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 PDEsystems of ellipticparabolic type with constraints. Such PDEsystems are investigated at WIAS concerning their analytically properties such as existence, uniqueness and regularity of solutions. In a further step, numerical simulations of the PDEsystems 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.
(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 CahnHilliard 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.
Highlights
Modeling, analysis, and numerics of a temperaturedependent 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 globalintime existence result is obtained by passing to the limit in a carefully devised timediscretization scheme.Publications
Monographs

M. Hintermüller, D. Wegner, Distributed and Boundary Control Problems for the Semidiscrete CahnHilliard/NavierStokes System with Nonsmooth GinzburgLandau Energies, in: Topological Optimization and Optimal Transport in the Applied Sciences, M. Bergounioux, E. Oudet, M. Rumpf, G. Carlier, Th. Champion, F. Santambrogio, eds., 17 of Radon Series on Computational and Applied Mathematics, De Gruyter, Berlin, 2017, pp. 4063, (Chapter Published).

H.Chr. Kaiser, D. Knees, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci, eds., PDE 2015: Theory and Applications of Partial Differential Equations, 10 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Science, Springfield, 2017, iv+933 pages, (Collection Published).

P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels, eds., Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs: In Honour of Prof. Gianni Gilardi, 22 of Springer INdAM Series, Springer International Publishing AG, Cham, 2017, xii+571 pages, (Collection Published).
Abstract
This volume gathers contributions in the field of partial differential equations, with a focus on mathematical models in phase transitions, complex fluids and thermomechanics. These contributions are dedicated to Professor Gianni Gilardi on the occasion of his 70th birthday. It particularly develops the following thematic areas: nonlinear dynamic and stationary equations; wellposedness of initial and boundary value problems for systems of PDEs; regularity properties for the solutions; optimal control problems and optimality conditions; feedback stabilization and stability results. Most of the articles are presented in a selfcontained manner, and describe new achievements and/or the state of the art in their line of research, providing interested readers with an overview of recent advances and future research directions in PDEs. 
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, J. Sprekels, Optimal velocity control of a convective CahnHilliard system with double obstacles and dynamic boundary conditions: A `deep quench' approach, Journal of Convex Analysis, 26 (2019), pp. 485514.
Abstract
In 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 socalled `deep quench approximation'. We derive results concerning the existence of optimal controls and the firstorder necessary optimality conditions in terms of a variational inequality and the associated adjoint system. 
G. Gilardi, J. Sprekels, Asymptotic limits and optimal control for the CahnHilliard system with convection and dynamic boundary conditions, Nonlinear Analysis. An International Mathematical Journal, 178 (2019), pp. 121.
Abstract
In this paper, we study initialboundary value problems for the CahnHilliard system with convection and nonconvex potential, where dynamic boundary conditions are assumed for both the associated order parameter and the corresponding chemical potential. While recent works addressed the case of viscous CahnHilliard systems, the `pure' nonviscous case is investigated here. In its first part, the paper deals with the asymptotic behavior of the solutions as time approaches infinity. It is shown that the $omega$limit of any trajectory can be characterized in terms of stationary solutions, provided the initial data are sufficiently smooth. The second part of the paper deals with the optimal control of the system by the fluid velocity. Results concerning existence and firstorder necessary optimality conditions are proved. Here, we have to restrict ourselves to the case of everywhere defined smooth potentials. In both parts of the paper, we start from corresponding known results for the viscous case, derive sufficiently strong estimates that are uniform with respect to the (positive) viscosity parameter, and then let the viscosity tend to zero to establish the sought results for the nonviscous case. 
J. Sprekels, H. Wu, Optimal distributed control of a CahnHilliardDarcy system with mass sources, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, published online on 24.01.2019, DOI 10.1007/s00245019095554 .
Abstract
In this paper, we study an optimal control problem for a twodimensional CahnHilliardDarcy system with mass sources that arises in the modeling of tumor growth. The aim is to monitor the tumor fraction in a finite time interval in such a way that both the tumor fraction, measured in terms of a tracking type cost functional, is kept under control and minimal harm is inflicted to the patient by administering the control, which could either be a drug or nutrition. We first prove that the optimal control problem admits a solution. Then we show that the controltostate operator is Fréchet differentiable between suitable Banach spaces and derive the firstorder necessary optimality conditions in terms of the adjoint variables and the usual variational inequality. 
S.P. Frigeri, C.G. Gal, M. Grasselli, J. Sprekels, Strong solutions to nonlocal 2D CahnHilliardNavierStokes systems with nonconstant viscosity, degenerate mobility and singular potential, Nonlinearity, 32 (2019), pp. 678727, DOI 10.1088/13616544/aaedd0 .
Abstract
We consider a nonlinear system which consists of the incompressible NavierStokes equations coupled with a convective nonlocal CahnHilliard equation. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids having the same density. We suppose that the viscosity depends smoothly on the order parameter as well as the mobility. Moreover, we assume that the mobility is degenerate at the pure phases and that the potential is singular (e.g. of logarithmic type). This system is endowed with noslip boundary condition for the (average) velocity and homogeneous Neumann boundary condition for the chemical potential. Thus the total mass is conserved. In the twodimensional case, this problem was already analyzed in some joint papers of the first three authors. However, in the present general case, only the existence of a global weak solution, the (conditional) weakstrong uniqueness and the existence of the global attractor were proven. Here we are able to establish the existence of a (unique) strong solution through an approximation procedure based on time discretization. As a consequence, we can prove suitable uniform estimates which allow us to show some smoothness of the global attractor. Finally, we discuss the existence of strong solutions for the convective nonlocal CahnHilliard equation, with a given velocity field, in the three dimensional case as well. 
P. Colli, G. Gilardi, J. Sprekels, On a CahnHilliard system with convection and dynamic boundary conditions, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica ``U. Dini'', Firenze; SpringerVerlag, Heidelberg. English, French, German, Italian, English abstracts., 197 (2018), pp. 14451475, DOI 10.1007/s1023101807321 .
Abstract
This paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of CahnHilliard type; an additional convective term with a forced velocity field, which could act as a control on the system, is also present in the equation. Either regular or singular potentials are admitted in the bulk and on the boundary. Both the viscous and pure CahnHilliard cases are investigated, and a number of results is proven about existence of solutions, uniqueness, regularity, continuous dependence, uniform boundedness of solutions, strict separation property. A complete approximation of the problem, based on the regularization of maximal monotone graphs and the use of a FaedoGalerkin scheme, is introduced and rigorously discussed. 
P. Colli, G. Gilardi, J. Sprekels, On the longtime behavior of a viscous CahnHilliard system with convection and dynamic boundary conditions, Journal of Elliptic and Parabolic Equations, 4 (2018), pp. 327347, DOI 10.1007/s4180801800216 .
Abstract
In this paper, we study the longtime asymptotic behavior of a phase separation process occurring in a threedimensional domain containing a fluid flow of given velocity. This process is modeled by a viscous convective CahnHilliard system, which consists of two nonlinearly coupled secondorder partial differential equations for the unknown quantities, the chemical potential and an order parameter representing the scaled density of one of the phases. In contrast to other contributions, in which zero Neumann boundary conditions were are assumed for both the chemical potential and the order parameter, we consider the case of dynamic boundary conditions, which model the situation when another phase transition takes place on the boundary. The phase transition processes in the bulk and on the boundary are driven by free energies functionals that may be nondifferentiable and have derivatives only in the sense of (possibly setvalued) subdifferentials. For the resulting initialboundary value system of CahnHilliard type, general wellposedness results have been established in piera recent contribution by the same authors. In the present paper, we investigate the asymptotic behavior of the solutions as times approaches infinity. More precisely, we study the ωlimit (in a suitable topology) of every solution trajectory. Under the assumptions that the viscosity coefficients are strictly positive and that at least one of the underlying free energies is differentiable, we prove that the omegalimit is meaningful and that all of its elements are solutions to the corresponding stationary system, where the component representing the chemical potential is a constant. 
P. Colli, G. Gilardi, J. Sprekels, Optimal distributed control of a generalized fractional CahnHilliard system, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, pp. published online on 15.11.2018, urlhttps://doi.org/10.1007/s0024501895407, DOI 10.1007/s0024501895407 .
Abstract
In the recent paper “Wellposedness and regularity for a generalized fractional CahnHilliard system” by the same authors, general wellposedness results have been established for a class of evolutionary systems of two equations having the structure of a viscous CahnHilliard system, in which nonlinearities of doublewell type occur. The operators appearing in the system equations are fractional versions in the spectral sense of general linear operators A,B, having compact resolvents, which are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. In this work we complement the results given in quoted paper by studying a distributed control problem for this evolutionary system. The main difficulty in the analysis is to establish a rigorous Fréchet differentiability result for the associated controltostate mapping. This seems only to be possible if the state stays bounded, which, in turn, makes it necessary to postulate an additional global boundedness assumption. One typical situation, in which this assumption is satisfied, arises when B is the negative Laplacian with zero Dirichlet boundary conditions and the nonlinearity is smooth with polynomial growth of at most order four. Also a case with logarithmic nonlinearity can be handled. Under the global boundedness assumption, we establish existence and firstorder necessary optimality conditions for the optimal control problem in terms of a variational inequality and the associated adjoint state system. 
P. Colli, G. Gilardi, J. Sprekels, Optimal velocity control of a viscous CahnHilliard system with convection and dynamic boundary conditions, SIAM Journal on Control and Optimization, 56 (2018), pp. 16651691, DOI 10.1137/17M1146786 .
Abstract
In 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 an additional difficulty for the analysis. The nonlinearities in the bulk and surface free energies are of logarithmic type, which entails that the thermodynamic forces driving the phase separation process may become singular. We show existence for the optimal control problem under investigation, prove the Fréchet differentiability of the associated controltostate mapping in suitable Banach spaces, and derive the firstorder necessary optimality conditions in terms of a variational inequality and the associated adjoint system. Due to the strong nonlinear couplings between state variables and control, the corresponding proofs require a considerable analytical effort. 
S. Frigeri, M. Grasselli, J. Sprekels, Optimal distributed control of twodimensional nonlocal CahnHilliardNavierStokes systems with degenerate mobility and singular potential, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, pp. published online on 24.09.2018, urlhttps://doi.org/10.1007/s0024501895247, DOI 10.1007/s0024501895247 .
Abstract
In this paper, we consider a twodimensional diffuse interface model for the phase separation of an incompressible and isothermal binary fluid mixture with matched densities. This model consists of the NavierStokes equations, nonlinearly coupled with a convective nonlocal CahnHilliard equation. The system rules the evolution of the volumeaveraged velocity of the mixture and the (relative) concentration difference of the two phases. The aim of this work is to study an optimal control problem for such a system, the control being a timedependent external force acting on the fluid. We first prove the existence of an optimal control for a given tracking type cost functional. Then we study the differentiability properties of the controltostate map, and we establish firstorder necessary optimality conditions. These results generalize the ones obtained by the first and the third authors jointly with E. Rocca in [19]. There the authors assumed a constant mobility and a regular potential with polynomially controlled growth. Here, we analyze the physically more relevant case of a degenerate mobility and a singular (e.g., logarithmic) potential. This is made possible by the existence of a unique strong solution which was recently proved by the authors and C. G. Gal in [14]. 
M. Hintermüller, M. Hinze, Ch. Kahle, T. Keil, A goaloriented dualweighted adaptive finite element approach for the optimal control of a nonsmooth CahnHilliardNavierStokes system, Optimization and Engineering. International Multidisciplinary Journal to Promote Optimization Theory & Applications in Engineering Sciences, 19 (2018), pp. 629662, DOI 10.1007/s1108101893936 .
Abstract
This paper is concerned with the development and implementation of an adaptive solution algorithm for the optimal control of a timediscrete CahnHilliardNavierStokes system with variable densities. The free energy density associated to the CahnHilliard system incorporates the doubleobstacle 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 NavierStokes equation. A dualweighed residual approach for goaloriented adaptive finite elements is presented which is based on the concept of Cstationarity. 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. 
S. Bergmann, D.A. BarraganYani, E. Flegel, K. Albe, B. Wagner, Anisotropic solidliquid interface kinetics in silicon: An atomistically informed phasefield model, Modelling and Simulation in Materials Science and Engineering, 25 (2017), pp. 065015/1065015/20, DOI 10.1088/1361651X/aa7862 .
Abstract
We present an atomistically informed parametrization of a phasefield model for describing the anisotropic mobility of liquidsolid interfaces in silicon. The model is derived from a consistent set of atomistic data and thus allows to directly link molecular dynamics and phase field simulations. Expressions for the free energy density, the interfacial energy and the temperature and orientation dependent interface mobility are systematically fitted to data from molecular dynamics simulations based on the StillingerWeber interatomic potential. The temperaturedependent interface velocity follows a VogelFulcher type behavior and allows to properly account for the dynamics in the undercooled melt. 
P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), pp. 25182546.
Abstract
In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by HawkinsDaruud et al. in citeHZO. The model consists of a CahnHilliard equation for the tumor cell fraction $vp$ coupled to a reactiondiffusion equation for a function $s$ representing the nutrientrich extracellular water volume fraction. The distributed control $u$ monitors as a righthand 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 controltostate operator is Fréchet differentiable between appropriate Banach spaces and derive the firstorder necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. 
P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Asymptotic analyses and error estimates for a CahnHilliard type phase field system modelling tumor growth, Discrete and Continuous Dynamical Systems, 10 (2017), pp. 3754.
Abstract
This paper is concerned with a phase field system of CahnHilliard 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 wellposedness, 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 
CH. Heinemann, Ch. Kraus, E. Rocca, R. Rossi, A temperaturedependent phasefield model for phase separation and damage, Archive for Rational Mechanics and Analysis, 225 (2017), pp. 177247.
Abstract
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 CahnHilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321359] and [C. Heinemann, C. Kraus: Existence results for diffuse interface models describing phase separation and damage. European J. Appl. Math. 24 (2013), 179211]), 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), 461490] in the framework of FourierNavierStokes 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), 13451369], [E. Rocca, R. Rossi: "Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage. SIAM J. Math. Anal., 47 (2015), 25192586] for the study of PDE systems for phase transition and damage. Our globalintime existence result is obtained by passing to the limit in a carefully devised timediscretization scheme. 
P. Krejčí, E. Rocca, J. Sprekels, Unsaturated deformable porous media flow with thermal phase transition, Mathematical Models & Methods in Applied Sciences, 27 (2017), pp. 26752710, DOI 10.1142/S0218202517500555 .
Abstract
In the present paper, a continuum model is introduced for fluid flow in a deformable porous medium, where the fluid may undergo phase transitions. Typically, such problems arise in modeling liquidsolid phase transformations in groundwater flows. The system of equations is derived here from the conservation principles for mass, momentum, and energy and from the ClausiusDuhem inequality for entropy. It couples the evolution of the displacement in the matrix material, of the capillary pressure, of the absolute temperature, and of the phase fraction. Mathematical results are proved under the additional hypothesis that inertia effects and shear stresses can be neglected. For the resulting highly nonlinear system of two PDEs, one ODE and one ordinary differential inclusion with natural initial and boundary conditions, existence of global in time solutions is proved by means of cutoff techniques and suitable Mosertype estimates. 
M. Hintermüller, T. Keil, D. Wegner, Optimal control of a semidiscrete CahnHilliardNavierStokes system with nonmatched fluid densities, SIAM Journal on Control and Optimization, 55 (2017), pp. 19541989.

S.P. Frigeri, Global existence of weak solutions for a nonlocal model for twophase flows of incompressible fluids with unmatched densities, Mathematical Models & Methods in Applied Sciences, 26 (2016), pp. 19571993.
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 NavierStokes type system coupled with a convective nonlocal CahnHilliard 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 doublewell 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. 189210.
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 NavierStokes 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 CahnHilliard equation with dynamic boundary conditions, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 73 (2016), pp. 195225, DOI 10.1007/s002450159299z .
Abstract
A boundary control problem for the viscous CahnHilliard 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 CahnHilliard type system modeling phase separation, Journal of Differential Equations, 260 (2016), pp. 79407964.
Abstract
This paper investigates a nonlocal version of a model for phase separation on an atomic lattice that was introduced by P. PodioGuidugli in Ric. Mat. 55 (2006) 105118. The model consists of an initialboundary 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 doubleobstacle potentials are admitted. In contrast to the local model, which was studied by P. PodioGuidugli 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 longrange 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. 11671187.
Abstract
Anisotropy is an essential feature of phasefield 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 phasefield 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  GinzburgLandau type phasefield model for dendritic growth. For the resulting derivativefree 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 vectorvalued AllenCahn variational inequalities, SIAM Journal on Control and Optimization, 54 (2016), pp. 129152.
Abstract
In this paper we consider a elastic vectorvalued AllenCahn 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 firstorder necessary optimality conditions for the original problem. 
S.P. Frigeri, E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal CahnHilliard/NavierStokes 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 NavierStokes system with a convective nonlocal CahnHilliard equation in two dimensions of space. We apply recently proved wellposedness and regularity results in order to establish existence of optimal controls as well as firstorder 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. 25192586.
Abstract
In this paper we analyze a PDE system modelling (nonisothermal) 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 righthand 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 globalintime 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 timediscrete analysis could be useful towards the numerical study of this model. 
E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal convective CahnHilliard equation by the velocity in three dimensions, SIAM Journal on Control and Optimization, 53 (2015), pp. 16541680.
Abstract
In this paper we study a distributed optimal control problem for a nonlocal convective CahnHilliard 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 wellposedness 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 firstorder necessary optimality conditions for the optimal control problem. 
S.P. Frigeri, M. Grasselli, E. Rocca, A diffuse interface model for twophase incompressible flows with nonlocal interactions and nonconstant mobility, Nonlinearity, 28 (2015), pp. 12571293.
Abstract
We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the NavierStokes system coupled with a convective nonlocal CahnHilliard equation with nonconstant mobility. We first prove the existence of a global weak solution in the case of nondegenerate 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 CahnHilliard equation with degenerate mobility and singular potential in dimension three. 
M. Heida, Existence of solutions for two types of generalized versions of the CahnHilliard equation, Applications of Mathematics, 60 (2015), pp. 5190.
Abstract
We show existence of solutions to two types of generalized anisotropic CahnHilliard 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 noflux 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 CahnHilliard and AllenCahn equations considered as gradient flows in Hilbert spaces, Journal of Mathematical Analysis and Applications, 423 (2015), pp. 410455.

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. 39283959.
Abstract
In this work, we analytically investigate a multicomponent 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 CahnHilliard type phase field system related to tumor growth, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 26 (2015), pp. 93108.
Abstract
In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of CahnHilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in [ColliGilardiHilhorst 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 CahnHilliard equation with dynamic boundary conditions, Advances in Nonlinear Analysis, 4 (2015), pp. 311325.
Abstract
A boundary control problem for the pure CahnHilliard equations with possibly singular potentials and dynamic boundary conditions is studied and firstorder necessary conditions for optimality are proved. 
P. Colli, M.H. Farshbaf Shaker, G. Gilardi, J. Sprekels, Optimal boundary control of a viscous CahnHilliard system with dynamic boundary condition and double obstacle potentials, SIAM Journal on Control and Optimization, 53 (2015), pp. 26962721.
Abstract
In this paper, we investigate optimal boundary control problems for CahnHilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the LaplaceBeltrami 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 firstorder necessary conditions of optimality. The general strategy, which follows the lines of the recent approach by Colli, FarshbafShaker, Sprekels (see Appl. Math. Optim., 2014) to the (simpler) AllenCahn 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 socalled “deep quench limit”. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired firstorder necessary optimality conditions for the case of (nondifferentiable) double obstacle potentials. 
P. Colli, M.H. Farshbaf Shaker, G. Gilardi, J. Sprekels, Secondorder analysis of a boundary control problem for the viscous CahnHilliard equation with dynamic boundary conditions, Annals of the Academy of Romanian Scientists. Mathematics and its Applications., 7 (2015), pp. 4166.
Abstract
In this paper we establish secondorder 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 CahnHilliard 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 AllenCahn equation with dynamic boundary conditions and double obstacles, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 71 (2015), pp. 124.
Abstract
In this paper, we investigate optimal control problems for AllenCahn variational inequalities with a dynamic boundary condition involving double obstacle potentials and the LaplaceBeltrami 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 firstorder 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 socalled “deep quench limit”. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired firstorder necessary optimality conditions for the case of (nondifferentiable) double obstacle potentials. 
P. Colli, J. Sprekels, Optimal control of an AllenCahn equation with singular potentials and dynamic boundary condition, SIAM Journal on Control and Optimization, 53 (2015), pp. 213234.
Abstract
In this paper, we investigate optimal control problems for AllenCahn equations with singular nonlinearities and a dynamic boundary condition involving singular nonlinearities and the LaplaceBeltrami 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 LaplaceBeltrami 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 wellposedness and regularity results for the state equation and then show the existence of optimal controls and that the controltostate mapping is twice continuously Fréchet differentiable between appropriate function spaces. Based on these results, we establish the firstorder necessary optimality conditions in terms of a variational inequality and the adjoint state equation, and we prove secondorder sufficient optimality conditions. 
E. Feireisl, E. Rocca, G. Schimperna, A. Zarnescu, Nonisothermal nematic liquid crystal flows with the BallMajumdar free energy, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica ``U. Dini'', Firenze; SpringerVerlag, Heidelberg. English, French, German, Italian, English abstracts., 194 (2015), pp. 12691299.
Abstract
In this paper we prove the existence of global in time weak solutions for an evolutionary PDE system modelling nonisothermal Landaude Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible NavierStokes system for the macroscopic velocity $vu$ is coupled to a nonlinear convective parabolic equation describing the evolution of the Qtensor $QQ$, namely a tensorvalued 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 Qtensor 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 blowup rate, is carried out. 
M.G. Hennessy, V.M. Burlakov, A. Münch, B. Wagner, A. Goriely, Controlled topological transitions in thinfilm phase separation, SIAM Journal on Applied Mathematics, 75 (2015), pp. 3860.
Abstract
In this paper the evolution of a binary mixture in a thinfilm 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 phasefield model and using matched asymptotic expansions a corresponding sharpinterface problem for the location of the interface is established. It is then argued that a thus created twolayer system is not at its energetic minimum but destabilizes into a controlled selfreplicating 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 thinfilm model for the free interfaces, which is derived asymptotically from the sharpinterface model. 
M.H. Farshbaf Shaker, A relaxation approach to vectorvalued AllenCahn MPEC problems, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 72 (2015), pp. 325351.

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. 217250.
Abstract
We introduce a complete damage model with a timedepending domain for linearelastically stressed solids under timevarying Dirichlet boundary conditions. The evolution of the system is described by a doubly nonlinear differential inclusion for the damage process and a quasistatic 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 localintime existence and globalintime 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 timedepending domain. In this context, two major challenges arise: Firstly, the timedependent 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 timedepending 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. 25652590.
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 hyperbolicparabolic 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, timediscretization and different variational techniques. 
CH. Heinemann, Ch. Kraus, Existence of weak solutions for a hyperbolicparabolic phase field system with mixed boundary conditions on nonsmooth domains, SIAM Journal on Mathematical Analysis, 47 (2015), pp. 20442073.
Abstract
The aim of this paper is to prove existence of weak solutions of hyperbolicparabolic 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 timediscretization, $H^2$regularization and variational techniques. 
CH. Heinemann, E. Rocca, Damage processes in thermoviscoelastic materials with damagedependent thermal expansion coefficients, Mathematical Methods in the Applied Sciences, 38 (2015), pp. 45874612.
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 damagedependent 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. 12651341.

G. Aki, W. Dreyer, J. Giesselmann, Ch. Kraus, A quasiincompressible diffuse interface model with phase transition, Mathematical Models & Methods in Applied Sciences, 24 (2014), pp. 827861.
Abstract
This work introduces a new thermodynamically consistent diffuse model for twocomponent 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 NavierStokes equations in the bulk phases equipped with admissible interfacial conditions. For the NavierStokes 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. PodioGuidugli, J. Sprekels, Analysis of a time discretization scheme for a nonstandard viscous CahnHilliard system, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), pp. 10611087.
Abstract
In this paper we propose a time discretization of a system of two parabolic equations describing diffusiondriven 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 boundaryvalue 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. 13181324.
Abstract
The present note deals with a nonstandard systems of differential equations describing a twospecies 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. 257275.
Abstract
We are concerned with a nonstandard phase field model of CahnHilliard type. The model, which was introduced by PodioGuidugli (Ric. Mat. 2006), describes twospecies phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated by Colli, Gilardi, PodioGuidugli, 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 initialboundary 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 CahnHilliard equation with dynamic boundary conditions and a dominating boundary potential, Journal of Mathematical Analysis and Applications, 419 (2014), pp. 972994.
Abstract
The CahnHilliard and viscous CahnHilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some wellposedness and regularity results are proved. 
S. Melchionna, E. Rocca, On a nonlocal CahnHilliard equation with a reaction term, Advances in Mathematical Sciences and Applications, 24 (2014), pp. 461497.
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 PenroseFife phasefield model with coupled dynamic boundary conditions, Discrete and Continuous Dynamical Systems, 34 (2014), pp. 42594290.

W. Dreyer, J. Giesselmann, Ch. Kraus, A compressible mixture model with phase transition, Physica D. Nonlinear Phenomena, 273274 (2014), pp. 113.
Abstract
We introduce a new thermodynamically consistent diffuse interface model of AllenCahn/NavierStokes type for multicomponent 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 nondissipative and a dissipative regime, where we recover in the sharp interface limit a generalized AllenCahn/Euler system for mixtures with chemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satify, for instance, a YoungLaplace and a Stefan type law. 
P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, An asymptotic analysis for a nonstandard CahnHilliard system with viscosity, Discrete and Continuous Dynamical Systems  Series S, 6 (2013), pp. 353368.
Abstract
This paper is concerned with a diffusion model of phasefield 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 wellposed and investigated the longtime 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 longtime behavior we characterize; in the proofs, we employ compactness and monotonicity arguments. 
P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, Global existence and uniqueness for a singular/degenerate CahnHilliard system with viscosity, Journal of Differential Equations, 254 (2013), pp. 42174244.
Abstract
Existence and uniqueness are investigated for a nonlinear diffusion problem of phasefield 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 twospecies 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 phasefield equation for $rho$ is coupled with a genuinely nonlinear diffusion equation for $mu$. The existence of a globalintime 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. PodioGuidugli, J. Sprekels, Continuous dependence for a nonstandard CahnHilliard system with nonlinear atom mobility, Rendiconti del Seminario Matematico. Universita e Politecnico Torino, 70 (2012), pp. 2752.
Abstract
This note is concerned with a nonlinear diffusion problem of phasefield 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 twospecies phase segregation on an atomic lattice [PodioGuidugli 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/PodioGuidugli/Sprekels 2012. 
P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, Distributed optimal control of a nonstandard system of phase field equations, Continuum Mechanics and Thermodynamics, 24 (2012), pp. 437459.
Abstract
We investigate a distributed optimal control problem for a phase field model of CahnHilliard type. The model describes twospecies 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 firstorder necessary conditions of optimality. 
P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, Global existence for a strongly coupled CahnHilliard system with viscosity, Bollettino della Unione Matematica Italiana. Serie 9, 5 (2012), pp. 495513.
Abstract
An existence result is proved for a nonlinear diffusion problem of phasefield 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 twospecies 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 phasefield 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. 119149.
Abstract
We investigate a nonstandard phase field model of CahnHilliard type. The model, which was introduced in PodioGuidugli (2006), describes twospecies phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been studied recently in Colli, Gilardi, PodioGuidugli, 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 wellposedness, and we study optimal boundary control problems. Existence of optimal controls is shown, and the firstorder 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. 461483.
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 multiphase 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 quasiincompressible flows: Sharp interface limits and numerics, ESAIM Proceedings, 38 (2012), pp. 5477.
Abstract
In this contribution, we investigate a diffuse interface model for quasiincompressible 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 CahnHilliard equation numerically by a Local Discontinuous Galerkin scheme. 
S. Bartels, R. Müller, Error control for the approximation of AllenCahn and CahnHilliard equations with a logarithmic potential, Numerische Mathematik, 119 (2011), pp. 409435.
Abstract
A fully computable upper bound for the finite element approximation error of AllenCahn and CahnHilliard 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 quasioptimal 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, Quasioptimal and robust a posteriori error estimates in $L^infty(L^2)$ for the approximation of AllenCahn equations past singularities, Mathematics of Computation, 80 (2011), pp. 761780.
Abstract
Optimal a posteriori error estimates in $L^infty(0,T;L^2(O))$ are derived for the finite element approximation of AllenCahn 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 multiphase system with nonconstant specific heat and heat conductivity, Journal of Differential Equations, 251 (2011), pp. 13541387.

P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, Wellposedness and longtime behavior for a nonstandard viscous CahnHilliard system, SIAM Journal on Applied Mathematics, 71 (2011), pp. 18491870.
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 globalintime smooth solution to the associated initial/boundaryvalue problem; moreover, we give a description of the relative $omega$limit set. 
K. Hermsdörfer, Ch. Kraus, D. Kröner, Interface conditions for limits of the NavierStokesKorteweg model, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 13 (2011), pp. 239254.
Abstract
In this contribution we will study the behaviour of the pressure across phase boundaries in liquidvapour flows. As mathematical model we will consider the static version of the NavierStokesKorteweg 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. 391407.

S. Bartels, R. Müller, A posteriori error controlled local resolution of evolving interfaces for generalized CahnHilliard equations, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 12 (2010), pp. 4573.

P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, A temperaturedependent phase segregation problem of the AllenCahn type, Advances in Mathematical Sciences and Applications, 20 (2010), pp. 219234.

P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, Existence and uniqueness of a globalintime solution to a phase segregation problem of the AllenCahn type, Mathematical Models & Methods in Applied Sciences, 20 (2010), pp. 519541.

H. Garcke, Ch. Kraus, An anisotropic, inhomogeneous, elastically modified GibbsThomson law as singular limit of a diffuse interface model, Advances in Mathematical Sciences and Applications, 20 (2010), pp. 511545.
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 GibbsThomson law in the sharp interface limit. To this end we show that one can pass to the limit in the weak formulation of the EulerLagrange equation of the diffuse phase field energy. 
W. Dreyer, Ch. Kraus, On the van der WaalsCahnHilliard phasefield model and its equilibria conditions in the sharp interface limit, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 140 A (2010), pp. 11611186.
Abstract
We study the equilibria of liquidvapor 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 WaalsCahnHilliard 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 twophase 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. 30283048.

P. Krejčí, E. Rocca, J. Sprekels, A bottle in a freezer, SIAM Journal on Mathematical Analysis, 41 (2009), pp. 18511873.

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. 411441.

P. Colli, P. Krejčí, E. Rocca, J. Sprekels, Nonlinear evolution inclusions arising from phase change models, Czechoslovak Mathematical Journal, 57 (2007), pp. 10671098.

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. 181194.

P. Krejčí, E. Rocca, J. Sprekels, A nonlocal phasefield model with nonconstant specific heat, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 9 (2007), pp. 285306.

P. Krejčí, E. Rocca, J. Sprekels, Nonlocal temperaturedependent phasefield models for nonisothermal phase transitions, Journal of the London Mathematical Society. Second Series, 76 (2007), pp. 197210.

P. Krejčí, J. Sprekels, Long time behaviour of a singular phase transition model, Discrete and Continuous Dynamical Systems, 15 (2006), pp. 11191135.

W. Dreyer, B. Wagner, Sharpinterface model for eutectic alloys. Part I: Concentration dependent surface tension, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 7 (2005), pp. 199227.

J. Griepentrog, On the unique solvability of a nonlocal phase separation problem for multicomponent systems, Banach Center Publications, 66 (2004), pp. 153164.

O. Klein, Asymptotic behaviour for a phasefield model with hysteresis in onedimensional thermoviscoplasticity, Applications of Mathematics, 49 (2004), pp. 309341.

P. Krejčí, J. Sprekels, Nonlocal phasefield models for nonisothermal phase transitions and hysteresis, Advances in Mathematical Sciences and Applications, 14 (2004), pp. 593612.

P. Krejčí, J. Sprekels, U. Stefanelli, Onedimensional thermoviscoplastic processes with hysteresis and phase transitions, Advances in Mathematical Sciences and Applications, 13 (2003), pp. 695712.

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. 755785.

O. Klein, C. Verdi, A posteriori error estimates for a time discrete scheme for a phasefield system of PenroseFife type, IMA Journal of Numerical Analysis, 23 (2003), pp. 5580.

J. Sprekels, S. Zheng, Global existence and asymptotic behaviour for a nonlocal phasefield model for nonisothermal phase transitions, Journal of Mathematical Analysis and Applications, 279 (2003), pp. 97110.

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. 4865.

N. Kenmochi, J. Sprekels, Phasefield systems with vectorial order parameters including diffusional hysteresis effects, Communications on Pure and Applied Analysis, 1 (2002), pp. 495511.

J. Sprekels, P. Krejčí, Phasefield systems for multidimensional PrandtlIshlinskii operators with nonpolyhedral characteristics, Mathematical Methods in the Applied Sciences, 25 (2002), pp. 309325.

P. Krejčí, J. Sprekels, S. Zheng, Asymptotic behaviour for a phasefield system with hysteresis, Journal of Differential Equations, 175 (2001), pp. 88107.

P. Krejčí, J. Sprekels, A hysteresis approach to phasefield models, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 39 (2000), pp. 569586.

P. Krejčí, J. Sprekels, Phasefield models with hysteresis, Journal of Mathematical Analysis and Applications, 252 (2000), pp. 198219.

G. Gilardi, P. Krejčí, J. Sprekels, Hysteresis in phasefield models with thermal memory, Mathematical Methods in the Applied Sciences, 23 (2000), pp. 909922.

W. Dreyer, W.H. Müller, A study of the coarsening in tin/lead solders, International Journal of Solids and Structures, 37 (2000), pp. 38413871.

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. 8394.
Contributions to Collected Editions

P. Colli, G. Gilardi, J. Sprekels, Limiting problems for a nonstandard viscous CahnHilliard system with dynamic boundary conditions, in: Proceedings of the INdAMISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin, eds., 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 217242, DOI 10.1007/9783319759401_11 .
Abstract
This note is concerned with a nonlinear diffusion problem of phasefield type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of twospecies phase segregation on an atomic lattice and was introduced by PodioGuidugli in Ric. Mat. 55 (2006), pp.105118. 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 LaplaceBeltrami 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 longtime behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omegalimit set in both cases. 
P. Colli, J. Sprekels, Optimal boundary control of a nonstandard CahnHilliard system with dynamic boundary condition and double obstacle inclusions, in: Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs: In Honour of Prof. Gianni Gilardi, P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels, eds., 22 of Springer INdAM Series, Springer International Publishing AG, Cham, 2017, pp. 151182, DOI 10.1007/9783319644899 .
Abstract
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.PodioGuidugli in Ric. Mat. 55 (2006), pp.105118. 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 LaplaceBeltrami 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. 3558, 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. 130, for the case of (differentiable) logarithmic potentials and perform a socalled "deep quench limit". Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired firstorder necessary optimality conditions for the case of (nondifferentiable) double obstacle potentials. 
A. Mielke, Free energy, free entropy, and a gradient structure for thermoplasticity, in: Innovative Numerical Approaches for MultiField and MultiScale 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. 135160.
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 nonisothermal 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 timeincremental minimization procedures, where the entropyproduction 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 boundaryvalue problem for general dissipative solids" (J. Mech. Physics Solids, 54, 401424, 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 twophase model, in: Scales in Plasticity, MiniWorkshop, November 814, 2015, G.A. Francfort, S. Luckhaus, eds., 12 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2015, pp. 30273030.

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. 189203.

P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, Global solution to a phase transition problem of the AllenCahn type, in: Nonlinear Evolution Equations and Mathematical Modeling, Proceedings of the RIMS Symposium held at the Research Institute for Mathematical Sciences, Kyoto University, October 2023, 2009, T. Aiki, ed., 1693 of Sūrikaisekikenkyūsho Kōkyūroku, Kyoto, 2010, pp. 104110.

P. Krejčí, E. Rocca, J. Sprekels, Liquidsolid phase transitions in a deformable container, in: Continuous Media with Microstructure, B. Albers, ed., Springer, Berlin/Heidelberg, 2010, pp. 285300.

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, 2830 May 2003, L. LopezDias, L. Torres, O. Alejos, eds., 343 of Physica B: Condensed Matter, Elsevier B.V., 2004, pp. 5358.

P. Krejčí, J. Sprekels, Phasefield 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. 295310.
Preprints, Reports, Technical Reports

O. Souček, M. Heida, J. Málek, On a thermodynamic framework for developing boundary conditions for Korteweg fluids, Preprint no. 2599, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2599 .
Abstract, PDF (2111 kByte)
We provide a derivation of several classes of boundary conditions for fluids of Kortewegtype using a simple and transparent thermodynamic approach that automatically guarentees that the derived boundary conditions are compatible with the second law of thermodynamics. The starting assumption of our approach is to describe the boundary of the domain as the membrane separating two different continua, one inside the domain, and the other outside the domain. With this viewpoint one may employ the framework of continuum thermodynamics involving singular surfaces. This approach allows us to identify, for various classes of surface Helmholtz free energies, the corresponding surface entropy production mechanisms. By establishing the constitutive relations that guarantee that the surface entropy production is nonnegative, we identify a new class of boundary conditions, which on one hand generalizes in a nontrivial manner the Navier's slip boundary conditions, and on the other hand describes dynamic and static contact angle conditions. We explore the general model in detail for a particular case of Korteweg fluid where the Helmholtz free energy in the bulk is that of a van der Waals fluid. We perform a series of numerical experiments to document the basic qualitative features of the novel boundary conditions and their practical applicability to model phenomena such as the contact angle hysteresis. 
P. Colli, G. Gilardi, J. Sprekels, Longtime behavior for a generalized CahnHilliard system with fractional operators, Preprint no. 2588, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2588 .
Abstract, PDF (248 kByte)
In this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the CahnHilliard system, with possibly singular potentials, which we recently investigated in the paper "Wellposedness and regularity for a generalized fractional CahnHilliard system". More precisely, we give a complete characterization of the Omegalimit of the phase parameter. The characterization depends on the first eigenvalue of one of the involved operators: if this eigenvalue is positive, then the chemical potential vanishes at infinity, and every element of the Omegalimit is a stationary solution to the phase equation; if it is zero instead, then every element of the Omegalimit solves a problem containing a real function which is related to the chemical potential. Such a function is nonunique and time dependent, in general, as we show by means of an example; however, we give sufficient conditions for it to be uniquely determined and constant. 
P. Colli, G. Gilardi, J. Sprekels, Deep quench approximation and optimal control of general CahnHilliard systems with fractional operators and double obstacle potentials, Preprint no. 2559, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2559 .
Abstract, PDF (349 kByte)
In the recent paper ”Wellposedness and regularity for a generalized fractional CahnHilliard system”, the same authors derived general wellposedness and regularity results for a rather general system of evolutionary operator equations having the structure of a CahnHilliard system. The operators appearing in the system equations were fractional versions in the spectral sense of general linear operators A and B having compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. The associated doublewell potentials driving the phase separation process modeled by the CahnHilliard system could be of a very general type that includes standard physically meaningful cases such as polynomial, logarithmic, and double obstacle nonlinearities. In the subsequent paper ”Optimal distributed control of a generalized fractional CahnHilliard system” (Appl. Math. Optim. (2018), https://doi.org/10.1007/s0024501895407) by the same authors, an analysis of distributed optimal control problems was performed for such evolutionary systems, where only the differentiable case of certain polynomial and logarithmic doublewell potentials could be admitted. Results concerning existence of optimizers and firstorder necessary optimality conditions were derived, where more restrictive conditions on the operators A and B had to be assumed in order to be able to show differentiability properties for the associated controltostate operator. In the present paper, we complement these results by studying a distributed control problem for such evolutionary systems in the case of nondifferentiable nonlinearities of double obstacle type. For such nonlinearities, it is well known that the standard constraint qualifications cannot be applied to construct appropriate Lagrange multipliers. To overcome this difficulty, we follow here the socalled ”deep quench” method. This technique, in which the nondifferentiable double obstacle nonlinearity is approximated by differentiable logarithmic nonlinearities, was first developed by P. Colli, M.H. FarshbafShaker and J. Sprekels in the paper ”A deep quench approach to the optimal control of an AllenCahn equation with dynamic boundary conditions and double obstacles” (Appl. Math. Optim. 71 (2015), pp. 124) and has proved to be a powerful tool in a number of optimal control problems with double obstacle potentials in the framework of systems of CahnHilliard type. We first give a general convergence analysis of the deep quench approximation that includes an error estimate and then demonstrate that its use leads in the double obstacle case to appropriate firstorder necessary optimality conditions in terms of a variational inequality and the associated adjoint state system. 
P. Colli, G. Gilardi, J. Sprekels, Wellposedness and regularity for a generalized fractional CahnHilliard system, Preprint no. 2509, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2509 .
Abstract, PDF (309 kByte)
In this paper, we investigate a rather general system of two operator equations that has the structure of a viscous or nonviscous CahnHilliard system in which nonlinearities of doublewell type occur. Standard cases like regular or logarithmic potentials, as well as nondifferentiable potentials involving indicator functions, are admitted. The operators appearing in the system equations are fractional versions of general linear operators A and B, where the latter are densely defined, unbounded, selfadjoint and monotone in a Hilbert space of functions defined in a smooth domain and have compact resolvents. In this connection, we remark the fact that our definition of the fractional power of operators uses the approach via spectral theory. Typical cases are given by standard secondorder elliptic differential operators (e.g., the Laplacian) with zero Dirichlet or Neumann boundary conditions, but also other cases like fourthorder systems or systems involving the Stokes operator are covered by the theory. We derive in this paper general wellposedness and regularity results that extend corresponding results which are known for either the nonfractional Laplacian with zero Neumann boundary condition or the fractional Laplacian with zero Dirichlet condition. These results are entirely new if at least one of the operators A and B differs from the Laplacian. It turns out that the first eigenvalue λ_{1} of A plays an important und not entirely obvious role: if λ_{1} is positive, then the operators A and B may be completely unrelated; if, however, λ_{1} equals zero, then it must be simple and the corresponding onedimensional eigenspace has to consist of the constant functions and to be a subset of the domain of definition of a certain fractional power of B. We are able to show general existence, uniqueness, and regularity results for both these cases, as well as for both the viscous and the nonviscous system. 
E. Meca Álvarez, A. Münch, B. Wagner, Localized instabilities and spinodal decomposition in driven systems in the presence of elasticity, Preprint no. 2387, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2387 .
Abstract, PDF (801 kByte)
We study numerically and analytically the instabilities associated with phase separation in a solid layer on which an external material flux is imposed. The first instability is localized within a boundary layer at the exposed free surface by a process akin to spinodal decomposition. In the limiting static case, when there is no material flux, the coherent spinodal decomposition is recovered. In the present problem stability analysis of the timedependent and nonuniform base states as well as numerical simulations of the full governing equations are used to establish the dependence of the wavelength and onset of the instability on parameter settings and its transient nature as the patterns eventually coarsen into a flat moving front. The second instability is related to the MullinsSekerka instability in the presence of elasticity and arises at the moving front between the two phases when the flux is reversed. Stability analyses of the full model and the corresponding sharpinterface model are carried out and compared. Our results demonstrate how interface and bulk instabilities can be analysed within the same framework which allows to identify and distinguish each of them clearly. The relevance for a detailed understanding of both instabilities and their interconnections in a realistic setting are demonstrated for a system of equations modelling the lithiation/delithiation processes within the context of Lithium ion batteries.
Talks, Poster

D. Peschka, Gradient structures for flows of concentrated suspensions  jamming and free boundaries, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S11 ``Interfacial Flows", February 18  22, 2019, Technische Universität Wien, Austria, February 20, 2019.

M. Hintermüller, Optimal control of multiphase fluids and droplets, Colloquium of the Mathematical Institute, University of Oxford, UK, June 7, 2019.

J. Sprekels, CahnHilliard systems with general fractionalorder operators, Workshop ``Special Materials and Complex Systems'' (SMACS 2018), June 18  22, 2018, University of Milan/University of Pavia, Gargnano, Italy, June 22, 2018.

M. Thomas, Analysis and simulation for a phasefield fracture model at finite strains based on modified invariants, Workshop ``Special Materials and Complex Systems'' (SMACS 2018), June 18  22, 2018, University of Milan/University of Pavia, Gargnano, Italy, June 18, 2018.

M. Thomas, Reliable error estimates for phasefield models of brittle fracture, MATH+ Center Days 2018, October 31  November 2, 2018, ZuseInstitut Berlin (ZIB), Berlin, October 31, 2018.

M. Hintermüller, Nonsmooth structures in PDE constrained optimization, Mathematisches Kolloquium, Universität Bielefeld, Fakultät für Mathematik, June 7, 2018.

T. Keil, Simulation and control of a nonsmooth CahnHilliard NavierStokes system with variable fluid densities (with Carmen Graessle), Annual Meeting of the DFG Priority Programme 1962, October 9  11, 2017, Kremmen (Sommerfeld), October 11, 2017.

T. Keil, Strong stationarity conditions for the optimal control of a CahnHilliardNavierStokes system, 14th International Conference on Free Boundary Problems: Theory and Applications, Theme Session 8 ``Optimization and Control of Interfaces'', July 9  14, 2017, Shanghai Jiao Tong University, China, July 10, 2017.

M. Hintermüller, Adaptive finite element solvers for MPECs in function space, SIAM Conference on Optimization, Minisymposium MS122 ``Recent Trends in PDEConstrained Optimization'', May 22  25, 2017, Vancouver, British Columbia, Canada, May 25, 2017.

M. Hintermüller, Nonsmooth structures in PDEconstrained optimization, Mathematisches Kolloquium, Universität DuisburgEssen, Fakultät für Mathematik, Essen, January 11, 2017.

M. Hintermüller, Optimal control of multiphase fluids and droplets, Kolloquium, FriedrichAlexanderUniversität ErlangenNürnberg, Department Mathematik, Erlangen, May 2, 2017.

M. Hintermüller, Optimal control of nonsmooth phasefield models, DFGAIMS Workshop on ``Shape Optimization, Homogenization and Control'', March 13  16, 2017, Mbour, Senegal, March 14, 2017.

M. Hintermüller, Recent trends in PDEconstrained optimization with nonsmooth structures, Fourth Conference on Numerical Analysis and Optimization (NAOIV2017), January 2  5, 2017, Sultan Qaboos University, Muscat, Oman, January 4, 2017.

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

M. Thomas, Rateindependent evolution of sets, INdAMISIMM 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 CahnHilliard 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 CahnHilliard/NavierStokes 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 damagedependent 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 CahnHilliard 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 EssenDuisburg, Fakultät für Mathematik, Essen, December 3, 2015.

J. Sprekels, Optimal boundary control problems for CahnHilliard 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, Rateindependent damage models with spatial BVregularization  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 AllenCahn 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, Multimaterial 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, Wellposedness 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 twophase model, MiniWorkshop ``Scales in Plasticity'', November 8  14, 2015, Mathematisches Forschungsinstitut Oberwolfach, November 11, 2015.

J. Sprekels, Optimal boundary control problems for CahnHilliard 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 CahnHilliard 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 CahnHilliard systems coupled with complete damage processes, DIMO2013  Diffuse Interface Models, Levico Terme, Italy, September 10  13, 2013.

CH. Heinemann, Degenerating CahnHilliard 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 NavierStokesKorteweg 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 reactiondiffusion 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 AllenCahn 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 AllenCahn 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, PrandtlIshlinskii operators and elastoplasticity, Spring School on ``Rateindependent 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 ratedependent complete damage processes, Materialmodellierungsseminar, WIAS, Berlin, October 31, 2012.

CH. Heinemann, Kopplung von Phasenseparation und Schädigung in elastischen Materialien, LeibnizDoktorandenForum 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, CahnLarché 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 CahnHilliard 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, HumboldtUniversität zu Berlin, Institut für Mathematik, November 21, 2011.

CH. Heinemann, Existence results for CahnHilliard 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 nonstandard phasefield system of CahnHilliard 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 CahnHilliard 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, Wellposedness, 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 UMIDMV 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 phasefield 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 phasefield model with hysteresis in thermoviscoplasticity, INdAM Workshop ``Dissipative Models in Phase Transitions'', September 5  11, 2004, Cortona, Italy, September 9, 2004.

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

J. Sprekels, On nonlocal phasefield 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 (HMM2003), May 28  30, 2003, Universidad de Salamanca, Departamento de Física Aplicada, Spain, May 30, 2003.

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

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

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

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

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

J. Sprekels, Phasefield 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, Phasefield 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. PodioGuidugli, J. Sprekels, A temperaturedependent phase segregation problem of the AllenCahn 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 CahnHilliard equations, Preprint no. 0902, HumboldtUniversität zu Berlin, Institut für Mathematik, 2009.

P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, Global solution and longtime behavior for a problem of phase segregation of the AllenCahn 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 AllenCahn equation with a memory term. Global existence and uniqueness of a smooth solution are proven and a characterization of the omegalimit set is given. 
F. Klopp, B. Metzger, The GrossPitaevskii 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 GrossPitaevskii energy functionals with a random background potential, we study the GrossPitaevskii ground state. We characterize a regime of interaction coupling when the GrossPitaevskii 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.
Applications
 Modeling of phase separation and damage in modern materials
 Modeling of thin films and nano structures on substrates
 Nonlinear material models, multifunctional materials and hysteresis in connection with elastoplastic processes
 Phase transition and hysteresis in the context of storage problems
 Photovoltaics