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Author: Colli, Pierluigi × DDC: 510 × Subject: Cahn–Hilliard systems × Subject: Fractional operators ×

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Now showing 1 - 3 of 3
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    Deep quench approximation and optimal control of general Cahn-Hilliard systems with fractional operators and double obstacle potentials
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In the recent paper Well-posedness and regularity for a generalized fractional CahnHilliard system, the same authors derived general well-posedness 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 double-well 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/s00245-018-9540-7) 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 double-well potentials could be admitted. Results concerning existence of optimizers and first-order 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 control-to-state 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 so-called 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. Farshbaf-Shaker 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. 1-24) 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 first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint state system.
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    Well-posedness and regularity for a generalized fractional Cahn-Hilliard system
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In this paper, we investigate a rather general system of two operator equations that has the structure of a viscous or nonviscous Cahn–Hilliard system in which nonlinearities of double-well type occur. Standard cases like regular or logarithmic potentials, as well as non-differentiable 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, self-adjoint 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 second-order elliptic differential operators (e.g., the Laplacian) with zero Dirichlet or Neumann boundary conditions, but also other cases like fourth-order 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 non-fractional 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 1 of A plays an important und not entirely obvious role: if 1 is positive, then the operators A and B may be completely unrelated; if, however, 1 equals zero, then it must be simple and the corresponding one-dimensional 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.
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    An asymptotic analysis for a generalized Cahn–Hilliard system with fractional operators
    (Basel : Springer, 2021) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In the recent paper “Well-posedness and regularity for a generalized fractional Cahn–Hilliard system” (Colli et al. in Atti Accad Naz Lincei Rend Lincei Mat Appl 30:437–478, 2019), the same authors have studied viscous and nonviscous Cahn–Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers A2r and B2σ (in the spectral sense) of general linear operators A and B, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space L2(Ω), for some bounded and smooth domain Ω⊂R3, and have compact resolvents. Existence, uniqueness, and regularity results have been proved in the quoted paper. Here, in the case of the viscous system, we analyze the asymptotic behavior of the solution as the parameter σ appearing in the operator B2σ decreasingly tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of B appears.