Willi Jäger

Diffusion, convection, adsorption, and reaction of chemicals in porous media

Mathematical modeling of reactive flow through porous media is of great importance for a lot of applications in physics, chemists, geology, and biology. The flow of a fluid transporting reacting species in a geometrically complicated medium poses a lot of problems for modeling and simulation. In order to understand the processes in a porous soil, catalyst, electrode, or in a biological membrane, computer-simulations have proven to be very useful. A special example where the method of this paper has been mathematically and chemically a success is a model for a gel-permeationchromatograph; see [23]. The model. was derived by the method of homogenization and the resulting equations could be applied to a real chromatograph with a surprising agreement between experimental data and results obtained by the theoretical model. Homogenization techniques in soil chemistry and for catalysis have been studied by the authors for several years [12, 15, 3, 4, 16, 13, 141. Homo~nization is a technique used to derive a macroscopic model from microscopic processes which are assumed to be theoretically treatable. The complexity of the microscopic system, however, does not allow numerical simulations. For instance, if the geometry of the problem in a porous medium is hopelessly complicated, one has to pass to a homogenized macroscopic model system; see [ 1, 2, 19, 24-26-J.

Effective Transmission Conditions for Reaction-Diffusion Processes in Domains Separated by an Interface

In this paper, we develop multiscale methods appropriate for the homogenization of processes in domains containing thin heterogeneous layers. Our model problem consists of a nonlinear reaction-diffusion system defined in such a domain, and properly scaled in the layer region. Both the period of the heterogeneities and the thickness of the layer are of order

On Periodicity of Solutions For Thermocontrol Problems with Hysteresis-Type Switches

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Homogenization Limit of a Model System for Interaction of Flow, Chemical Reactions, and Mechanics in Cell Tissues

By performing an asymptotic analysis with respect to the scale parameter

A quantitative model of population dynamics in malaria with drug treatment

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On the existence of periodic solutions to some nonlinear thermal control problems

we derive an effective model which consists of the reaction-diffusion equations on two domains separated by an interface together with appropriate transmission conditions across this interface. These conditions are determined by solving local problems on the standard periodicity cell in the layer. Our asymptotic analysis is based on weak and strong two-scale convergence results for sequences of functions defined on thin heterogeneous layers. For the derivation of the transmission conditions, we develop a new method based on test functions of boundary l...

Analysis of a Free Boundary Problem Modeling Thrombus Growth

Mathematical models of thermocontrol processes occurring in chemical reactors and climate control systems are considered. In the models under consideration, the temperature inside a domain is controlled by a thermostat acting on the boundary. The feedback is based on temperature measurements performed by thermal sensors inside the domain. The solvability of the corresponding nonlinear nonlocal problems and the periodicity of solutions are studied.

Multiscale Analysis of Processes in Complex Media

In this article we obtain rigorously the homogenization limit for a fluid-structure-reactive flow system. It consists of cell tissue and intercellular liquid, transporting solutes. The cell tissue is assumed to be linearly elastic and deforming with a viscous nonstationary flow. The elastic moduli of the tissue change with cumulative concentration value. In the limit, when the scale parameter goes to zero, we obtain the quasi-static Biot system, coupled with the upscaled reactive flow. Effective Biots coefficients depend on the reactant concentration. In addition to the weak two-scale convergence results, we prove convergence of the elastic and viscous energies. This then implies a strong two-scale convergence result for the fluid-structure variables. Next we establish the regularity of the solutions for the upscaled equations. To the best of our knowledge, ours is the only known study of the regularity of solutions to the quasi-static Biot system. The regularity is used to prove the uniqueness for the u...

Preface: IWR Special Issue on Scientific Computing: Dedicated to Hans Georg Bock on the Occasion of His 70th Birthday

In this paper we build a population dynamics of malaria including drug treatment. By taking into account both sensitive and resistant parasites, we want to see the effect of treatments on resistance phenomenon and prevent it from overspreading. Our main results include a new dynamics model, its mathematical properties which are found through analysis, the determination of unknown parameters with help of a data set for malaria from Burkina Faso and the numerical simulations of the fitted model. Based on these results, treatment strategies to reduce drug resistance can be elaborated.

Drug Resistance in Infectious Diseases: Modeling, Parameter Estimation and Numerical Simulation

1 We consider the heat equation with a boundary condition involving a control function. The control function satisfies an ordinary differential equation with a right-hand side containing a nonlinear functional that provides the hysteresis phenomenon. The dependence of the functional on the “mean” temperature over the domain causes nonlocal effects. The problem under consideration occurs in the modeling of thermal control processes in chemical reactors and climate control systems. The solvability of the problem and the periodicity of its solutions are considered. 1. In chemical reactors and climate control systems, there arises the problem of temperature control inside a volume by means of some thermal elements on the boundary of the volume. We consider a mathematical model for such a thermal control process. In our model, the temperature distribution inside the domain obeys the heat equation, while the boundary condition involves a control function. The control function satisfies an ordinary differential equation whose right-hand side is given by a nonlinear functional depending on the “mean” temperature over the domain, which provides the so-called hysteresis phenomenon. The existence and uniqueness of solutions to twophase Stefan problems involving a boundary hysteresis control were studied in [1‐3]. In the present paper, we establish an existence and uniqueness result for the heat equation and investigate the periodicity of its solutions. We suggest the concepts of a strong periodic solution 1 The article was translated by the authors. and a mean-periodic solution and show that the existence of a mean-periodic solution implies the existence of a strong periodic solution with the same period. We also give an example in which a unique mean-periodic solution (hence, a unique strong periodic solution) exists.