Global existence theorem for a model governing the motion of two cell populations

Price, Brock C.; Xu, Xiangsheng

doi:10.3934/krm.2020042

Cited by 9 publications

(10 citation statements)

References 17 publications

(39 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Nevertheless, even when dealing with the same constant mobility coefficients, the nature of the multi-species system (at least for dimension greater than one) usually requires strong compactness on the pressure gradient. We refer the reader to [17,29] for existence results of the two-species model without membrane conditions.…”

Section: Discussionmentioning

confidence: 99%

Effective interface conditions for a porous medium type problem

Ciavolella,

David,

Poulain

2021

Preprint

View full text Add to dashboard Cite

Motivated by biological applications on tumour invasion through thin membranes, we study a porous-medium type equation where the density of the cell population evolves under Darcy's law, assuming continuity of both the density and flux velocity on the thin membrane which separates two domains. The drastically different scales and mobility rates between the membrane and the adjacent tissues lead to consider the limit as the thickness of the membrane approaches zero. We are interested in recovering the effective interface problem and the transmission conditions on the limiting zero-thickness surface, formally derived by Chaplain et al., [10], which are compatible with nonlinear generalized Kedem-Katchalsky ones. Our analysis relies on a priori estimates and compactness arguments as well as on the construction of a suitable extension operator which allows to deal with the degeneracy of the mobility rate in the membrane, as its thickness tends to zero.

show abstract

Section: Discussionmentioning

confidence: 99%

Effective interface conditions for a porous medium type problem

Ciavolella,

David,

Poulain

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Indeed, any regularity better than bounded variation cannot be expected, [8,19,31]. A different approach to prove the existence of solutions relies on the strong compactness of the pressure gradient, either by variations of the Aronson-Bénilan estimate [19,43,45] or convergence of the norm [34,44,49,57]. Historically, System (3) first appeared in [23] in the context of epidemiological modelling of polymorphic populations and in [42] with applications to population dynamics that avoid overcrowding.…”

Section: Cross-interaction Systemsmentioning

confidence: 99%

A Degenerate Cross-Diffusion System as the Inviscid Limit of a Nonlocal Tissue Growth Model

David¹,

Dębiec²,

Mandal³

et al. 2023

Preprint

View full text Add to dashboard Cite

In recent years, there has been a spike in the interest in multi-phase tissue growth models. Depending on the type of tissue, the velocity is linked to the pressure through Stoke's law, Brinkman's law or Darcy's law. While each of these velocitypressure relations has been studied in the literature, little emphasis has been placed on the fine relationship between them. In this paper, we want to address this dearth in the literature providing a rigorous argument that bridges the gap between a viscoelastic tumour model (of Brinkman type) and an inviscid tumour model (of Darcy type).

show abstract

“…A fundamental new ingredient is a uniform L 1 estimate on the time derivative of pressure. With this new estimate, another possible route to establish the complementarity condition, the hard part of the problem, would be to use the pure compactness method in [55,63]. Still another possible route is through the obstacle problem, see [38] and the references therein.…”

Section: Conclusion Extensions and Perspectivesmentioning

confidence: 99%

“…Liu and Xu investigated the existence and incompressible limit of a tissue growth model with autophagy cf. [55], and it should be pointed out that the authors in [55] obtained the complementarity relation by the approach of Price and Xu [63] without using the time derivative estimates of pressure, neither the Aronson-Bénilan estimates.…”

Section: Introductionmentioning

confidence: 99%

Incompressible limits of Patlak-Keller-Segel model and its stationary state

He¹,

Li²,

Perthame³

2022

Preprint

View full text Add to dashboard Cite

The incompressible limits of both the n-dimensional (n ≥ 3) Patlak-Keller-Segel (PKS) model and its stationary state are investigated. We derive a geometric free boundary problem of Hele-Shaw type for describing the limit density which relates the compressible (PKS) model and the incompressible (Hele-Shaw) model, and establish the complementarity relation which describes the limit pressure by a degenerate elliptic equation in the saturated zone. To this end, the novel components are to establish the uniform L 3 estimate of the pressure gradient, the uniform Aronson-Bénilan estimates in L 1 &L 2 &L 3 settings, and the uniform L 1 estimate for the time derivative of pressure. Furthermore, for the Hele-Shaw problem, we prove both the uniqueness of solutions, meaning that the incompressible limit of the PKS model is unique, the finite speed of propagation, and the limit energy functional. In addition, we establish the corresponding incompressible limit of the stationary state for the PKS model with a given mass, where, different from the case of PKS model, we obtain the uniform bound of pressure and the uniformly bounded support of density.

show abstract

Global existence theorem for a model governing the motion of two cell populations

Cited by 9 publications

References 17 publications

Effective interface conditions for a porous medium type problem

Effective interface conditions for a porous medium type problem

A Degenerate Cross-Diffusion System as the Inviscid Limit of a Nonlocal Tissue Growth Model

Incompressible limits of Patlak-Keller-Segel model and its stationary state

Contact Info

Product

Resources

About