We will discuss structural stability of reaction-diffusion waves in the bistable and in the monostable cases. We will show existence of generalized travelling waves in a heterogeneous medium and of reaction-diffusion-convection waves in incompressible fluids. We will develop mathematical tools to study some non Fredholm operators and apply them for combustion waves with the Lewis number different from 1.

The talk will be devoted to nonlinear propagation of pulsating fronts for reaction-diffusion equations in periodic domains. Such problems involve linear and nonlinear elliptic and parabolic equations with periodic coefficients. It is of interest to study the effects of various phenomena such as diffusion, reaction, advection, as well as the geometry of the domain, on the speed of fronts. Some asymptotics and some applications to a periodic patch model in ecology will be given. This talk is based on some works with H. Berestycki, N. Nadirashvili and L. Roques.

We consider competition-diffusion systems for two species when there is large inter-species competition and the inhomogeneous species densities are specified on the boundary of the spatial domain. We discuss the limit of solutions of this system in the large competition limit, showing convergence to a free-boundary problem on a finite time interval using integral estimates. Moreover, in the special case when the diffusion coefficients of the two species are the same, convergence on unbounded time intervals can be established using a "blow-up" method, which enables aspects of the long-time behaviour of the competition-diffusion system with sufficiently large competition to be deduced from that of solutions of the limiting problem.

Chalcopyrite disease within sphalerite is probably the best-known example for so-called diffusion induced segregation (DIS) phenomena. In this talk a mathematical formulation of the physical process as reaction-diffusion equations is given. Very generally it is shown how reaction terms for systems undergoing phase changes have to be formulated to obtain a thermodynamically consistent model. The existence of weak solutions is shown, some quantitative numerical computations are presented. In the final simulation step, the physical free energy is simulated by molecular dynamics techniques to have numerical simulations closer to reality.

The classical theory by Lifshitz, Slyozov and Wagner describes diffusion limited coarsening of particles in the limit of vanishing volume fraction $\phi$. Recently there has been a large interest in identifying higher order correction terms due to some shortcomings of the LSW theory. We first present a rigorous mathematical analysis in a stochastic setting which identifies the scaling of the first order correction to the LSW theory. The order of the relative deviation of the coarsening rate from the LSW theory shows a cross--over between $\phi^{1/3}$ and $\phi^{1/2}$ when screening effects become important. Second, we discuss a self-consistent derivation of the expected growth rate of a particle under certain assumptions on the statistics of the system. In particular we study the the influence of correlations.

At first we study a convection-diffusion problem with so called cellular convection. In the homogenization limit we estimate the effective diffusivity. In particular for strong convection, which is a singular limit, these estimates have the correct scaling. Secondly we study the speed enhancement for travelling waves of reaction-diffusion-convection problems of KPP and combustion type. For shear flow convection these estimates have the correct scaling in several asymptotic regimes, i.e. strong convection, small diffusion and rapid oscillations.

We analyze asymptotic laws of front speed enhancement in stationary ergodic random shears for the Fisher-KPP nonlinearity on the entire plane. This is possible through a Hamilton-Jacobi reduction and Liapunov exponent analysis of the so called parabolic Anderson problem. Numerical computation of ensemble averaged front speeds in channel domains shows enhancement behavior for more general nonlinearities in the presence of random shears. We show modified asymptotic laws observed numerically, and speed statistics.

We consider travelling waves in spatially quasi-periodic media. We first define the notion of travelling waves and discuss their uniquness and stability properties in the case of bistable nonlinearities. Much of the discussions go in parallel to the case of periodic spatial inhomogeneity, but in the quasi-periodic case, some difficulties related to the small divisor problem also occur. We then study travelling waves in a periodically or quasi-periodically racheted cylinder. Our goal is to estimate their avarage speed near the homogenization limit. Surprisingly, the limit speed of the travelling wave depends only on the maximal opening angle of the rachet, despite its complex behavior near the boundary.

We discuss the process of ``glueing'' (as developed by Freidlin and Wentzell) within the framework of stochastic averaging. We understand a singular perturbations problem, the solvability of which is directly related to glueing conditions. We give some precise bounds on the PDE solution of this glueing condition. As time permits, we will also apply these results to some new problems.

We consider Hamilton-Jacobi equations with or without viscosity with periodic boundary conditions under the influence of additive noise, white in time and periodic and twice differentiable in space. We show for certain Hamiltonians the existence of a solution global in time, which is unique up to constants and attracts up to constants any other solution. This solution can be used to construct measures which are invariant under the evolution. We assume superlinear growth, but no convexity of the Hamiltonian.

Some of the most striking effects that dominate the behaviour of microscopic systems are caused by interfacial energies. One example are surface-relaxation patterns. For a simplistic atomistic model based on pair-potentials which describes the elastic behavior of a monatomic crystal it can be proven that the first correction term of the total energy can be written as a surface integral. The integrand is implicitly defined by a family of algebraic equations that determine the relaxation pattern. Numerical computations suggest that the associated Frank-diagram is differentiable only for irrational surface normals.

Modelling and efficient simulation of reactive transport in porous media plays a major role, in particular concerning the fate of contaminants in soils and aquifers. This is a formidable task, as one has to deal with a microscopically heterogeneous system consisting of the phases fluid-air-solid.Reactions to be considered often are heterogeneous, i.e., they involve two phases. We present a comprehensive model taking saturated-unsaturated fluid flow and homogeneous and heterogeneous reactions for multiple components, both in quasi-static equilibrium and in kinetic non-equilibrium, into account.More presisely the model is capable to account for fluid flow in the saturated and vadose zone to simulate transient and steady state scenarios; solute transport with equilibrium or kinetic sorption isotherms of general shape; degradation processes of zeroth or first order; biodegradation with multiplicative Monod kinetics including arbitrary electron donators, electron acceptors and microbial species acting in different respirative pathways; geochemical transformations like redox reactions or aqueous complexation of equilibrium or kinetic type. carrier facilitation; surfactant transport.In this way ariving at a highly nonlinear time-dependent problem in 10 or more varables, at least in three spatial dimensions a reduction in complexity is desired. Therefore various transformation techniques are presentd with the aim to decouple the problem as far as possible. The remaining nonlinear system is approximated with Newton's method using the multigrid method for the arising linear subproblems. As spatial discretization hybridized mixed finite elements are chosen due to their advantageous qualitative properties. As we have to deal with global nonlinear problems, local nonlinear problems have to be solved in the process of static condensation. Large gradients and small reaction zones require grid adaptation based on a posteriori error indicators. In the time discretization the stiffness of the system has to be taken into account, e.g., by appropriate multistep methods.

A broad range of scientific and engineering problems involve multiple scales and multi-scale phenomena. In particular, the physical properties that control many hydrologic subsurface processes vary dramatically over a range of spatial scales. Such heterogeneity occurs in many geological media where depositional processes that act over different characteristic time scales induce spatial patterns having different characteristic spatial scales. In this talk, we focus on flow and transport processes as they take place in the subsurface and discuss coarse scale models derived from a detailed small scale model, bypassing the necessity of empirical modelling and, on the other hand, overcoming the limitations of global homogenization procedures. Employing systematic coarsening procedures that yield models at intermediate scales and resolution will lead us to adaptive coarsening procedures.

Utilizing a focused laser beam manipulated through computer-controlled mirrors, and capable of "writing" spatiotemporal temperature fields on a surface, we explore the fundamental impact of localized spatiotemporal perturbations on a simple reaction-diffusion system (1). Our two-dimensional model system is the low-pressure catalytic oxidation of CO on Pt(110), a reaction exhibiting well-understood spatiotemporal patterns. In the simplest case the laser spot causes the ignition of a reaction wave by a single critical "kick" at a selected surface location. The cooperativeness between two local sub-critical perturbations separated in time and/or space is then explored (2). A temperature heterogeneity moving along a line may ignite waves along its path, or can drag preexisting pulses. In particular, we studied how a traveling chemical pulse is ''dragged'' by a localized, moving temperature heterogeneity as a function of its intensity and speed. The acceleration and eventual ''detachment'' of the wave from the heterogeneity is explored through simulation and stability analysis (3). Additionally we demonstrated how pulses, the basic building blocks of chemical patterns, can be modified, guided, and erased and how the overall reaction rate can be increased through localized actuation. Computational studies supplement and rationalize these experimental findings. Finally we studied ultra thin (~200 nm) Pt(110) metal single crystals enables, which small thermal capacity allowed the exploration of catalytic reaction energetics at low pressures. We discovered a new chemo-thermo-mechanical instability in this regime, in which catalytic reaction kinetics interact with heat transfer and mechanical buckling to create oscillations (4). These interacting components are separated and explored through experimentation, mathematical modeling, and scientific computation, and from their synthesis an explanation of the phenomenon emerges.1. J. Wolff, A. G. Papathanasiou, I. G. Kevrekidis, H. H. Rotermund, G. Ertl, Science 294, 134-137 (2001).2. J. Wolff, A. G. Papathanasiou, H. H. Rotermund, G. Ertl, M. Katsoulakis, X. Li, I. G. Kevrekidis, Phys. Rev. Lett. 90, 148301-4 (2003).3. J. Wolff, A. G. Papathanasiou, H. H. Rotermund, G. Ertl, X. Li, I. G. Kevrekidis, Phys. Rev. Lett. 90, 018302 1-4 (2003). 4. Fehmi Cirak, Jaime E. Cisternas, Alberto M. Cuitino, Gerhard Ertl, Philip Holmes, Ioannis G. Kevrekidis, Michael Ortiz, Harm Hinrich Rotermund, Michael Schunack, Janpeter Wolff, Science 300, 1932 (2003)

We prove limit theorems for small-scale pair dispersion in synthetic velocity fields with power-law spatial spectra and wave-number dependent correlation times. These limit theorems are related to a family of generalized Richardson's laws with a limiting case corresponding to Richardson's t3 and 4/3-laws.

We consider reaction-diffusion systems on infinite cylinders and look for standing or pinned waves. While in the spatial homogeneous situation pinned waves may only exist for exceptional parameter values, one would expect, that waves are pinned at periodic heterogenities for large parameter regions. Using a spatial dynamics approach and an exponential homogenization techniques, we show that these parameter regions are in fact exponentially small in the period of the heterogenities.

We consider a model of tumor growth suggested by [Luckhaus-Triolo], which emphasizes the competition for space rather than for nutrients. In their work a coupled system of a parabolic and an ordinary differential equation is derived describing the density evolution of mutant and the nearly stationary healthy cells. Especially, the question is raised if healthy tissue is able to prevent mutant cells to invade. In our project we consider the asymptotic propagation of initially separated cell populations. Here we present the traveling wave problem for the model described above and show existence of waves by a vanishing viscosity approach. In particular, a unique monotone wave exists, which, depending on an associated Lyapunov functional, either invades the healthy tissue or is stationary and discontinuous. However, the discontinuous wave is not stable.
This is an ongoing joint work with Steffen Heinze and Ben Schweizer.

We consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses continuous spectrum up to the imaginary axis, for all values of the bifurcation parameter, and that a pair of imaginary eigenvalues crosses the imaginary axis. For a reaction-diffusion-convection system we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations, prove the occurrence of a Hopf bifurcation and the nonlinear stability of the bifurcating time-periodic solutions, again with respect to spatially localized perturbations. This is joint work with Markus Kunze.