Global existence for a singular phase field system related to a sliding mode control problem

In the present contribution we consider a singular phase field system located in a smooth and bounded three-dimensional domain. The entropy balance equation is perturbed by a logarithmic nonlinearity and by the presence of an additional term involving a possibly nonlocal maximal monotone operator and arising from a class of sliding mode control problems. The second equation of the system accounts for the phase dynamics, and it is deduced from a balance law for the microscopic forces that are responsible for the phase transition process. The resulting system is highly nonlinear; the main difficulties lie in the contemporary presence of two nonlinearities, one of which under time derivative, in the entropy balance equation. Consequently, we are able to prove only the existence of solutions. To this aim, we will introduce a backward finite differences scheme and argue on this by proving uniform estimates and passing to the limit on the time step.Comment: Key words: Phase field system; maximal monotone nonlinearities; nonlocal terms; initial and boundary value problem; existence of solution

Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition

In this paper, we investigate optimal control problems for Allen-Cahn equations with singular nonlinearities and a dynamic boundary condition involving singular nonlinearities and the Laplace-Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. Parabolic problems with nonlinear dynamic boundary conditions involving the Laplace-Beltrami operation have recently drawn increasing attention due to their importance in applications, while their optimal control was apparently never studied before. In this paper, we first extend known well-posedness and regularity results for the state equation and then show the existence of optimal controls and that the control-to-state mapping is twice continuously Fr\'echet differentiable between appropriate function spaces. Based on these results, we establish the first-order necessary optimality conditions in terms of a variational inequality and the adjoint state equation, and we prove second-order sufficient optimality conditions.Comment: Key words: optimal control; parabolic problems; dynamic boundary conditions; optimality condition

From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation

A rigorous proof is given for the convergence of the solutions of a viscous Cahn-Hilliard system to the solution of the regularized version of the forward-backward parabolic equation, as the coefficient of the diffusive term goes to 0. Non-homogenous Neumann boundary condition are handled for the chemical potential and the subdifferential of a possible non-smooth double-well functional is considered in the equation. An error estimate for the difference of solutions is also proved in a suitable norm and with a specified rate of convergence.Comment: Key words and phrases: Cahn-Hilliard system, forward-backward parabolic equation, viscosity, initial-boundary value problem, asymptotic analysis, well-posednes

Convergence properties for a generalization of the Caginalp phase field system

We are concerned with a phase field system consisting of two partial differential equations in terms of the variables thermal displacement, that is basically the time integration of temperature, and phase parameter. The system is a generalization of the well-known Caginalp model for phase transitions, when including a diffusive term for the thermal displacement in the balance equation and when dealing with an arbitrary maximal monotone graph, along with a smooth anti-monotone function, in the phase equation. A Cauchy-Neumann problem has been studied for such a system in arXiv:1107.3950v2 [math.AP], by proving well-posedness and regularity results, as well as convergence of the problem as the coefficient of the diffusive term for the thermal displacement tends to zero. The aim of this contribution is rather to investigate the asymptotic behaviour of the problem as the coefficient in front of the Laplacian of the temperature goes to 0: this analysis is motivated by the types III and II cases in the thermomechanical theory of Green and Naghdi. Under minimal assumptions on the data of the problems, we show a convergence result. Then, with the help of uniform regularity estimates, we discuss the rate of convergence for the difference of the solutions in suitable norms.Comment: Key words: phase field model, initial-boundary value problem, regularity of solutions, convergence, error estimate

Global existence for a phase separation system deduced from the entropy balance

This paper is concerned with a thermomechanical model describing phase separation phenomena in terms of the entropy balance and equilibrium equations for the microforces. The related system is highly nonlinear and admits singular potentials in the phase equation. Both the viscous and the non-viscous cases are considered in the Cahn--Hilliard relations characterizing the phase dynamics. The entropy balance is written in terms of the absolute temperature and of its logarithm, appearing under time derivative. The initial and boundary value problem is considered for the system of partial differential equations. The existence of a global solution is proved via some approximations involving Yosida regularizations and a suitable time discretization

Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials

The well-posedness of a system of partial differential equations and dynamic boundary conditions, both of Cahn-Hilliard type, is discussed. The existence of a weak solution and its continuous dependence on the data are proved using a suitable setting for the conservation of a total mass in the bulk plus the boundary. A very general class of double-well like potentials is allowed. Moreover, some further regularity is obtained to guarantee the strong solution

Optimal distributed control of a generalized fractional Cahn-Hilliard system

In the recent paper `Well-posedness and regularity for a generalized fractional Cahn-Hilliard system' (arXiv:1804.11290) by the same authors, general well-posedness results have been established for a a class of evolutionary systems of two equations having the structure of a viscous Cahn-Hilliard system, in which nonlinearities of double-well 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 arXiv:1804.11290 by studying a distributed control problem for this evolutionary system. The main difficulty in the analysis is to establish a rigorous Frechet differentiability result for the associated control-to-state 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 first-order necessary optimality conditions for the optimal control problem in terms of a variational inequality and the associated adjoint state system.Comment: Key words: fractional operators, Cahn-Hilliard systems, optimal control, necessary optimality condition