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Author: Gilardi, Gianni × Author: Sprekels, Jürgen × Type: Text × Subject: regularity of solutions ×

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Now showing 1 - 5 of 5
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    Regularity of the solution to a nonstandard system of phase field equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    A nonstandard systems of differential equations describing two-species phase segregation is considered. This system naturally arises in the asymptotic analysis recently done by Colli, Gilardi, Krej¡cí, and Sprekels as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. In particular, a well-posedness result is proved for the limit system. This paper deals with the above limit problem in a less general but still very significant framework and provides a very simple proof of further regularity for the solution. As a byproduct, a simple uniqueness proof is given as well.
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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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    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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    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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    On a Cahn-Hilliard system with convection and dynamic boundary conditions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    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.