Page 473 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 473
CA18232: VARIATIONAL METHODS AND EQUATIONS ON GRAPHS (MS-40)
Kinetic and macroscopic diffusion models for gas mixtures in the context
of respiration
Bérénice Grec, berenice.grec@u-paris.fr
University of Paris, France
In this talk, I will first discuss shortly the context of respiration, and in particular the need to
describe accurately the diffusion of respiratory gases in the lower part of the lung. At the macro-
scopic level, diffusion processes for mixtures are often modelled using cross-diffusion models.
In order both to determine the regime in which such models are valid, and to compute the binary
diffusion coefficients, it is of particular interest to derive these models from a description at the
mesoscopic level by means of kinetic equations.
More precisely, we consider the Boltzmann equations for mixtures with general cross-
sections (i.e. for any kind of molecules interactions), and obtain the so-called Maxwell-Stefan
equations by performing a Hilbert asymptotic expansion at low Knudsen and Mach numbers.
This allows us to compute the values of the Maxwell-Stefan diffusion coefficients with explicit
formulae with respect to the cross-sections. We also justify the specific ansatz we use thanks to
the so-called moment method.
This is a joint work with Laurent Boudin and Vincent Pavan.
Mathematical modeling of traffic flow
Helge Holden, helge.holden@ntnu.no
Norwegian University of Science and Technology, Norway
Traditionally, there are two types of mathematical models for vehicular traffic, namely the
Follow-the-Leader (FtL) models and the continuum models, using variants of the classical
Lighthill–Whitham–Richards (LWR) models. In the FtL models individual vehicles are tracked,
and this leads to a system of ordinary differential equations. On the other hands, in LWR mod-
els, traffic is represented by the density of vehicles, and the resulting equation is a first order
hyperbolic conservation law. We will study the continuum limit of the FtL model when traffic
becomes dense. We will also mention the problem of modeling traffic flow on networks.
In the second part of the talk, we will discuss a novel model for multi-lane traffic within
the LWR framework. The talk is based on joint work with Nils Henrik Risebro (University of
Oslo).
Hidden symmetries in non-self-adjoint graphs
Amru Hussein, hussein@mathematik.uni-kl.de
TU Kaiserslautern, Germany
On finite metric graphs Laplace operators subject to general non-self-adjoint matching condi-
tions imposed at graph vertices are considered. A regularity criterium related to the Cayley
transform of boundary conditions is discussed and spectral properties of such regular opera-
tors are investigated, in particular similarity transforms to self-adjoint operators and generation
of C0-semigroups. Concrete examples are discussed exhibiting that non-self-adjoint boundary
conditions can yield to unexpected spectral features.
471
Kinetic and macroscopic diffusion models for gas mixtures in the context
of respiration
Bérénice Grec, berenice.grec@u-paris.fr
University of Paris, France
In this talk, I will first discuss shortly the context of respiration, and in particular the need to
describe accurately the diffusion of respiratory gases in the lower part of the lung. At the macro-
scopic level, diffusion processes for mixtures are often modelled using cross-diffusion models.
In order both to determine the regime in which such models are valid, and to compute the binary
diffusion coefficients, it is of particular interest to derive these models from a description at the
mesoscopic level by means of kinetic equations.
More precisely, we consider the Boltzmann equations for mixtures with general cross-
sections (i.e. for any kind of molecules interactions), and obtain the so-called Maxwell-Stefan
equations by performing a Hilbert asymptotic expansion at low Knudsen and Mach numbers.
This allows us to compute the values of the Maxwell-Stefan diffusion coefficients with explicit
formulae with respect to the cross-sections. We also justify the specific ansatz we use thanks to
the so-called moment method.
This is a joint work with Laurent Boudin and Vincent Pavan.
Mathematical modeling of traffic flow
Helge Holden, helge.holden@ntnu.no
Norwegian University of Science and Technology, Norway
Traditionally, there are two types of mathematical models for vehicular traffic, namely the
Follow-the-Leader (FtL) models and the continuum models, using variants of the classical
Lighthill–Whitham–Richards (LWR) models. In the FtL models individual vehicles are tracked,
and this leads to a system of ordinary differential equations. On the other hands, in LWR mod-
els, traffic is represented by the density of vehicles, and the resulting equation is a first order
hyperbolic conservation law. We will study the continuum limit of the FtL model when traffic
becomes dense. We will also mention the problem of modeling traffic flow on networks.
In the second part of the talk, we will discuss a novel model for multi-lane traffic within
the LWR framework. The talk is based on joint work with Nils Henrik Risebro (University of
Oslo).
Hidden symmetries in non-self-adjoint graphs
Amru Hussein, hussein@mathematik.uni-kl.de
TU Kaiserslautern, Germany
On finite metric graphs Laplace operators subject to general non-self-adjoint matching condi-
tions imposed at graph vertices are considered. A regularity criterium related to the Cayley
transform of boundary conditions is discussed and spectral properties of such regular opera-
tors are investigated, in particular similarity transforms to self-adjoint operators and generation
of C0-semigroups. Concrete examples are discussed exhibiting that non-self-adjoint boundary
conditions can yield to unexpected spectral features.
471
