Abstracts

The Fisher-KPP equation is probably the simplest model of combustion and population biology, among other things. It proved to be a useful tool for understanding of, e.g. ways infectious diseases spread, and nuclear combustion in stars. In these lectures we describe behavior of solutions of the Fisher-KPP equation in spatially and temporally periodic media.

1. Introduction to conservation laws, characteristics, shocks, entropy solutions.
2. Numerical methods for one dimensional scalar equations
3. Extension to systems of equations in one dimension, Riemann solvers.
4. Extensions to multiple dimensions.  Applications to the shallow water equations.

In this series of lectures, I will discuss the development of several interesting new constitutive relations that open up the possibility to resolve some important open issues in both solid and fluid mechanics.

We will review models for flow and reactive transport in porous media, ranging from groundwater to multiphase flow and from tracer transport to transport with general equilibrium/kinetic reactions also with pore changing mechanisms leading eventually to a full nonlinear coupling of all processes. Some aspects of analysis (existence, qualitative behavior) will be elucidated and tools for numerical simulation of real world problems (carbon dioxide sequestration, …) in terms of discretization and nonlinear solvers will be addressed. Being so far always on a macro (field) scale without explicit geometric resolution of the pore scale, also the upscaling process via periodic homogenization from pore scale models will be discussed leading also to new types of coupled micro-macro models.

Recently the fractional Laplacian operator $(−∆)^s$ and more general integro-differential operators with singular kernels of fractional differentiability of order $s \in (0,1)$ have been intensively studied.  After being investigated first from the perspective of potential theory and harmonic analysis, during the last decade also the regularity theory has obtained great interest. The aim of this minicourse is to go through some recent developments in the nonlocal regularity theory. The approach will be suitable also for many nonlinear problems.