93 research outputs found
Flow Characteristics in a Crowded Transport Model
The aim of this paper is to discuss the appropriate modelling of in- and
outflow boundary conditions for nonlinear drift-diffusion models for the
transport of particles including size exclusion and their effect on the
behaviour of solutions. We use a derivation from a microscopic asymmetric
exclusion process and its extension to particles entering or leaving on the
boundaries. This leads to specific Robin-type boundary conditions for inflow
and outflow, respectively. For the stationary equation we prove the existence
of solutions in a suitable setup. Moreover, we investigate the flow
characteristics for small diffusion, which yields the occurence of a maximal
current phase in addition to well-known one-sided boundary layer effects for
linear drift-diffusion problems. In a one-dimensional setup we provide rigorous
estimates in terms of , which confirm three different phases.
Finally, we derive a numerical approach to solve the problem also in multiple
dimensions. This provides further insight and allows for the investigation of
more complicated geometric setups
Recommended from our members
On some partial differential equation models in socio-economic contexts - analysis and numerical simulations
This thesis deals with the analysis and numerical simulation of different partial differential equation models arising in socioeconomic sciences. It is divided into two parts: The first part deals with a mean-field price formation model introduced by Lasry and
Lions in 2007. This model describes the dynamic behaviour of the price of a good being traded between a group of buyers and a group of vendors. Existence (locally in time) of smooth solutions is established, and obstructions to proving a global existence result are examined. Also, properties of a regularised version of the model are explored and numerical examples are shown. Furthermore, the possibility of reconstructing the initial datum given a number of observations, regarding the price and the transaction rate, is considered. Using a variational approach, the problem can be expressed as a non-linear constrained minimization problem. We show that the initial datum is uniquely determined by the price (identifiability). Furthermore, a numerical scheme is implemented and a variety of examples are presented.
The second part of this thesis treats two different models describing the motion of (large) human crowds. For the first model, introduced by R.L. Hughes in 2002, several regularised versions are considered. Existence and uniqueness of entropy solutions are proven using the technique of vanishing viscosity. In one space dimension, the dynamic behaviour of solutions of the original model is explored for some special cases. These results are compared to numerical simulations. Moreover, we consider a discrete cellular automaton model introduced by A. Kirchner and A. Schadschneider in 2002.
By (formally) passing to the continuum limit, we obtain a system of partial differential equations. Some analytical properties, such as linear stability of stationary states, are
examined and extensive numerical simulations show capabilities and limitations of the model in both the discrete and continuous setting
On the uniqueness of nonlinear diffusion coefficients in the presence of lower order terms
We consider the identification of nonlinear diffusion coefficients of the
form or in quasi-linear parabolic and elliptic equations.
Uniqueness for this inverse problem is established under very general
assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof
of our main result relies on the construction of a series of appropriate
Dirichlet data and test functions with a particular singular behavior at the
boundary. This allows us to localize the analysis and to separate the principal
part of the equation from the remaining terms. We therefore do not require
specific knowledge of lower order terms or initial data which allows to apply
our results to a variety of applications. This is illustrated by discussing
some typical examples in detail
Simultaneous identification of diffusion and absorption coefficients in a quasilinear elliptic problem
In this work we consider the identifiability of two coefficients and
in a quasilinear elliptic partial differential equation from observation
of the Dirichlet-to-Neumann map. We use a linearization procedure due to Isakov
[On uniqueness in inverse problems for semilinear parabolic equations. Archive
for Rational Mechanics and Analysis, 1993] and special singular solutions to
first determine and for . Based on this partial
result, we are then able to determine for by an
adjoint approach.Comment: 10 pages; Proof of Theorem 4.1 correcte
Numerical identification of a nonlinear diffusion law via regularization in Hilbert scales
We consider the reconstruction of a diffusion coefficient in a quasilinear
elliptic problem from a single measurement of overspecified Neumann and
Dirichlet data. The uniqueness for this parameter identification problem has
been established by Cannon and we therefore focus on the stable solution in the
presence of data noise. For this, we utilize a reformulation of the inverse
problem as a linear ill-posed operator equation with perturbed data and
operators. We are able to explicitly characterize the mapping properties of the
corresponding operators which allow us to apply regularization in Hilbert
scales. We can then prove convergence and convergence rates of the regularized
reconstructions under very mild assumptions on the exact parameter. These are,
in fact, already needed for the analysis of the forward problem and no
additional source conditions are required. Numerical tests are presented to
illustrate the theoretical statements.Comment: 17 pages, 2 figure
Identification of Chemotaxis Models with Volume Filling
Chemotaxis refers to the directed movement of cells in response to a chemical
signal called chemoattractant. A crucial point in the mathematical modeling of
chemotactic processes is the correct description of the chemotactic sensitivity
and of the production rate of the chemoattractant. In this paper, we
investigate the identification of these non-linear parameter functions in a
chemotaxis model with volume filling. We also discuss the numerical realization
of Tikhonov regularization for the stable solution of the inverse problem. Our
theoretical findings are supported by numerical tests.Comment: Added bibfile missing in v2, no changes on conten
A PDE model for bleb formation and interaction with linker proteins
The aim of this paper is to further develop mathematical models for bleb
formation in cells, including cell-membrane interactions with linker proteins.
This leads to nonlinear reaction-diffusion equations on a surface coupled to
fluid dynamics in the bulk. We provide a detailed mathematical analysis and
investigate some singular limits of the model, connecting it to previous
literature. Moreover, we provide numerical simulations in different scenarios,
confirming that the model can reproduce experimental results on bleb initation
Identification of nonlinear heat conduction laws
We consider the identification of nonlinear heat conduction laws in
stationary and instationary heat transfer problems. Only a single additional
measurement of the temperature on a curve on the boundary is required to
determine the unknown parameter function on the range of observed temperatures.
We first present a new proof of Cannon's uniqueness result for the stationary
case, then derive a corresponding stability estimate, and finally extend our
argument to instationary problems
- …