arXiv subjects

Markus Schmidtchen

Publications and source records attributed to Markus Schmidtchen.

26 records · Page 2Linked to original sources

arXiv2020

The Aronson-Bénilan Estimate in Lebesgue Spaces

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 its publication, the theory of porous medium flow has expanded relentlessly with applications including thermodynamics, gas flow, ground water flow as well as ecological population dynamics. The purpose of this paper is to clarify the use of recent extensions of the Aronson and Bénilan estimate in Lp spaces, of some modifications and improvements, as well as to show certain limitations of their strategy.

math.AP↗

arXiv2020

Impacts of California Proposition 47 on Crime in Santa Monica, CA

We examine crime patterns in Santa Monica, California before and after passage of Proposition 47, a 2014 initiative that reclassified some non-violent felonies to misdemeanors. We also study how the 2016 opening of four new light rail stations, and how more community-based policing starting in late 2018, impacted crime. A series of statistical analyses are performed on reclassified (larceny, fraud, possession of narcotics, forgery, receiving/possessing stolen property) and non-reclassified crimes by probing publicly available databases from 2006 to 2019. We compare data before and after passage of Proposition 47, city-wide and within eight neighborhoods. Similar analyses are conducted within a 450 meter radius of the new transit stations. Reports of monthly reclassified crimes increased city-wide by approximately 15% after enactment of Proposition 47, with a significant drop observed in late 2018. Downtown exhibited the largest overall surge. The reported incidence of larceny intensified throughout the city. Two new train stations, including Downtown, reported significant crime increases in their vicinity after service began. While the number of reported reclassified crimes increased after passage of Proposition 47, those not affected by the new law decreased or stayed constant, suggesting that Proposition 47 strongly impacted crime in Santa Monica. Reported crimes decreased in late 2018 concurrent with the adoption of new policing measures that enhanced outreach and patrolling. These findings may be relevant to law enforcement and policy-makers. Follow-up studies needed to confirm long-term trends may be affected by the COVID-19 pandemic that drastically changed societal conditions.

stat.AP↗

arXiv2020

Convergence of a Finite Volume Scheme for a System of Interacting Species with Cross-Diffusion

In this work we present the convergence of a positivity preserving semi-discrete finite volume scheme for a coupled system of two non-local partial differential equations with cross-diffusion. The key to proving the convergence result is to establish positivity in order to obtain a discrete energy estimate to obtain compactness. We numerically observe the convergence to reference solutions with a first order accuracy in space. Moreover we recover segregated stationary states in spite of the regularising effect of the self-diffusion. However, if the self-diffusion or the cross-diffusion is strong enough, mixing occurs while both densities remain continuous.

math.NA↗

arXiv2020

Trend to Equilibrium for Systems with Small Cross-Diffusion

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 dynamics of a variety of large systems of interacting particles. Under suitable assumptions, we prove existence of classical solutions and we show exponential convergence in time to the stationary state. Furthermore, we consider the special case of one mobile and one immobile species, for which the system reduces to a nonlinear equation of Fokker-Planck type. In this framework, we improve the convergence result obtained for the general system and we derive sharper $L^{\infty}$-bounds for the solutions in two spatial dimensions. We conclude by illustrating the behaviour of solutions with numerical experiments in one and two spatial dimensions.

math.AP↗

arXiv2019

Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension

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 the context of visco-elastic media, where the velocity field is linked to the total population pressure via an elliptic equation. The same model for only one species has been studied by Perthame and Vauchelet in the past. The first part of this paper is dedicated to establishing existence of solutions to the problem, while the second part deals with the incompressible limit as the stiffness of the pressure law tends to infinity. Here we present a novel approach in one spatial dimension that differs from the kinetic reformulation used in the aforementioned study and, instead, relies on uniform BV-estimates.

math.AP↗

arXiv2019

Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues

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 and hyperbolic type which extend the standard compressible porous medium equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary problem. We establish the complementarity relation and also a segregation result. Several major mathematical difficulties arise in the two species case. Firstly, the system structure makes comparison principles fail. Secondly, segregation and internal layers limit the regularity available on some quantities to BV. Thirdly, the Aronson-B{é}nilan estimates cannot be established in our context. We are lead, as it is classical, to add correction terms. This procedure requires technical manipulations based on BV estimates only valid in one space dimension. Another novelty is to establish an L1 version in place of the standard upper bound.

math.AP↗

arXiv2017

Splitting Schemes & Segregation In Reaction-(Cross-)Diffusion Systems

One of the most fascinating phenomena observed in reaction-diffusion systems is the emergence of segregated solutions, i.e. population densities with disjoint supports. We analyse such a reaction cross-diffusion system. In order to prove existence of weak solutions for a wide class of initial data without restriction about their supports or their positivity, we propose a variational splitting scheme combining ODEs with methods from optimal transport. In addition, this approach allows us to prove conservation of segregation for initially segregated data even in the presence of vacuum.

math.AP↗

arXiv2016

A multiscale approach for spatially inhomogeneous disease dynamics

In this paper we introduce an agent-based epidemiological model that generalizes the classical SIR model by Kermack and McKendrick. We further provide a multiscale approach to the derivation of a macroscopic counterpart via the mean-field limit. The chain of equations acquired via the multiscale approach are investigated, analytically as well as numerically. The outcome of these results provide strong evidence of the models' robustness and justifies their applicability in describing disease dynamics, in particularly when mobility is involved.

math.DS↗