundary control problem for the viscous Cahn--Hilliard equation with dynamic boundary conditions, Annals of the Academy of Romanian Scientists. Mathematics and its Applications., 7 (2015) pp. 41--66.
    Abstract
    In this paper we establish second-order sufficient optimality conditions for a boundary control problem that has been introduced and studied by three of the authors in the preprint arXiv:1407.3916. This control problem regards the viscous Cahn--Hilliard equation with possibly singular potentials and dynamic boundary conditions.

  • P. Colli, M.H. Farshbaf Shaker, J. Sprekels, A deep quench approach to the optimal control of an Allen--Cahn equation with dynamic boundary conditions and double obstacles, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 71 (2015) pp. 1--24.
    Abstract
    In this paper, we investigate optimal control problems for Allen-Cahn variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy is the following: we use the results that were recently established by two of the authors for the case of (differentiable) logarithmic potentials and perform a so-called ``deep quench limit''. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (non-differentiable) double obstacle potentials.

  • P. Colli, J. Sprekels, Optimal control of an Allen--Cahn equation with singular potentials and dynamic boundary condition, SIAM Journal on Control and Optimization, 53 (2015) pp. 213--234.
    Abstract
    In this paper, we investigate optimal control problems for Allen--Cahn equations with singular nonlinearities and a dynamic boundary condition involving singular nonlinearities and the Laplace--Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. Parabolic problems with nonlinear dynamic boundary conditions involving the Laplace--Beltrami operation have recently drawn increasing attention due to their importance in applications, while their optimal control was apparently never studied before. In this paper, we first extend known well-posedness and regularity results for the state equation and then show the existence of optimal controls and that the control-to-state mapping is twice continuously Fréchet differentiable between appropriate function spaces. Based on these results, we establish the first-order necessary optimality conditions in terms of a variational inequality and the adjoint state equation, and we prove second-order sufficient optimality conditions.

  • E. Feireisl, E. Rocca, G. Schimperna, A. Zarnescu, Nonisothermal nematic liquid crystal flows with the Ball--Majumdar free energy, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica Ü. Dini", Firenze; Springer-Verlag, Heidelberg. English, French, German, Italian, English abstracts., 194 (2015) pp. 1269--1299.
    Abstract
    In this paper we prove the existence of global in time weak solutions for an evolutionary PDE system modelling nonisothermal Landau-de Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible Navier-Stokes system for the macroscopic velocity $vu$ is coupled to a nonlinear convective parabolic equation describing the evolution of the Q-tensor $QQ$, namely a tensor-valued variable representing the normalized second order moments of the probability distribution function of the LC molecules. The effects of the (absolute) temperature $vt$ are prescribed in the form of an energy balance identity complemented with a global entropy production inequality. Compared to previous contributions, we can consider here the physically realistic singular configuration potential $f$ introduced by Ball and Majumdar. This potential gives rise to severe mathematical difficulties since it introduces, in the Q-tensor equation, a term which is at the same time singular in $QQ$ and degenerate in $vt$. To treat it a careful analysis of the properties of $f$, particularly of its blow-up rate, is carried out.

  • M.G. Hennessy, V.M. Burlakov, A. Münch, B. Wagner, A. Goriely, Controlled topological transitions in thin-film phase separation, SIAM Journal on Applied Mathematics, 75 (2015) pp. 38--60.
    Abstract
    In this paper the evolution of a binary mixture in a thin-film geometry with a wall at the top and bottom is considered. Bringing the mixture into its miscibility gap so that no spinodal decomposition occurs in the bulk, a slight energetic bias of the walls towards each one of the constituents ensures the nucleation of thin boundary layers that grow until the constituents have moved into one of the two layers. These layers are separated by an interfacial region, where the composition changes rapidly. Conditions that ensure the separation into two layers with a thin interfacial region are investigated based on a phase-field model and using matched asymptotic expansions a corresponding sharp-interface problem for the location of the interface is established. It is then argued that a thus created two-layer system is not at its energetic minimum but destabilizes into a controlled self-replicating pattern of trapezoidal vertical stripes by minimizing the interfacial energy between the phases while conserving their area. A quantitative analysis of this mechanism is carried out via a new thin-film model for the free interfaces, which is derived asymptotically from the sharp-interface model.

  • M.H. Farshbaf Shaker, A relaxation approach to vector-valued Allen--Cahn MPEC problems, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 72 (2015) pp. 325--351.

  • CH. Heinemann, Ch. Kraus, Complete damage in linear elastic materials -- Modeling, weak formulation and existence results, Calculus of Variations and Partial Differential Equations, 54 (2015) pp. 217--250.
    Abstract
    We introduce a complete damage model with a time-depending domain for linear-elastically stressed solids under time-varying Dirichlet boundary conditions. The evolution of the system is described by a doubly nonlinear differential inclusion for the damage process and a quasi-static balance equation for the displacement field. For the introduced complete damage model, we propose a classical formulation and a corresponding suitable weak formulation in an $SBV$-framework. We show that the classical differential inclusion can be regained from the notion of weak solutions under certain regularity assumptions. The main aim of this work is to prove local-in-time existence and global-in-time existence in some weaker sense for the introduced model. In contrast to incomplete damage theories, the material can be exposed to damage in the proposed model in such a way that the elastic behavior may break down on the damaged parts of the material, i.e. we loose coercivity properties of the free energy. This leads to several mathematical difficulties. For instance, it might occur that not completely damaged material regions are isolated from the Dirichlet boundary. In this case, the deformation field cannot be controlled in the transition from incomplete to complete damage. To tackle this problem, we consider the evolution process on a time-depending domain. In this context, two major challenges arise: Firstly, the time-dependent domain approach leads to jumps in the energy which have to be accounted for in the energy inequality of the notion of weak solutions. To handle this problem, several energy estimates are established by $Gamma$-convergence techniques. Secondly, the time-depending domain might have bad smoothness properties such that Korn's inequality cannot be applied. To this end, a covering result for such sets with smooth compactly embedded domains has been shown.

  • CH. Heinemann, Ch. Kraus, Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects, Discrete and Continuous Dynamical Systems, 35 (2015) pp. 2565--2590.
    Abstract
    In 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 generalize the notion of weak solution introduced in WIAS 1520. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques.

  • CH. Heinemann, Ch. Kraus, Existence of weak solutions for a hyperbolic-parabolic phase field system with mixed boundary conditions on non-smooth domains, SIAM Journal on Mathematical Analysis, 47 (2015) pp. 2044--2073.
    Abstract
    The aim of this paper is to prove existence of weak solutions of hyperbolic-parabolic evolution inclusions defined on Lipschitz domains with mixed boundary conditions describing, for instance, damage processes and elasticity with inertial effects. To this end, we first present a suitable weak formulation in order to deal with such evolution inclusions. Then, existence of weak solutions is proven by utilizing time-discretization, $H^2$--regularization and variational techniques.

  • CH. Heinemann, E. Rocca, Damage processes in thermoviscoelastic materials with damage-dependent thermal expansion coefficients, Mathematical Methods in the Applied Sciences, 38 (2015) pp. 4587--4612.
    Abstract
    In this paper we prove existence of global in time weak solutions for a highly nonlinear PDE system arising in the context of damage phenomena in thermoviscoelastic materials. The main novelty of the present contribution with respect to the ones already present in the literature consists in the possibility of taking into account a damage-dependent thermal expansion coefficient. This term implies the presence of nonlinear couplings in the PDE system, which make the analysis more challenging.

  • E. Rocca, R. Rossi, A degenerating PDE system for phase transitions and damage, Mathematical Models & Methods in Applied Sciences, 24 (2014) pp. 1265--1341.

  • G. Aki, W. Dreyer, J. Giesselmann, Ch. Kraus, A quasi-incompressible diffuse interface model with phase transition, Mathematical Models & Methods in Applied Sciences, 24 (2014) pp. 827--861.
    Abstract
    This work introduces a new thermodynamically consistent diffuse model for two-component flows of incompressible fluids. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. To this end, we consider two scaling regimes where in one case we recover the Euler equations and in the other case the Navier-Stokes equations in the bulk phases equipped with admissible interfacial conditions. For the Navier-Stokes regime, we further assume the densities of the fluids are close to each other in the sense of a small parameter which is related to the interfacial thickness of the diffuse model.

  • P. Colli, G. Gilardi, P. Krejčí, P. Podio-Guidugli, J. Sprekels, Analysis of a time discretization scheme for a nonstandard viscous Cahn--Hilliard system, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014) pp. 1061--1087.
    Abstract
    In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of nonlinearities. Moreover, for the difference of discrete and continuous solutions we prove an error estimate of order one with respect to the time step.

  • P. Colli, G. Gilardi, P. Krejčí, J. Sprekels, A continuous dependence result for a nonstandard system of phase field equations, Mathematical Methods in the Applied Sciences, 37 (2014) pp. 1318--1324.
    Abstract
    The present note deals with a nonstandard systems 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.

  • P. Colli, G. Gilardi, P. Krejčí, J. Sprekels, A vanishing diffusion limit in a nonstandard system of phase field equations, Evolution Equations and Control Theory, 3 (2014) pp. 257--275.
    Abstract
    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.

  • P. Colli, G. Gilardi, J. Sprekels, On the Cahn--Hilliard equation with dynamic boundary conditions and a dominating boundary potential, Journal of Mathematical Analysis and Applications, 419 (2014) pp. 972--994.
    Abstract
    The Cahn--Hilliard and viscous Cahn--Hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well-posedness and regularity results are proved.

  • S. Melchionna, E. Rocca, On a nonlocal Cahn--Hilliard equation with a reaction term, Advances in Mathematical Sciences and Applications, 24 (2014) pp. 461--497.
    Abstract
    We prove existence, uniqueness, regularity and separation properties for a nonlocal Cahn- Hilliard equation with a reaction term. We deal here with the case of logarithmic potential and degenerate mobility as well an uniformly lipschitz in u reaction term g(x, t, u).

  • A. Miranville, E. Rocca, G. Schimperna, A. Segatti, The Penrose--Fife phase-field model with coupled dynamic boundary conditions, Discrete and Continuous Dynamical Systems, 34 (2014) pp. 4259--4290.

  • W. Dreyer, J. Giesselmann, Ch. Kraus, A compressible mixture model with phase transition, Physica D. Nonlinear Phenomena, 273--274 (2014) pp. 1--13.
    Abstract
    We introduce a new thermodynamically consistent diffuse interface model of Allen-Cahn/Navier-Stokes type for multi-component flows with phase transitions and chemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e. a non-dissipative and a dissipative regime, where we recover in the sharp interface limit a generalized Allen-Cahn/Euler system for mixtures with chemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satify, for instance, a Young-Laplace and a Stefan type law.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, An asymptotic analysis for a nonstandard Cahn--Hilliard system with viscosity, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013) pp. 353--368.
    Abstract
    This 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.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn--Hilliard system with viscosity, Journal of Differential Equations, 254 (2013) pp. 4217--4244.
    Abstract
    Existence 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.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Continuous dependence for a nonstandard Cahn--Hilliard system with nonlinear atom mobility, Rendiconti del Seminario Matematico. Universita e Politecnico Torino, 70 (2012) pp. 27--52.
    Abstract
    This 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.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Distributed optimal control of a nonstandard system of phase field equations, Continuum Mechanics and Thermodynamics, 24 (2012) pp. 437--459.
    Abstract
    We 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.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Global existence for a strongly coupled Cahn--Hilliard system with viscosity, Bollettino della Unione Matematica Italiana. Serie 9, 5 (2012) pp. 495--513.
    Abstract
    An 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.

  • P. Colli, G. Gilardi, J. Sprekels, Analysis and optimal boundary control of a nonstandard system of phase field equations, Milan Journal of Mathematics, 80 (2012) pp. 119--149.
    Abstract
    We investigate a nonstandard phase field model of Cahn-Hilliard type. The model, which was introduced in Podio-Guidugli (2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been studied recently in Colli, Gilardi, Podio-Guidugli, and Sprekels (2011a and b) 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. For the resulting system, we show well-posedness, and we study optimal boundary control problems. Existence of optimal controls is shown, and the first-order necessary optimality conditions are derived. Owing to the strong nonlinear couplings in the PDE system, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional will be of standard type.

  • W. Dreyer, F. Duderstadt, M. Hantke, G. Warnecke, Bubbles in liquids with phase transition, Continuum Mechanics and Thermodynamics, 24 (2012) pp. 461--483.
    Abstract
    We consider a bubble of vapor and inert gas surrounded by the corresponding liquid phase. We study the behavior of the bubble due to phase change, i.e. condensation and evaporation, at the interface. Special attention is given to the effects of surface tension and heat production on the bubble dynamics as well as the propagation of acoustic elastic waves by including slight compressibility of the liquid phase. Separately we study the influence of the three phenomena heat conduction, elastic waves, and phase transition on the evolution of the bubble. The objective is to derive relations including the mass, momentum, and energy transfer between the phases. We find ordinary differential equations, in the cases of heat transfer and the emission of acoustic waves partial differential equations, that describe the bubble dynamics. From numerical evidence we deduce that the effect of phase transition and heat transfer on the behavior of the radius of the bubble is negligible. It turns out that the elastic waves in the liquid are of greatest importance to the dynamics of the bubble radius. The phase transition has a strong influence on the evolution of the temperature, in particular at the interface. Furthermore the phase transition leads to a drastic change of the water content in the bubble, so that a rebounding bubble is only possible, if it contains in addition an inert gas. In a forthcoming paper the equations derived are sought in order to close equations for multi-phase mixture balance laws for dispersed bubbles in liquids involving phase change. Also the model is used to make comparisons with experimental data on the oscillation of a laser induced bubble. For this case it was necessary to include the effect of an inert gas in the thermodynamic modeling of the phase transition.

  • G. Aki, J. Daube, W. Dreyer, J. Giesselmann, M. Kränkel, Ch. Kraus, A diffuse interface model for quasi-incompressible flows: Sharp interface limits and numerics, ESAIM Proceedings, 38 (2012) pp. 54--77.
    Abstract
    In this contribution, we investigate a diffuse interface model for quasi-incompressible flows. We determine corresponding sharp interface limits of two different scalings. The sharp interface limit is deduced by matched asymptotic expansions of the fields in powers of the interface. In particular, we study solutions of the derived system of inner equations and discuss the results within the general setting of jump conditions for sharp interface models. Furthermore, we treat, as a subproblem, the convective Cahn-Hilliard equation numerically by a Local Discontinuous Galerkin scheme.

  • S. Bartels, R. Müller, Error control for the approximation of Allen--Cahn and Cahn--Hilliard equations with a logarithmic potential, Numerische Mathematik, 119 (2011) pp. 409--435.
    Abstract
    A fully computable upper bound for the finite element approximation error of Allen-Cahn and Cahn-Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold.

  • S. Bartels, R. Müller, Quasi-optimal and robust a posteriori error estimates in $L^infty(L^2)$ for the approximation of Allen--Cahn equations past singularities, Mathematics of Computation, 80 (2011) pp. 761--780.
    Abstract
    Optimal a posteriori error estimates in $L^infty(0,T;L^2(O))$ are derived for the finite element approximation of Allen-Cahn equations. The estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface. The error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming, nonconforming, mixed, and discontinuous Galerkin methods. Numerical experiments illustrate the theoretical results.

  • P. Colli, P. Krejčí, E. Rocca, J. Sprekels, A nonlocal quasilinear multi-phase system with nonconstant specific heat and heat conductivity, Journal of Differential Equations, 251 (2011) pp. 1354--1387.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn--Hilliard system, SIAM Journal on Applied Mathematics, 71 (2011) pp. 1849--1870.
    Abstract
    We study a diffusion model of phase field type, consisting of a system of two partial differential equations encoding the balances of microforces and microenergy; the two unknowns are the order parameter and the chemical potential. By a careful development of uniform estimates and the deduction of certain useful boundedness properties, we prove existence and uniqueness of a global-in-time smooth solution to the associated initial/boundary-value problem; moreover, we give a description of the relative $omega$-limit set.

  • K. Hermsdörfer, Ch. Kraus, D. Kröner, Interface conditions for limits of the Navier--Stokes--Korteweg model, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 13 (2011) pp. 239--254.
    Abstract
    In this contribution we will study the behaviour of the pressure across phase boundaries in liquid-vapour flows. As mathematical model we will consider the static version of the Navier-Stokes-Korteweg model which belongs to the class of diffuse interface models. From this static equation a formula for the pressure jump across the phase interface can be derived. If we perform then the sharp interface limit we see that the resulting interface condition for the pressure seems to be inconsistent with classical results of hydrodynamics. Therefore we will present two approaches to recover the results of hydrodynamics in the sharp interface limit at least for special situations.

  • P. Krejčí, E. Rocca, J. Sprekels, Phase separation in a gravity field, Discrete and Continuous Dynamical Systems -- Series S, 4 (2011) pp. 391-407.

  • S. Bartels, R. Müller, A posteriori error controlled local resolution of evolving interfaces for generalized Cahn--Hilliard equations, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 12 (2010) pp. 45--73.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, A temperature-dependent phase segregation problem of the Allen--Cahn type, Advances in Mathematical Sciences and Applications, 20 (2010) pp. 219-234.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Existence and uniqueness of a global-in-time solution to a phase segregation problem of the Allen--Cahn type, Mathematical Models & Methods in Applied Sciences, 20 (2010) pp. 519-541.

  • H. Garcke, Ch. Kraus, An anisotropic, inhomogeneous, elastically modified Gibbs--Thomson law as singular limit of a diffuse interface model, Advances in Mathematical Sciences and Applications, 20 (2010) pp. 511--545.
    Abstract
    We consider the sharp interface limit of a diffuse phase field model with prescribed total mass taking into account a spatially inhomogeneous anisotropic interfacial energy and an elastic energy. The main aim is the derivation of a weak formulation of an anisotropic, inhomogeneous, elastically modified Gibbs-Thomson law in the sharp interface limit. To this end we show that one can pass to the limit in the weak formulation of the Euler-Lagrange equation of the diffuse phase field energy.

  • W. Dreyer, Ch. Kraus, On the van der Waals--Cahn--Hilliard phase-field model and its equilibria conditions in the sharp interface limit, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 140 A (2010) pp. 1161--1186.
    Abstract
    We study the equilibria of liquid--vapor phase transitions of a single substance at constant temperature and relate the sharp interface model of classical thermodynamics to a phase field model that determines the equilibria by the stationary van der Waals--Cahn--Hilliard theory.
    For two reasons we reconsider this old problem. 1. Equilibria in a two phase system can be established either under fixed total volume of the system or under fixed external pressure. The latter case implies that the domain of the two--phase system varies. However, in the mathematical literature rigorous sharp interface limits of phase transitions are usually considered under fixed volume. This brings the necessity to extend the existing tools for rigorous sharp interface limits to changing domains since in nature most processes involving phase transitions run at constant pressure. 2. Thermodynamics provides for a single substance two jump conditions at the sharp interface, viz. the continuity of the specific Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. The existing estimates for rigorous sharp interface limits show only the first condition. We identify the cause of this phenomenon and develop a strategy that yields both conditions up to the first order.
    The necessary information on the equilibrium conditions are achieved by an asymptotic expansion of the density which is valid for an arbitrary double well potential. We establish this expansion by means of local energy estimates, uniform convergence results of the density and estimates on the Laplacian of the density.

  • J. Sprekels, H. Wu, A note on parabolic equation with nonlinear dynamical boundary condition, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 72 (2010) pp. 3028--3048.

  • P. Krejčí, E. Rocca, J. Sprekels, A bottle in a freezer, SIAM Journal on Mathematical Analysis, 41 (2009) pp. 1851-1873.

  • TH. Böhme, W. Dreyer, W.H. Müller, Determination of stiffness and higher gradient coefficients by means of the embedded atom method: An approach for binary alloys, Continuum Mechanics and Thermodynamics, 18 (2007) pp. 411--441.

  • P. Colli, P. Krejčí, E. Rocca, J. Sprekels, Nonlinear evolution inclusions arising from phase change models, Czechoslovak Mathematical Journal, 57 (2007) pp. 1067-1098.

  • C. Lefter, J. Sprekels, Optimal boundary control of a phase field system modeling nonisothermal phase transitions, Advances in Mathematical Sciences and Applications, 17 (2007) pp. 181-194.

  • P. Krejčí, E. Rocca, J. Sprekels, A nonlocal phase-field model with nonconstant specific heat, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 9 (2007) pp. 285--306.

  • P. Krejčí, E. Rocca, J. Sprekels, Nonlocal temperature-dependent phase-field models for non-isothermal phase transitions, Journal of the London Mathematical Society. Second Series, 76 (2007) pp. 197-210.

  • P. Krejčí, J. Sprekels, Long time behaviour of a singular phase transition model, Discrete and Continuous Dynamical Systems, 15 (2006) pp. 1119-1135.

  • W. Dreyer, B. Wagner, Sharp-interface model for eutectic alloys. Part I: Concentration dependent surface tension, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 7 (2005) pp. 199--227.

  • J. Griepentrog, On the unique solvability of a nonlocal phase separation problem for multicomponent systems, Banach Center Publications, 66 (2004) pp. 153-164.

  • O. Klein, Asymptotic behaviour for a phase-field model with hysteresis in one-dimensional thermo-visco-plasticity, Applications of Mathematics, 49 (2004) pp. 309--341.

  • P. Krejčí, J. Sprekels, Nonlocal phase-field models for non-isothermal phase transitions and hysteresis, Advances in Mathematical Sciences and Applications, 14 (2004) pp. 593--612.

  • P. Krejčí, J. Sprekels, U. Stefanelli, One-dimensional thermo-visco-plastic processes with hysteresis and phase transitions, Advances in Mathematical Sciences and Applications, 13 (2003) pp. 695-712.

  • O. Klein, P. Krejčí, Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 4 (2003) pp. 755--785.

  • O. Klein, C. Verdi, A posteriori error estimates for a time discrete scheme for a phase-field system of Penrose-Fife type, IMA Journal of Numerical Analysis, 23 (2003) pp. 55--80.

  • J. Sprekels, S. Zheng, Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions, Journal of Mathematical Analysis and Applications, 279 (2003) pp. 97-110.

  • E. Bonetti, P. Colli, W. Dreyer, G. Gilardi, G. Schimperna, J. Sprekels, On a model for phase separation in binary alloys driven by mechanical effects, Physica D. Nonlinear Phenomena, 165 (2002) pp. 48--65.

  • N. Kenmochi, J. Sprekels, Phase-field systems with vectorial order parameters including diffusional hysteresis effects, Communications on Pure and Applied Analysis, 1 (2002) pp. 495-511.

  • J. Sprekels, P. Krejčí, Phase-field systems for multi-dimensional Prandtl-Ishlinskii operators with non-polyhedral characteristics, Mathematical Methods in the Applied Sciences, 25 (2002) pp. 309-325.

  • P. Krejčí, J. Sprekels, S. Zheng, Asymptotic behaviour for a phase-field system with hysteresis, Journal of Differential Equations, 175 (2001) pp. 88--107.

  • P. Krejčí, J. Sprekels, A hysteresis approach to phase-field models, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 39 (2000) pp. 569--586.

  • P. Krejčí, J. Sprekels, Phase-field models with hysteresis, Journal of Mathematical Analysis and Applications, 252 (2000) pp. 198--219.

  • G. Gilardi, P. Krejčí, J. Sprekels, Hysteresis in phase-field models with thermal memory, Mathematical Methods in the Applied Sciences, 23 (2000) pp. 909--922.

  • W. Dreyer, W.H. Müller, A study of the coarsening in tin/lead solders, International Journal of Solids and Structures, 37 (2000) pp. 3841--3871.

  • W. Dreyer, W.H. Müller, Computer modeling of micromorphological change by phase field models: Applications to metals and ceramics, Journal of the Australasian Ceramic Society, 36 (2000) pp. 83--94.

  Contributions to Collected Editions

  • A. Mielke, Free energy, free entropy, and a gradient structure for thermoplasticity, in: Innovative Numerical Approaches for Multi-Field and Multi-Scale Problems. In Honor of Michael Ortiz's 60th Birthday, K. Weinberg, A. Pandolfi, eds., 81 of Lecture Notes in Applied and Computational Mechanics, Springer International Publishing Switzerland, Cham, 2016, pp. 135--160.
    Abstract
    In the modeling of solids the free energy, the energy, and the entropy play a central role. We show that the free entropy, which is defined as the negative of the free energy divided by the temperature, is similarly important. The derivatives of the free energy are suitable thermodynamical driving forces for reversible (i.e. Hamiltonian) parts of the dynamics, while for the dissipative parts the derivatives of the free entropy are the correct driving forces. This difference does not matter for isothermal cases nor for local materials, but it is relevant in the non-isothermal case if the densities also depend on gradients, as is the case in gradient thermoplasticity.

    Using the total entropy as a driving functional, we develop gradient structures for quasistatic thermoplasticity, which again features the role of the free entropy. The big advantage of the gradient structure is the possibility of deriving time-incremental minimization procedures, where the entropy-production potential minus the total entropy is minimized with respect to the internal variables and the temperature.

    We also highlight that the usage of an auxiliary temperature as an integrating factor in Yang/Stainier/Ortiz "A variational formulation of the coupled thermomechanical boundary-value problem for general dissipative solids" (J. Mech. Physics Solids, 54, 401-424, 2006) serves exactly the purpose to transform the reversible driving forces, obtained from the free energy, into the needed irreversible driving forces, which should have been derived from the free entropy. This reconfirms the fact that only the usage of the free entropy as driving functional for dissipative processes allows us to derive a proper variational formulation.

  • A. Mielke, Evolutionary relaxation of a two-phase model, in: Scales in Plasticity, Mini-Workshop, November 8--14, 2015, G.A. Francfort, S. Luckhaus, eds., 12 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2015, pp. 3027--3030.

  • D. Knees, R. Kornhuber, Ch. Kraus, A. Mielke, J. Sprekels, C3 -- Phase transformation and separation in solids, in: MATHEON -- Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 189--203.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Global solution to a phase transition problem of the Allen--Cahn type, in: Nonlinear Evolution Equations and Mathematical Modeling, Proceedings of the RIMS Symposium held at the Research Institute for Mathematical Sciences, Kyoto University, October 20--23, 2009, T. Aiki, ed., 1693 of Sūrikaisekikenkyūsho Kōkyūroku, Kyoto, 2010, pp. 104-110.

  • P. Krejčí, E. Rocca, J. Sprekels, Liquid-solid phase transitions in a deformable container, in: Continuous Media with Microstructure, B. Albers, ed., Springer, Berlin/Heidelberg, 2010, pp. 285-300.

  • O. Klein, P. Krejčí, Asymptotic behaviour of evolution equations involving outwards pointing hysteresis operators, in: Proceedings of the Fourth International Symposium on Hysteresis and Micromagnetic Modeling, Salamanca, Spain, 28--30 May 2003, L. Lopez-Dias, L. Torres, O. Alejos, eds., 343 of Physica B: Condensed Matter, Elsevier B.V., 2004, pp. 53-58.

  • P. Krejčí, J. Sprekels, Phase-field systems and vector hysteresis operators, in: Free Boundary Problems: Theory and Applications II, N. Kenmochi, ed., 14 of Gakuto Int. Series Math. Sci. & Appl., Gakkōtosho, Tokyo, 2000, pp. 295--310.

  Preprints, Reports, Technical Reports

  • P. Colli, J. Sprekels, Optimal boundary control of a nonstandard Cahn--Hilliard system with dynamic boundary condition and double obstacle inclusions, Preprint no. 2370, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2370 .
    Abstract, PDF (268 kByte)
    In this paper, we study an optimal boundary control problem for a model for phase separation taking place in a spatial domain that was introduced by P.Podio-Guidugli in Ric. Mat. 55 (2006), pp.105-118. The model consists of a strongly coupled system of nonlinear parabolic differential inclusions, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically; the system is complemented by initial and boundary conditions. For the order parameter of the phase separation process, a dynamic boundary condition involving the Laplace-Beltrami operator is assumed, which models an additional nonconserving phase transition occurring on the surface of the domain. We complement in this paper results that were established in the recent contribution appeared in Evol. Equ. Control Theory 6 (2017), pp. 35-58, by the two authors and Gianni Gilardi. In contrast to that paper, in which differentiable potentials of logarithmic type were considered, we investigate here the (more difficult) case of nondifferentiable potentials of double obstacle type. For such nonlinearities, the standard techniques of optimal control theory to establish the existence of Lagrange multipliers for the state constraints are known to fail. To overcome these difficulties, we employ the following line of approach: we use the results contained in the preprint arXiv:1609.07046 [math.AP] (2016), pp. 1-30, for the case of (differentiable) logarithmic potentials and perform a so-called "deep quench limit". Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (nondifferentiable) double obstacle potentials.

  • P. Colli, G. Gilardi, J. Sprekels, Limiting problems for a nonstandard viscous Cahn--Hilliard system with dynamic boundary conditions, Preprint no. 2369, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2369 .
    Abstract, PDF (353 kByte)
    This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice and was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the Laplace-Beltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested to some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions to the corresponding solutions for the limit problem is proven. Then, we study the long-time behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omega-limit set in both cases.

  • M. Hintermüller, M. Hinze, Ch. Kahle, T. Keil, A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn--Hilliard--Navier--Stokes system, Preprint no. 2311, WIAS, Berlin, 2016.
    Abstract, PDF (640 kByte)
    This paper is concerned with the development and implementation of an adaptive solution algorithm for the optimal control of a time-discrete Cahn--Hilliard--Navier--Stokes system with variable densities. The free energy density associated to the Cahn--Hilliard system incorporates the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier--Stokes equation. A dual-weighed residual approach for goal-oriented adaptive finite elements is presented which is based on the concept of C-stationarity. The overall error representation depends on primal residual weighted by approximate dual quantities and vice versa as well as various complementary mismatch errors. Details on the numerical realization of the adaptive concept and a report on numerical tests are given.

  • P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Preprint no. 2228, WIAS, Berlin, 2016.
    Abstract, PDF (294 kByte)
    In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins--Daruud et al. in citeHZO. The model consists of a Cahn--Hilliard equation for the tumor cell fraction $vp$ coupled to a reaction-diffusion equation for a function $s$ representing the nutrient-rich extracellular water volume fraction. The distributed control $u$ monitors as a right-hand side the equation for $s$ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

  • CH. Heinemann, K. Sturm, Shape optimisation for a class of semilinear variational inequalities with applications to damage models, Preprint no. 2209, WIAS, Berlin, 2016.
    Abstract, PDF (588 kByte)
    The present contribution investigates shape optimisation problems for a class of semilinear elliptic variational inequalities with Neumann boundary conditions. Sensitivity estimates and material derivatives are firstly derived in an abstract operator setting where the operators are defined on polyhedral subsets of reflexive Banach spaces. The results are then refined for variational inequalities arising from minimisation problems for certain convex energy functionals considered over upper obstacle sets in $H^1$. One particularity is that we allow for dynamic obstacle functions which may arise from another optimisation problems. We prove a strong convergence property for the material derivative and establish state-shape derivatives under regularity assumptions. Finally, as a concrete application from continuum mechanics, we show how the dynamic obstacle case can be used to treat shape optimisation problems for time-discretised brittle damage models for elastic solids. We derive a necessary optimality system for optimal shapes whose state variables approximate desired damage patterns and/or displacement fields.

  • CH. Heinemann, Ch. Kraus, E. Rocca, R. Rossi, A temperature-dependent phase-field model for phase separation and damage, Preprint no. 2164, WIAS, Berlin, 2015.
    Abstract, Postscript (1675 kByte), PDF (742 kByte)
    In this paper we study a model for phase separation and damage in thermoviscoelastic materials. The main novelty of the paper consists in the fact that, in contrast with previous works in the literature (cf., e.g., [C. Heinemann, C. Kraus: Existence results of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321--359] and [C. Heinemann, C. Kraus: Existence results for diffuse interface models describing phase separation and damage. European J. Appl. Math. 24 (2013), 179--211]), we encompass in the model thermal processes, nonlinearly coupled with the damage, concentration and displacement evolutions. More in particular, we prove the existence of "entropic weak solutions", resorting to a solvability concept first introduced in [E. Feireisl: Mathematical theory of compressible, viscous, and heat conducting fluids. Comput. Math. Appl. 53 (2007), 461--490] in the framework of Fourier-Navier-Stokes systems and then recently employed in [E. Feireisl, H. Petzeltová, E. Rocca: Existence of solutions to a phase transition model with microscopic movements. Math. Methods Appl. Sci. 32 (2009), 1345--1369], [E. Rocca, R. Rossi: "Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage. SIAM J. Math. Anal., 47 (2015), 2519--2586] for the study of PDE systems for phase transition and damage. Our global-in-time existence result is obtained by passing to the limit in a carefully devised time-discretization scheme.

  • M. Dai, E. Feireisl, E. Rocca, G. Schimperna, M.E. Schonbek , Analysis of a diffuse interface model of multispecies tumor growth, Preprint no. 2150, WIAS, Berlin, 2015.
    Abstract, PDF (203 kByte)
    We consider a diffuse interface model for tumor growth recently proposed in citecwsl. In this new approach sharp interfaces are replaced by narrow transition layers arising due to adhesive forces among the cell species. Hence, a continuum thermodynamically consistent model is introduced. The resulting PDE system couples four different types of equations: a Cahn-Hilliard type equation for the tumor cells (which include proliferating and dead cells), a Darcy law for the tissue velocity field, whose divergence may be different from 0 and depend on the other variables, a transport equation for the proliferating (viable) tumor cells, and a quasi-static reaction diffusion equation for the nutrient concentration. We establish existence of weak solutions for the PDE system coupled with suitable initial and boundary conditions. In particular, the proliferation function at the boundary is supposed to be nonnegative on the set where the velocity $vu$ satisfies $vucdotnu>0$, where $nu$ is the outer normal to the boundary of the domain. We also study a singular limit as the diffuse interface coefficient tends to zero.

  • E. Bonetti, E. Rocca, Generalized gradient flow structure of internal energy driven phase field systems, Preprint no. 2144, WIAS, Berlin, 2015.
    Abstract, PDF (186 kByte)
    In this paper we introduce a general abstract formulation of a variational thermomechanical model, by means of a unified derivation via a generalization of the principle of virtual powers for all the variables of the system, including the thermal one. In particular, choosing as thermal variable the entropy of the system, and as driving functional the internal energy, we get a gradient flow structure (in a suitable abstract setting) for the whole nonlinear PDE system. We prove a global in time existence of (weak) solutions result for the Cauchy problem associated to the abstract PDE system as well as uniqueness in case of suitable smoothness assumptions on the functionals.

  • V. Barbu, P. Colli, G. Gilardi, G. Marinoschi, E. Rocca, Sliding modes for a phase-field system, Preprint no. 2133, WIAS, Berlin, 2015.
    Abstract, PDF (295 kByte)
    In the present contribution the sliding mode control (SMC) problem for a phase-field model of Caginalp type is considered. First we prove the well-posedness and some regularity results for the phase-field type state systems modified by the state- feedback control laws. Then, we show that the chosen SMC laws force the system to reach within finite time the sliding manifold (that we chose in order that one of the physical variables or a combination of them remains constant in time). We study three different types of feedback control laws: the first one appears in the internal energy balance and forces a linear combination of the temperature and the phase to reach a given (space dependent) value, while the second and third ones are added in the phase relation and lead the phase onto a prescribed target  $phi^*$. While the control law is non-local in space for the first two problems, it is local in the third one, i.e., its value at any point and any time just depends on the value of the state.

  • E. Bonetti, E. Rocca, G. Schimperna, R. Scala, On the strongly damped wave equation with constraint, Preprint no. 2094, WIAS, Berlin, 2015.
    Abstract, PDF (246 kByte)
    A weak formulation for the so-called semilinear strongly damped wave equation with constraint is introduced and a corresponding notion of solution is de?ned. The main idea in this approach consists in the use of duality techniques in Sobolev-Bochner spaces, aimed at providing a suitable "relaxation" of the constraint term. A global in time existence result is proved under the natural condition that the initial data have finite "physical" energy.

  Talks, Poster

  • M. Hintermüller, Optimal control of nonsmooth phase-field models, DFG-AIMS Workshop on ``Shape Optimization, Homogenization and Control'', March 13 - 16, 2017, Mbour, Senegal, March 14, 2017.

  • M. Hintermüller, Non-smooth structures in PDE-constrained optimization, Mathematisches Kolloquium, Universität Duisburg-Essen, Fakultät für Mathematik, Essen, January 11, 2017.

  • M. Hintermüller, Recent trends in PDE-constrained optimization with non-smooth structures, Fourth Conference on Numerical Analysis and Optimization (NAOIV-2017), January 2 - 5, 2017, Sultan Qaboos University, Muscat, Oman, January 4, 2017.

  • M. Thomas, From adhesive contact to brittle delamination in visco-elastodynamics, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22 - 26, 2016, WIAS Berlin, Berlin, February 26, 2016.

  • M. Thomas, Rate-independent evolution of sets, INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, September 5 - 8, 2016, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Rome, Italy, September 6, 2016.

  • S.P. Frigeri, On a diffuse interface model of tumor growth, 9th European Conference on Elliptic and Parabolic Problems, May 23 - 27, 2016, University of Zurich, Institute of Mathematics, Gaeta, Italy, May 23, 2016.

  • E. Rocca, Optimal control of a nonlocal convective Cahn--Hilliard equation by the velocity, Numerical Analysis Seminars, Durham University, UK, March 13, 2015.

  • S.P. Frigeri, On a diffuse interface model of tumor growth, INdAM Workshop ``Special Materials in Complex Systems -- SMaCS 2015'', May 18 - 22, 2015, Rome, Italy, May 22, 2015.

  • S.P. Frigeri, On a nonlocal diffuse interface model for binary incompressible fluids with different densities, Mathematical Thermodynamics of Complex Fluids, June 28 - July 3, 2015, Fondazione CIME ``Roberto Conti'' (International Mathematical Summer Center), Cetraro, Italy, July 2, 2015.

  • S.P. Frigeri, Recent results on optimal control for Cahn--Hilliard/Navier--Stokes systems with nonlocal interactions, Control Theory and Related Topics, April 13 - 14, 2015, Politecnico di Milano, Italy, April 13, 2015.

  • CH. Heinemann, Damage processes in thermoviscoelastic materials with damage-dependent thermal expansion coefficients, 3rd Workshop of the GAMM Activity Group Analysis of Partial differential Equations, September 30 - October 2, 2015, Universität Kassel, Institut für Mathematik, October 1, 2015.

  • CH. Heinemann, On elastic Cahn--Hilliard systems coupled with evolution inclusions for damage processes, 86th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2015), Young Researchers' Minisymposium 2, March 23 - 27, 2015, Lecce, Italy, March 23, 2015.

  • CH. Heinemann, Solvability of differential inclusions describing damage processes and applications to optimal control problems, Universität Essen-Duisburg, Fakultät für Mathematik, Essen, December 3, 2015.

  • J. Sprekels, Optimal boundary control problems for Cahn--Hilliard systems with dynamic boundary conditions, INdAM Workshop ``Special Materials in Complex Systems -- SMaCS 2015'', May 18 - 22, 2015, Rome, Italy, May 21, 2015.

  • M. Thomas, Rate-independent damage models with spatial BV-regularization --- Existence & fine properties of solutions, Oberseminar ``Angewandte Analysis'', Universität Freiburg, Abteilung für Angewandte Mathematik, Freiburg, February 10, 2015.

  • M.H. Farshbaf Shaker, A deep quench approach to the optimal control of an Allen--Cahn equation with dynamic boundary conditions, National Institute for Mathematical Sciences, Division of Computational Mathematics, Daejeon, Korea (Republic of), May 20, 2015.

  • M.H. Farshbaf Shaker, Multi-material phase field approach to structural topology optimization and its relation to sharp interface approach, University of Tokyo, Graduate School of Mathematical Sciences, Japan, October 6, 2015.

  • M.H. Farshbaf Shaker, Relating phase field and sharp interface approaches to structural topology optimization, National Institute for Mathematical Sciences, Division of Computational Mathematics, Daejeon, Korea (Republic of), May 13, 2015.

  • M.H. Farshbaf Shaker, Relating phase field and sharp interface approaches to structural topology optimization, Technische Universität Berlin, Institut für Mathematik, February 5, 2015.

  • CH. Heinemann, Well-posedness of strong solutions for a damage model in 2D, Universitá di Brescia, Department DICATAM -- Section of Mathematics, Italy, March 13, 2015.

  • A. Mielke, Evolutionary relaxation of a two-phase model, Mini-Workshop ``Scales in Plasticity'', November 8 - 14, 2015, Mathematisches Forschungsinstitut Oberwolfach, November 11, 2015.

  • J. Sprekels, Optimal boundary control problems for Cahn--Hilliard systems with singular potentials and dynamic boundary conditions, Romanian Academy, Simeon Stoilow Institute of Mathematics, Bucharest, March 18, 2015.

  • E. Rocca, ``Entropic'' solutions to a thermodynamically consistent PDE system for phase transitions and damage, Symposium on Trends in Application of Mathematics to Mechanics (STAMM), September 8 - 11, 2014, International Society for the Interaction of Mechanics and Mathematics (ISIMM), Poitiers, France, September 9, 2014.

  • CH. Heinemann, Analysis of degenerating Cahn--Hilliard systems coupled with complete damage processes, 2013 CNA Summer School, Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, USA, May 30 - June 7, 2013.

  • CH. Heinemann, Degenerating Cahn--Hilliard systems coupled with complete damage processes, DIMO2013 -- Diffuse Interface Models, Levico Terme, Italy, September 10 - 13, 2013.

  • CH. Heinemann, Degenerating Cahn--Hilliard systems coupled with mechanical effects and complete damage processes, Equadiff13, MS27 -- Recent Results in Continuum and Fracture Mechanics, August 26 - 30, 2013, Prague, Czech Republic, August 27, 2013.

  • CH. Heinemann, On a PDE system describing damage processes and phase separation, Oberseminar Analysis, Universität Augsburg, July 11, 2013.

  • H. Abels, J. Daube, Ch. Kraus, D. Kröner, Sharp interface limit for the Navier--Stokes--Korteweg model, DIMO2013 -- Diffuse Interface Models, Levico Terme, Italy, September 10 - 13, 2013.

  • CH. Kraus, Damage and phase separation processes: Modeling and analysis of nonlinear PDE systems, DIMO2013 -- Diffuse Interface Models, September 10 - 13, 2013, Levico Terme, Italy, September 11, 2013.

  • CH. Kraus, Modeling and analysis of a nonlinear PDE system for phase separation and damage, Università di Pavia, Dipartimento di Matematica, Italy, January 22, 2013.

  • A. Mielke, On the geometry of reaction-diffusion systems: Optimal transport versus reaction, Recent Trends in Differential Equations: Analysis and Discretisation Methods, November 7 - 9, 2013, Technische Universität Berlin, Institut für Mathematik, November 9, 2013.

  • J. Sprekels, Optimal control of Allen--Cahn equations with singular potentials and dynamic boundary conditions, DIMO2013 -- Diffuse Interface Models, September 10 - 13, 2013, Levico Terme, Italy, September 11, 2013.

  • J. Sprekels, Optimal control of the Allen--Cahn equation with dynamic boundary condition and double obstacle potentials: A ``deep quench'' approach, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica ``F. Casorati'', Italy, September 17, 2013.

  • J. Sprekels, Prandtl--Ishlinskii operators and elastoplasticity, Spring School on ``Rate-independent Evolutions and Hysteresis Modelling'', May 27 - 31, 2013, Politecnico di Milano, Università degli Studi di Milano, Italy.

  • CH. Heinemann, Complete damage in linear elastic materials, Variational Models and Methods for Evolution, Levico, Italy, September 10 - 12, 2012.

  • CH. Heinemann, Damage processes coupled with phase separation in elastically stressed alloys, GAMM Jahrestagung 2012 (83rd Annual Meeting), March 26 - 30, 2012, Technische Universität Darmstadt, March 27, 2012.

  • CH. Heinemann, Existence of weak solutions for rate-dependent complete damage processes, Materialmodellierungsseminar, WIAS, Berlin, October 31, 2012.

  • CH. Heinemann, Kopplung von Phasenseparation und Schädigung in elastischen Materialien, Leibniz-Doktoranden-Forum der Sektion D, Berlin, June 7 - 8, 2012.

  • CH. Kraus, A nonlinear PDE system for phase separation and damage, Universität Freiburg, Abteilung Angewandte Mathematik, November 13, 2012.

  • CH. Kraus, Cahn--Larché systems coupled with damage, Università degli Studi di Milano, Dipartimento di Matematica, Italy, November 28, 2012.

  • CH. Kraus, Phase field systems for phase separation and damage processes, 12th International Conference on Free Boundary Problems: Theory and Applications, June 11 - 15, 2012, Frauenchiemsee, June 12, 2012.

  • CH. Kraus, Phasenfeldsysteme für Entmischungs- und Schädigungsprozesse, Mathematisches Kolloquium, Universität Stuttgart, Fachbereich Mathematik, May 15, 2012.

  • J. Sprekels, A time discretization for a nonstandard viscous Cahn--Hilliard system, INDAM Workshop PDEs for Multiphase Advanced Materials (ADMAT2012), September 17 - 21, 2012, Cortona, Italy, September 19, 2012.

  • J.A. Griepentrog, On nonlocal phase separation processes in multicomponent systems, 10th GAMM Seminar on Microstructures, January 20 - 22, 2011, Technische Universität Darmstadt, Fachbereich Mathematik, January 22, 2011.

  • J.A. Griepentrog, The role of nonsmooth regularity theory in the analysis of phase separation processes, Ehrenkolloquium anlässlich des 60. Geburtstages von PD Dr. habil. Lutz Recke, Humboldt-Universität zu Berlin, Institut für Mathematik, November 21, 2011.

  • CH. Heinemann, Existence results for Cahn-Hilliard equations coupled with elasticity and damage, Workshop on Phase Separation, Damage and Fracture, September 21 - 23, 2011, WIAS, September 23, 2011.

  • CH. Kraus, Diffuse interface systems for phase separation and damage, Seminar on Partial Differential Equations, Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague, May 3, 2011.

  • CH. Kraus, Phase separation systems coupled with elasticity and damage, ICIAM 2011, July 18 - 22, 2011, Vancouver, Canada, July 18, 2011.

  • J. Sprekels, A non-standard phase-field system of Cahn--Hilliard type for diffusive phase segregation, Schwerpunktkolloquium``Analysis und Numerik'', Universität Konstanz, Fachbereich Mathematik und Statistik, July 14, 2011.

  • J. Sprekels, A nonstandard phase field system of Cahn--Hilliard type for diffusive phase segregation, Seminario Matematico e Fisico di Milano, Università degli Studi di Milano, Dipartimento di Matematica, Italy, September 21, 2011.

  • J. Sprekels, Phase field models and hysteresis operators, Trends in Thermodynamics and Materials Theory 2011, December 15 - 17, 2011, Technische Universität Berlin, December 16, 2011.

  • J. Sprekels, Well-posedness, asymptotic behavior and optimal control of a nonstandard phase field model for diffusive phase segregation, Workshop on Optimal Control of Partial Differential Equations, November 28 - December 1, 2011, Wasserschloss Klaffenbach, Chemnitz, November 30, 2011.

  • J. Sprekels, Models of phase transitions and hysteresis operators, Joint International Meeting UMI-DMV 2007, Minisymposium ``Phase Transitions and Hysteresis in Free Boundary Problems'', June 18 - 22, 2007, Università degli Studi di Perugia, Dipartimento di Matematica e Informatica, Italy, June 21, 2007.

  • J. Sprekels, Nonlocal phase-field models for nonisothermal phase transitions, International Conference ``Free Boundary Problems: Theory and Applications'', June 7 - 12, 2005, Coimbra, Portugal, June 8, 2005.

  • O. Klein, Asymptotic behaviour for a phase-field model with hysteresis in thermo-visco-plasticity, INdAM Workshop ``Dissipative Models in Phase Transitions'', September 5 - 11, 2004, Cortona, Italy, September 9, 2004.

  • O. Klein, Long-time behaviour of solutions to equations involving outwards pointing hysteresis operators, International Workshop on Hysteresis & Multi-Scale Asymptotics (HAMSA 2004), March 17 - 21, 2004, University College Cork, Ireland, March 19, 2004.

  • J. Sprekels, On nonlocal phase-field models, Workshop ``Thermodynamische Materialtheorien'', December 12 - 15, 2004, Mathematisches Forschungsinstitut Oberwolfach, December 14, 2004.

  • O. Klein, Asymptotic behaviour of evolution equations involving outwards pointing hysteresis operators, 4th International Symposium on Hysteresis and Micromagnetic Modeling (HMM-2003), May 28 - 30, 2003, Universidad de Salamanca, Departamento de Física Aplicada, Spain, May 30, 2003.

  • J. Sprekels, On nonlocal models for non-isothermal phase transitions, Academy of Sciences of the Czech Republic, Institute of Mathematics, Prague, February 19, 2002.

  • J. Sprekels, On nonlocal phase transition models for non-conserved order parameters, Università degli Studi di Firenze, Dipartimento di Matematica ``U. Dini'', Italy, October 14, 2002.

  • J. Sprekels, Hysteresis operators in phase-field modelling, Workshop ``Phasenübergänge'', April 29 - May 4, 2001, Mathematisches Forschungsinstitut Oberwolfach, May 1, 2001.

  • J. Sprekels, Hysteresis operators in phase-field systems, University of Warsaw, Interdisciplinary Centre for Mathematics and Computer Modelling, Poland, January 5, 2001.

  • J. Sprekels, On nonlocal phase transition models for non-conserved order parameters, Università di Pavia, Dipartimento di Matematica, Italy, September 19, 2001.

  • J. Sprekels, Phase-field systems and vector hysteresis operators, Workshop "`Phase Transitions and Interfaces in Evolution Equations: Analysis, Control and Approximation"', February 8 - 12, 2000, Santa Margherita Ligure, Italy, February 11, 2000.

  • J. Sprekels, Phase-field systems with hysteresis, Konferenz "`Evolution Equations 2000: Applications to Physics, Industry, Life Sciences and Economics"', October 30 - 31, 2000, Levico, Italy, October 30, 2000.

  External Preprints

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, A temperature-dependent phase segregation problem of the Allen--Cahn type, Preprint no. arXiv:1005.0911, Cornell University Library, arXiv.org, 2010.

  • S. Bartels, R. Müller, Error controlled local resolution of evolving interfaces for generalized Cahn-Hilliard equations, Preprint no. 09--02, Humboldt-Universität zu Berlin, Institut für Mathematik, 2009.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Global solution and long-time behavior for a problem of phase segregation of the Allen--Cahn type (electronic only), Preprint no. arXiv:0902.4741, Cornell University Library, arXiv.org, 2009.
    Abstract
    In this paper we study a model for phase segregation consisting in a sistem of a partial and an ordinary differential equation. By a careful definition of maximal solution to the latter equation, this system reduces to an Allen-Cahn equation with a memory term. Global existence and uniqueness of a smooth solution are proven and a characterization of the omega-limit set is given.

  • F. Klopp, B. Metzger, The Gross--Pitaevskii functional with a random background potential and condensation in the single particle ground state, Preprint no. arXiv:0910.2896, Cornell University Library, arXiv.org, 2009.
    Abstract
    For discrete and continuous Gross-Pitaevskii energy functionals with a random background potential, we study the Gross-Pitaevskii ground state. We characterize a regime of interaction coupling when the Gross-Pitaevskii ground state and the ground state of the random background Hamiltonian asymptotically coincide.

  • F. Lanzara, V. Maz'ya, G. Schmidt, On the fast computation of high dimensional volume potentials (electronic only), Preprint no. arXiv:0911.0443, Cornell University Library, arXiv.org, 2009.

  • P. Krejčí, E. Rocca, J. Sprekels, Phase separation in a gravity field (electronic only), Preprint no. arXiv:0905.2131, Cornell University Library, arXiv.org, 2009.