Cahn–Hilliard–Brinkman systems for tumour growth

Ebenbeck, Matthias; Garcke, Harald; Nürnberg, Robert

doi:10.3934/dcdss.2021034

Cited by 19 publications

(29 citation statements)

References 62 publications

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“…Note that this is the standard inductionless MHD equations in each subdomain. We now analyse (17) according to the mobilities introduced in (22). For the cases of M (ϕ) = m 0 and M (ϕ) = m 0 1 − ϕ 2 + , the chemical potential does not contribute to the equations at zeroth order.…”

Section: Outer Expansionmentioning

confidence: 99%

“…In matched asymptotic techniques, inner and outer quantities are linked together by matching conditions. For this end, we employ the matching conditions [25,24,17]:…”

Section: Inner Expansionmentioning

confidence: 99%

“…This is a solvability condition for φ1 , the so-called Gibbs-Thomas equation [4,17]. Now we analyse (16) only for the mobilities (22) for the Case I.…”

Section: Inner Expansion To Higher Ordermentioning

confidence: 99%

“…As a result, after the phase field model is obtained, one of important and natural issue is to investigate whether the diffuse interface model can be related to the corresponding sharp interface model when the interfacial width tends to zero. The sharp interface limits of some different diffusion interface models can be found in [4,20,21,31,16,24,17,5] for instance. Nevertheless, up to our knowledge, there seems to exist no corresponding rigorous results on sharp interface limits of diffuse interface methhod for two-phase incompressible inductionless MHD equations.…”

Section: Introductionmentioning

confidence: 99%

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Sharp-interface limits of the diffuse interface model for two-phase inductionless magnetohydrodynamic fluids

Zhang

2021

Preprint

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In this paper, we propose and analyze a diffuse interface model for inductionless magnetohydrodynamic fluids. The model couples a convective Cahn-Hilliard equation for the evolution of the interface, the Navier-Stokes system for fluid flow and the possion equation for electrostatics. The model is derived from Onsager's variational principle and conservation laws systematically. We perform formally matched asymptotic expansions and develop several sharp interface models in the limit when the interfacial thickness tends to zero. It is shown that the sharp interface limit of the models are the standard incompressible inductionless magnetohydrodynamic equations coupled with several different interface conditions for different choice of the mobilities. Numerical results verify the convergence of the diffuse interface model with different mobilitiess.

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Section: Outer Expansionmentioning

confidence: 99%

“…In matched asymptotic techniques, inner and outer quantities are linked together by matching conditions. For this end, we employ the matching conditions [25,24,17]:…”

Section: Inner Expansionmentioning

confidence: 99%

“…This is a solvability condition for φ1 , the so-called Gibbs-Thomas equation [4,17]. Now we analyse (16) only for the mobilities (22) for the Case I.…”

Section: Inner Expansion To Higher Ordermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Sharp-interface limits of the diffuse interface model for two-phase inductionless magnetohydrodynamic fluids

Zhang

2021

Preprint

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“…A two-component Cahn-Hilliard-Brinkman model for tumor growth (including a reactiondiffusion type equation to describe the nutrient density) was proposed and analyzed in [28]. A simplified variant of this model was studied in [29][30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

Existence of weak solutions to multiphase Cahn-Hilliard-Darcy and Cahn-Hilliard-Brinkman models for stratified tumor growth with chemotaxis and general source terms

Knopf,

Signori

2021

Preprint

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We investigate a multiphase Cahn-Hilliard model for tumor growth with general source terms. The multiphase approach allows us to consider multiple cell types and multiple chemical species (oxygen and/or nutrients) that are consumed by the tumor. Compared to classical two-phase tumor growth models, the multiphase model can be used to describe a stratified tumor exhibiting several layers of tissue (e.g., proliferating, quiescent and necrotic tissue) more precisely. Our model consists of a convective Cahn-Hilliard type equation to describe the tumor evolution, a velocity equation for the associated volume-averaged velocity field, and a convective reaction-diffusion type equation to describe the density of the chemical species. The velocity equation is either represented by Darcy's law or by the Brinkman equation. We first construct a global weak solution of the multiphase Cahn-Hilliard-Brinkman model. After that, we show that such weak solutions of the system converge to a weak solution of the multiphase Cahn-Hilliard-Darcy system as the viscosities tend to zero in some suitable sense. This means that the existence of a global weak solution to the Cahn-Hilliard-Darcy system is also established.

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From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

Elbar

Mason

Perthame

et al. 2023

Commun. Math. Phys.

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We provide a rigorous mathematical framework to establish the hydrodynamic limit of the Vlasov model introduced in Takata and Noguchi (J. Stat. Phys. 172:880-903, 2018) by Noguchi and Takata in order to describe phase transition of fluids by kinetic equations. We prove that, when the scale parameter tends to 0, this model converges to a nonlocal Cahn-Hilliard equation with degenerate mobility. For our analysis, we introduce apropriate forms of the short and long range potentials which allow us to derive Helmhotlz free energy estimates. Several compactness properties follow from the energy, the energy dissipation and kinetic averaging lemmas. In particular we prove a new weak compactness bound on the flux.

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Cahn–Hilliard–Brinkman systems for tumour growth

Cited by 19 publications

References 62 publications

Sharp-interface limits of the diffuse interface model for two-phase inductionless magnetohydrodynamic fluids

Sharp-interface limits of the diffuse interface model for two-phase inductionless magnetohydrodynamic fluids

Existence of weak solutions to multiphase Cahn-Hilliard-Darcy and Cahn-Hilliard-Brinkman models for stratified tumor growth with chemotaxis and general source terms

From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

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