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We investigate the Keller--Segel--(Navier--)Stokes system posed in a smooth bounded domain \(\Omega \subset \mathbb{R}^N\) with \(N = 2,3\): \begin{equation*} \begin{cases} n_t + u \cdot \nabla n = \Delta n - \nabla \cdot \big( n S(n)\nabla…

Analysis of PDEs · Mathematics 2026-01-21 Minh Le

Chemotaxis describes the intricate interplay of cellular motion in response to a chemical signal. We here consider the case of slab geometry which models chemotactic motion between two infinite membranes. Like previous works, we are…

Analysis of PDEs · Mathematics 2026-01-16 Herbert Egger , Kathrin Hellmuth , Nora Philippi , Matthias Schlottbom

This manuscript deals with the three-dimensional version of a flux-limited Keller-Segel system coupled to the incompressible Stokes equations through transport and buoyancy. The main goal consists in verifying that within a certain…

Analysis of PDEs · Mathematics 2020-09-16 Michael Winkler

We consider a simplified chemotaxis model of tumor angiogenesis, described by a Keller-Segel system on the two dimensional infinite cylindrical domain $(x, y) \in \mathbb{R} \times {\mathbf S^{\lambda}}$, where $ \mathbf S^{\lambda}$ is the…

Analysis of PDEs · Mathematics 2019-03-12 Myeongju Chae , Kyudong Choi

This paper is concerned with the diffusion limit (as $\va\rightarrow 0$) of radial solutions to a chemotaxis system with logarithmic singular sensitivity in a bounded interval with mixed Dirichlet and Robin boundary conditions. We use a…

Analysis of PDEs · Mathematics 2020-10-12 Qianqian Hou

A novel trait-structured Keller-Segel model that explores the dynamics of a migrating cell population guided by chemotaxis in response to an external ligand concentration is derived and analysed. Unlike traditional Keller-Segel models, this…

Cell Behavior · Quantitative Biology 2025-02-27 Viktoria Freingruber , Tommaso Lorenzi , Kevin J. Painter , Mariya Ptashnyk

The Keller-Segel model is a system of partial differential equations modelling chemotactic aggregation in cellular systems. This model has blowing up solutions for large enough initial conditions in dimensions d >= 2, but all the solutions…

Analysis of PDEs · Mathematics 2009-11-11 Carlos Escudero

We consider a generalised Keller-Segel model with non-linear porous medium type diffusion and non-local attractive power law interaction, focusing on potentials that are more singular than Newtonian interaction. We show uniqueness of…

Analysis of PDEs · Mathematics 2020-06-16 Vincent Calvez , Jose Antonio Carrillo , Franca Hoffmann

In bounded $n$-dimensional domains with $n\ge 3$, this manuscript considers an initial-boundary problem for a quasilinear chemotaxis system with indirect attractant production, as arising, inter alia, in the modeling of effects due to…

Analysis of PDEs · Mathematics 2024-11-12 Youshan Tao , Michael Winkler

We present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased random walk. In order to take into account…

Analysis of PDEs · Mathematics 2020-03-27 Federica Bubba , Tommaso Lorenzi , Fiona R Macfarlane

We are concerned with the hyperbolic Keller-Segel model with quorum sensing, a model describing the collective cell movement due to chemical signalling with a flux limitation for high cell densities. This is a first order quasilinear…

Analysis of PDEs · Mathematics 2007-05-23 Benoit Perthame , Anne-Laure Dalibard

Chemotaxis phenomena govern the directed movement of micro-organisms in response to chemical stimuli. In this paper, we investigate two Keller--Segel systems of reaction-advection-diffusion equations modeling chemotaxis on thin networks.…

Analysis of PDEs · Mathematics 2024-04-02 Hewan Shemtaga , Wenxian Shen , Selim Sukhtaiev

Consider the coupled Keller-Segel-Navier-Stokes or the chemotaxis-consumption-Navier-Stokes system in bounded Lipschitz domains for general coupling terms which, e.g., include buoyancy forces. It is shown that these systems admit local…

Analysis of PDEs · Mathematics 2025-05-08 Matthias Hieber , Hideo Kozono , Sylvie Monniaux , Patrick Tolksdorf

In this paper, the well-posedness of two-dimensional signal-dependent Keller-Segel system and its mean-field derivation from a interacting particle system on the whole space are investigated. The signal dependence effect is reflected by the…

Analysis of PDEs · Mathematics 2025-06-05 Lukas Bol , Li Chen , Yue Li

Chemotaxis systems of Keller--Segel type constitute one of the central mathematical frameworks for understanding aggregation phenomena in biological and ecological systems. Over the past decades, the theory has evolved from the classical…

Analysis of PDEs · Mathematics 2026-03-06 Kolade M Owolabi , Eben Mare , Clara O Ijalana , Kolawole S Adegbie

Keller-Segel systems in two and three space dimensions with an additional cross-diffusion term in the equation for the chemical concentration are analyzed. The cross-diffusion term has a stabilizing effect and leads to the global-in-time…

Analysis of PDEs · Mathematics 2019-07-29 Ansgar Jüngel , Oliver Leingang , Shu Wang

We consider the singular limit of a chemotaxis model of bacterial collective motion recently introduced in arXiv:2009.11048 [math.AP]. The equation models aggregation-diffusion phenomena with advection that is discontinuous and depends…

Analysis of PDEs · Mathematics 2025-08-28 Maria Gualdani , Mikel Ispizua , Nicola Zamponi

The coupled quasilinear Keller-Segel-Navier-Stokes system is considered under Neumann boundary conditions for $n$ and $c$ and no-slip boundary conditions for $u$ in three-dimensional bounded domains $\Omega\subseteq \mathbb{R}^3$ with…

Analysis of PDEs · Mathematics 2017-04-11 Jiashan Zheng

This paper is devoted to the analysis of the classical Keller-Segel system over $\mathbb{R}^d$, $d\geq 3$. We describe as much as possible the dynamics of the system characterized by various criteria, both in the parabolic-elliptic case and…

Analysis of PDEs · Mathematics 2010-03-23 Vincent Calvez , Lucilla Corrias , Mohammed Abderrahman Ebde

We consider a Keller-Segel model with non-linear porous medium type diffusion and non-local attractive power law interaction, focusing on potentials that are less singular than Newtonian interaction. Here, the nonlinear diffusion is chosen…

Analysis of PDEs · Mathematics 2024-12-18 Shen Bian , Yichen Zou