37 citations · 91 across the 23 of their papers we have counts for
21 papers · 1 filter
Incompressible limit for a two-species Brinkman model with drift
Tomasz Dębiec, Valentin Vincent Neumann, Markus Schmidtchen
We study a two-species model for tissue growth in which both populations are transported by an external drift and by a velocity potential determined through Brinkman's law. The pre…
Convergence of a discrete-in-time Approximation to a Degenerate Parabolic-Hyperbolic System
Julia Hauser, Hideki Murakawa, Markus Schmidtchen
In this paper we consider an implicit semi-discrete approximation of a degenerate reaction-cross-diffusion system. Due to the symmetry in the parabolic part, this system is known t…
Nonlocal approximation of an anisotropic cross-diffusion system
Tomasz Dębiec, Markus Schmidtchen
Localisation limits and nonlocal approximations of degenerate parabolic systems have experienced a renaissance in recent years. However, only few results cover anisotropic systems.…
From Finite to Continuous Phenotypes in (Visco-)Elastic Tissue Growth Models
Tomasz Dębiec, Mainak Mandal, Markus Schmidtchen
In this study, we explore a mathematical model for tissue growth focusing on the interplay between multiple cell subpopulations with distinct phenotypic characteristics. The model…
Existence of weak solutions for a volume-filling model of cell invasion into extracellular matrix
Rebecca M. Crossley, Jan-Frederik Pietschmann, Markus Schmidtchen
We study the existence of weak solutions for a model of cell invasion into the extracellular matrix (ECM), which consists of a non-linear partial differential equation for the dens…
Uniform regularity estimates for nonlinear diffusion-advection equations in the hard-congestion limit
Noemi David, Filippo Santambrogio, Markus Schmidtchen
We present regularity results for nonlinear drift-diffusion equations of porous medium type (together with their incompressible limit). We relax the assumptions imposed on the drif…
The approximation of the quadratic porous medium equation via nonlocal interacting particles subject to repulsive Morse potential
Marco Di Francesco, Valeria Iorio, Markus Schmidtchen
We propose a deterministic particle method for a one-dimensional nonlocal equation with interactions through the repulsive Morse potential. We show that the particle method converg…
Gradient Flow Solutions For Porous Medium Equations with Nonlocal Lévy-type Pressure
Guy Foghem, David Padilla-Garza, Markus Schmidtchen
We study a porous medium-type equation whose pressure is given by a nonlocal Lévy operator associated to a symmetric jump Lévy kernel. The class of nonlocal operators under conside…
A Degenerate Cross-Diffusion System as the Inviscid Limit of a Nonlocal Tissue Growth Model
Noemi David, Tomasz Dębiec, Mainak Mandal +1
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…
Derivation of a macroscopic model for Brownian hard needles
Maria Bruna, S. Jon Chapman, Markus Schmidtchen
We study the role of anisotropic steric interactions in a system of hard Brownian needles. Despite having no volume, non-overlapping needles exclude a volume in configuration space…
Nonlocal cross-interaction systems on graphs: Energy landscape and dynamics
Georg Heinze, Jan-Frederik Pietschmann, Markus Schmidtchen
We explore the dynamical behavior and energetic properties of a model of two species that interact nonlocally on finite graphs. The authors recently introduced the model in the con…
On a novel gradient flow structure for the aggregation equation
A. Esposito, R. S. Gvalani, A. Schlichting +1
The aggregation equation arises naturally in kinetic theory in the study of granular media, and its interpretation as a 2-Wasserstein gradient flow for the nonlocal interaction ene…
Nonlocal cross-interaction systems on graphs: Nonquadratic Finslerian structure and nonlinear mobilities
Georg Heinze, Jan-Frederik Pietschmann, Markus Schmidtchen
We study the evolution of a system of two species with nonlinear mobility and nonlocal interactions on a graph whose vertices are given by an arbitrary, positive measure. To this e…
On the Incompressible Limit for a Tumour Growth Model incorporating Convective Effects
Noemi David, Markus Schmidtchen
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 ad…
Many-particle limit for a system of interaction equations driven by Newtonian potentials
Marco Di Francesco, Antonio Esposito, Markus Schmidtchen
We consider a discrete particle system of two species coupled through nonlocal interactions driven by the one-dimensional Newtonian potential, with repulsive self-interaction and a…
The Aronson-Bénilan Estimate in Lebesgue Spaces
Giulia Bevilacqua, Benoît Perthame, Markus Schmidtchen
In a celebrated three-pages long paper in 1979, Aronson and Bénilan obtained a remarkable estimate on second order derivatives for the solution of the porous media equation. Since…
Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension
Tomasz Dębiec, Markus Schmidtchen
We present a two-species model with applications in tumour modelling. The main novelty is the coupling of both species through the so-called Brinkman law which is typically used in…
Segregation and Gap Formation in Cross-Diffusion Models
M. Burger, J. A. Carrillo, J. -F. Pietschmann +1
In this paper we analyse a class of nonlinear cross-diffusion systems for two species with local repulsive interactions that exhibit a formal gradient flow structure with respect t…
Trend to Equilibrium for Systems with Small Cross-Diffusion
Luca Alasio, Helene Ranetbauer, Markus Schmidtchen +1
This paper presents new analytical results for a class of nonlinear parabolic systems of partial different equations with small cross-diffusion which describe the macroscopic dynam…
Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues
Federica Bubba, Benoît Perthame, Camille Pouchol +1
Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic…
Zoology of a non-local cross-diffusion model for two species
J. A. Carrillo, Y. Huang, M. Schmidtchen
We study a non-local two species cross-interaction model with cross-diffusion. We propose a positivity preserving finite volume scheme based on the numerical method introduced in R…