A Hele-Shaw Limit Without Monotonicity

Guillen, Nestor; Kim, In-Won; Mellet, Antoine

doi:10.1007/s00205-021-01750-4

Cited by 17 publications

(15 citation statements)

References 21 publications

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“…(b) We show that for all t > 0 the pressure p(t) can be defined as the unique solution of a simple obstacle problem. This result is similar to a result obtained in [12] in the context of tumour growth (without the potential ϕ). The derivation of this obstacle problem relies on the variational nature of the JKO scheme and is thus very different from the proof presented in [12], which involved the porous media type equation (1.1).…”

Section: Motivations and Overview Of The Paper's Objectivessupporting

confidence: 90%

“…This result is similar to a result obtained in [12] in the context of tumour growth (without the potential ϕ). The derivation of this obstacle problem relies on the variational nature of the JKO scheme and is thus very different from the proof presented in [12], which involved the porous media type equation (1.1). But it does not require any technical a priori estimates and it yields the complementarity condition:…”

Section: Motivations and Overview Of The Paper's Objectivessupporting

confidence: 90%

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A density-constrained model for chemotaxis

Kim

Mellet

2023

Nonlinearity

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We consider a model of congestion dynamics with chemotaxis: the density of cells follows a chemical signal it generates, while subject to an incompressibility constraint. The incompressibility constraint results in the formation of patches, describing regions where the maximal density has been reached. The dynamics of these patches can be described by either Hele-Shaw or Richards equation type flow (depending on whether we consider the model with diffusion or the model with pure advection). Our focus in this paper is on the construction of weak solutions for this problem via a variational discrete time scheme of JKO type. We also establish the uniqueness of these solutions. In addition, we make more rigorous the connection between this incompressible chemotaxis model and the free boundary problems describing the motion of the patches in terms of the density and associated pressure variable. In particular, we obtain new results characterising the pressure variable as the solution of an obstacle problem and prove that in the pure advection case the dynamic preserves patches.

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Section: Motivations and Overview Of The Paper's Objectivessupporting

confidence: 90%

Section: Motivations and Overview Of The Paper's Objectivessupporting

confidence: 90%

A density-constrained model for chemotaxis

Kim

Mellet

2023

Nonlinearity

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“…Moreover, our approach provides an answer to several open problems proposed in [40]: The first question the authors raise concerns the monotonicity assumption on

G(p) + \Delta \Phi &gt; 0

, which in our case is not necessary. An improvement in this direction has also been obtained very recently, [34]. We stress that the growth rate in [40] does not depend on the pressure but on space and time, only. The next question concerns the class of initial data.…”

Section: Introductionmentioning

confidence: 89%

“…Besides, explicit solutions to the limit problem are presented in [44] for initial data of the form of an indicator of a bounded set. Recently, interesting progress have been made in [34] where the authors are able to establish the incompressible limit and the complementarity relation without relying on any Aronson‐Bénilan‐type estimates. Instead, their approach is based on viscosity solutions and establishing the equivalence between the complementarity relation and an obstacle problem.…”

Section: Introductionmentioning

confidence: 99%

On the incompressible limit for a tumour growth model incorporating convective effects

David,

Schmidtchen

2023

Comm Pure Appl Math

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In this work we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quirós, and Vázquez proposed in 2014 but incorporates the advective effects caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self‐propulsion. The main result of this work is the incompressible limit of this model which builds a bridge between the density‐based model and a geometry free‐boundary problem by passing to a singular limit in the pressure law. The limiting objects are then proven to be unique.

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“…Many research (e.g. [10,12,20,27,28,39]) indicate that the porous medium type functions have a Hele-Shaw type asymptote as the power m tends to infinity. In particular, the solution of (P m ) tends to the solution of (see Theorem 3.3 for precise description):…”

Section: 1mentioning

confidence: 99%

A unified Bayesian inversion approach for a class of tumor growth models with different pressure laws

Liu

Zhou

2023

Preprint

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In this paper, we use the Bayesian inversion approach to study the data assimilation problem for a family of tumor growth models described by porous-medium type equations. The models contain uncertain parameters and are indexed by a physical parameter $m$, which characterizes the constitutive relation between density and pressure. Based on these models, we employ the Bayesian inversion framework to infer parametric and nonparametric unknowns that affect tumor growth from noisy observations of tumor cell density. We establish the well-posedness and the stability theories for the Bayesian inversion problem and further prove the convergence of the posterior distribution in the so-called incompressible limit, $m \rightarrow \infty$. Since the posterior distribution across the index regime $m\in[2,\infty)$ can thus be treated in a unified manner, such theoretical results also guide the design of the numerical inference for the unknown. We propose a generic computational framework for such inverse problems, which consists of a typical sampling algorithm and an asymptotic preserving solver for the forward problem. With extensive numerical tests, we demonstrate that the proposed method achieves satisfactory accuracy in the Bayesian inference of the tumor growth models, which is uniform with respect to the constitutive relation.

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A Hele-Shaw Limit Without Monotonicity

Cited by 17 publications

References 21 publications

A density-constrained model for chemotaxis

A density-constrained model for chemotaxis

On the incompressible limit for a tumour growth model incorporating convective effects

A unified Bayesian inversion approach for a class of tumor growth models with different pressure laws

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