Analysis of a tumor model as a multicomponent deformable porous medium

Krejčı́, Pavel; Rocca, Elisabetta; Sprekels, Juergen

doi:10.4171/ifb/472

Cited by 5 publications

(6 citation statements)

References 47 publications

(46 reference statements)

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“…The most important property is the positive definiteness of D BD (c) on L(c); see (27). This property implies the a priori estimates proved in the following subsection.…”

Section: Properties Of the Mobility Matrix And A Priori Estimatesmentioning

confidence: 55%

“…We derive first the energy inequality. To this end, we multiply equation ( 25) for c i by µ i = (∂E/∂c i )(c), integrate over Ω, integrate by parts (using the boundary conditions ( 26)), and take into account the lower bound (27) for D BD (c):…”

Section: 2mentioning

confidence: 99%

“…Only few works are concerned with the multi-species situation, and all of them require additional conditions. The mobility matrix in [5,28] is assumed to be diagonal and that one in [27] has constant entries, while the works [11,13] suppose a particular (but nondiagonal) structure of the mobility matrix. We also mention the works [2,3] on related models with free energies of the type H.…”

Section: Introductionmentioning

confidence: 99%

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Existence and weak-strong uniqueness for Maxwell-Stefan-Cahn-Hilliard systems

Huo¹,

Jüngel²,

Tzavaras³

2022

Preprint

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A Maxwell-Stefan system for fluid mixtures with driving forces depending on Cahn-Hilliard-type chemical potentials is analyzed. The corresponding parabolic crossdiffusion equations contain fourth-order derivatives and are considered in a bounded domain with no-flux boundary conditions. The main difficulty of the analysis is the degeneracy of the diffusion matrix, which is overcome by proving the positive definiteness of the matrix on a subspace and using the Bott-Duffin matrix inverse. The global existence of weak solutions and a weak-strong uniqueness property are shown by a careful combination of (relative) energy and entropy estimates, yielding H 2 (Ω) bounds for the densities, which cannot be obtained from the energy or entropy inequalities alone.

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“…The most important property is the positive definiteness of D BD (c) on L(c); see (27). This property implies the a priori estimates proved in the following subsection.…”

Section: Properties Of the Mobility Matrix And A Priori Estimatesmentioning

confidence: 55%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Existence and weak-strong uniqueness for Maxwell-Stefan-Cahn-Hilliard systems

Huo¹,

Jüngel²,

Tzavaras³

2022

Preprint

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show abstract

“…The balance laws and the constitutive assumptions in [27] are formulated in a fixed reference configuration, assuming an infinitesimal elastic deformation and neglecting inertia effects. A similar approach is carried out in [7,20,21,25,29,33], where a Cahn-Hilliard equation coupled with infinitesimal elasticity is introduced and analyzed. In these latter studies, the models are derived from a multiphasic mixture.…”

Section: Introductionmentioning

confidence: 99%

A Cahn-Hilliard model coupled to viscoelasticity with large deformations

Agosti¹,

Colli²,

Garcke³

et al. 2022

Preprint

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We propose a new class of phase field models coupled to viscoelasticity with large deformations, obtained from a diffuse interface mixture model composed by a phase with elastic properties and a liquid phase. The model is formulated in the Eulerian configuration and it is derived by imposing the mass balance for the mixture components and the momentum balance that comes from a generalized form of the principle of virtual powers. The latter considers the presence of a system of microforces and microstresses associated to the microscopic interactions between the mixture's constituents together with a system of macroforces and macrostresses associated to their viscoelastic behavior, taking into account also the friction between the phases. The free energy density of the system is given as the sum of a Cahn-Hilliard term and an elastic polyconvex term, with a coupling between the phase field variable and the elastic deformation gradient in the elastic contribution. General constitutive assumptions complying with a mechanical version of the second law of thermodynamics in isothermal situations are taken. We study the global existence of a weak solution for a simplified and regularized version of the general model, which considers an incompressible elastic free energy of Neo-Hookean type with elastic coefficients depending on the phase field variable. The regularization is properly designed to deal with the coupling between the phase field variable and the elastic deformation gradient in the elastic energy density. The analysis is made both in two and three space dimensions.

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“…To cope with the case of singular potential, some authors (see [6]) suggested to include suitable relaxations. Besides, we refer to [19,20,28] and the references therein, for related nonlocal versions, to [1,17,24] for the additional coupling with elasticity, and to [27] for the coupling with the Keller-Segel equation.…”

Section: Introductionmentioning

confidence: 99%

Cahn-Hilliard-Brinkman model for tumor growth with possibly singular potentials

Colli¹,

Gilardi²,

Signori³

et al. 2022

Preprint

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We analyze a phase field model for tumor growth consisting of a Cahn-Hilliard-Brinkman system, ruling the evolution of the tumor mass, coupled with an advectionreaction-diffusion equation for a chemical species acting as a nutrient. The main novelty of the paper concerns the discussion of the existence of weak solutions to the system covering all the meaningful cases for the nonlinear potentials; in particular, the typical choices given by the regular, the logarithmic, and the double obstacle potentials are admitted in our treatise. Compared to previous results related to similar models, we suggest, instead of the classical no-flux condition, a Dirichlet boundary condition for the chemical potential appearing in the Cahn-Hilliard-type equation. Besides, abstract growth conditions for the source terms that may depend on the solution variables are postulated.

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Analysis of a tumor model as a multicomponent deformable porous medium

Cited by 5 publications

References 47 publications

Existence and weak-strong uniqueness for Maxwell-Stefan-Cahn-Hilliard systems

Existence and weak-strong uniqueness for Maxwell-Stefan-Cahn-Hilliard systems

A Cahn-Hilliard model coupled to viscoelasticity with large deformations

Cahn-Hilliard-Brinkman model for tumor growth with possibly singular potentials

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