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Author: Gilardi, Gianni × Has files: Yes × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: Cahn--Hilliard systems ×

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Now showing 1 - 5 of 5
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    Well-posedness and regularity for a fractional tumor growth model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalization of a phase field system of Cahn--Hilliard type modelling tumor growth that has been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3--24) and investigated in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn--Hilliard equation for the tumor cell fraction φ, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. The generalization investigated in this paper is motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type. Under rather general assumptions, well-posedness and regularity results are shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth constributions of logarithmic or of double obstacle type to the energy density can be admitted.
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    An asymptotic analysis for a generalized Cahn--Hilliard system with fractional operators
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In a recent paper the same authors have proved existence, uniqueness and regularity results for a class of 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 in the spectral sense of general linear operators, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space of square-integrable functions on a bounded and smooth three-dimensional domain, and have compact resolvents. Here, for the case of the viscous system, we analyze the asymptotic behavior of the solution as the fractional power coefficient of the second operator 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 the second operator appears.
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    A distributed control problem for a fractional tumor growth model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three selfadjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn--Hilliard type phase field system modeling tumor growth that goes back to Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3--24.) The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in the recent work Adv. Math. Sci. Appl. 28 (2019), 343--375 by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type.
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    Asymptotic analysis of a tumor growth model with fractional operators
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn--Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3--24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn--Hilliard equation for the tumor cell fraction φ, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases.
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    Longtime behavior for a generalized Cahn--Hilliard system with fractional operators
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the Cahn--Hilliard system, with possibly singular potentials, which we recently investigated in the paper "Well-posedness and regularity for a generalized fractional CahnHilliard system". More precisely, we give a complete characterization of the Omega-limit 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 Omega-limit is a stationary solution to the phase equation; if it is zero instead, then every element of the Omega-limit 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.