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- ItemOptimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in 3D(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Rocca, Elisabetta; Sprekels, JürgenIn this paper we study a distributed optimal control problem for a nonlocal convective Cahn-Hilliard equation with degenerate mobility and singular potential in three dimensions of space. While the cost functional is of standard tracking type, the control problem under investigation cannot easily be treated via standard techniques for two reasons: The state system is a highly nonlinear system of PDEs containing singular and degenerating terms, and the control variable, which is given by the velocity of the motion occurring in the convective term, is nonlinearly coupled to the state variable. The latter fact makes it necessary to state rather special regularity assumptions for the admissible controls, which, while looking a bit nonstandard, are however quite natural in the corresponding analytical framework. In fact, they are indispensable prerequisites to guarantee the well-posedness of the associated state system. In this contribution, we employ recently proved existence, uniqueness and regularity results for the solution to the associated state system in order to establish the existence of optimal controls and appropriate first-order necessary optimality conditions for the optimal control problem.
- ItemAsymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Colli, Pierluigi; Gilardi, Gianni; Rocca, Elisabetta; Sprekels, JürgenThis paper is concerned with a phase field system of Cahn-Hilliard type that is related to a tumor growth model and consists of three equations in gianni terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent papers citeCGH and citeCGRS gianni from the viewpoint of well-posedness, long time bhv and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in citeCGRS by showing two independent sets of results as just one of the coefficents tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates
- ItemOptimal distributed control of a diffuse interface model of tumor growth(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Colli, Pierluigi; Gilardi, Gianni; Rocca, Elisabetta; Sprekels, JürgenIn 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 [25]. The model consists of a CahnHilliard equation for the tumor cell fraction 'coupled to a reaction-diffusion equation for a function phi representing the nutrientrich extracellular water volume fraction. The distributed control u monitors as a right-hand side the equation for sigma and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.
- ItemAnalysis of a tumor model as a multicomponent deformable porous medium(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Krejčí, Pavel; Rocca, Elisabetta; Sprekels, JürgenWe propose a diffuse interface model to describe tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phase-dependent elasticity coefficients. The resulting PDE system couples two Cahn--Hilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reaction-diffusion type equation for the nutrient proportion, and a quasistatic momentum balance. We prove here that the corresponding initial-boundary value problem has a solution in appropriate function spaces.
- ItemA nonlocal phase-field model with nonconstant specific heat(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Krejčí, Pavel; Rocca, Elisabetta; Sprekels, JürgenWe prove the existence, uniqueness, thermodynamic consistency, global boundedness from both above and below, and continuous data dependence for a strong solution to an integrodifferential model for nonisothermal phase transitions under nonhomogeneous mixed boundary conditions. The specific heat is allowed to depend on the order parameter, and the convex component of the free energy may or may not be singular.
- ItemVanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Colli, Pierluigi; Gilardi, Gianni; Rocca, Elisabetta; Sprekels, JürgenIn this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn--Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in [Colli-Gilardi-Hilhorst 2015], letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.
- ItemUnsaturated deformable porous media flow with phase transition(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Krejčí, Pavel; Rocca, Elisabetta; Sprekels, JürgenIn 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 liquid-solid phase transformations in groundwater flows. The system of equations is derived here from the conservation principles for mass, momentum, and energy and from the Clausius-Duhem 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 cut-off techniques and suitable Moser-type estimates.
- ItemWell-posedness and optimal control for a Cahn--Hilliard--Oono system with control in the mass term(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Colli, Pierluigi; Gilardi, Gianni; Rocca, Elisabetta; Sprekels, JürgenThe paper treats the problem of optimal distributed control of a Cahn--Hilliard--Oono system in Rd, 1 ≤ d ≤ 3 with the control located in the mass term and admitting general potentials that include both the case of a regular potential and the case of some singular potential. The first part of the paper is concerned with the dependence of the phase variable on the control variable. For this purpose, suitable regularity and continuous dependence results are shown. In particular, in the case of a logarithmic potential, we need to prove an ad hoc strict separation property, and for this reason we have to restrict ourselves to the case d = 2. In the rest of the work, we study the necessary first-order optimality conditions, which are proved under suitable compatibility conditions on the initial datum of the phase variable and the time derivative of the control, at least in case of potentials having unbounded domain
- ItemA nonlocal quasilinear multi-phase system with nonconstant specific heat and heat conductivity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Colli, Pierluigi; Krejˇcí, Pavel; Rocca, Elisabetta; Sprekels, JürgenIn this paper, we prove the existence and global boundedness from above for a solution to an integrodifferential model for nonisothermal multi-phase transitions under nonhomogeneous third type boundary conditions. The system couples a quasilinear internal energy balance ruling the evolution of the absolute temperature with a vectorial integro-differential inclusion governing the (vectorial) phase-parameter dynamics. The specific heat and the heat conductivity $k$ are allowed to depend both on the order parameter $chi$ and on the absolute temperature $theta$ of the system, and the convex component of the free energy may or may not be singular. Uniqueness and continuous data dependence are also proved under additional assumptions.
- ItemOptimal distributed control of a nonlocal Cahn-Hilliard/Navier-Stokes system in 2D(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Frigeri, Sergio Pietro; Rocca, Elisabetta; Sprekels, JürgenWe study a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids coupling the Navier-Stokes system with a convective nonlocal Cahn-Hilliard equation in two dimensions of space. We apply recently proved well-posedness and regularity results in order to establish existence of optimal controls as well as first-order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid flow.