Dr. Katharina Hopf

Weierstrass Institute for Applied Analysis and Stochastics
Weierstrass Group Multi-species Balance Laws

Teaching

Summer term 2024: X-Student Research Seminar

Supported by the Berlin University Alliance within the programme StuROPx.

Title: Beyond Fick's Law: Pressure-Driven Phenomena in Multi-Species Diffusion

Date and location: Thursdays, 13:00-15:00; Mathematics Institute, HU Berlin.

Relevant links:
Official AGNES entry
Details on registration and participation
AGNES entry with date and location
Project description at Berlin University Alliance webpage

Description:
The phenomenon of diffusion is ubiquitous in nature — everyday examples include the spread of perfume or smoke through the surrounding air, and the dispersion of a droplet of ink in a glass of water. In this seminar, we will go beyond the classical paradigm of Fick's law of diffusion. Our focus lies on a cross-diffusion system for two species driven by a population pressure that induces a spreading effect. The ultimate goal is to understand the effective dynamics of this system in a certain stiff-pressure regime. To this end, we employ a variety of mathematical methods from Analysis, Geometry, and Numerics.

Target group:
MSc and advanced BSc students in Mathematics or a related discipline. Interested doctoral students are also very welcome.

Prerequisites:
Required is a basic knowledge of Partial Differential Equations. A good background in mathematical analysis, elementary differential geometry, or numerical simulation of differential equations is beneficial.

Winter term 2022/23: Nonlinear Partial Differential Equations
HU Berlin
Description:
This course is an advanced Mathematics course (MSc) in partial differential equations. Lecture: 4SWS; Exercise class: 2SWS.

Content:
  1. Fixed point theorems
  2. Variational inequalities and quasilinear elliptic equations
  3. Quasilinear parabolic equations and degenerate diffusion
  4. Reaction-diffusion systems
  5. Introduction to hyperbolic systems of conservation laws
Prerequisites:
Content of the Bachelor modules 'Functional Analysis' (M17) and 'Partial Differential Equations' (M18) at HU Berlin. This includes in particular Banach spaces and their dual spaces, weak and strong convergence, compactness, Sobolev spaces (incl. embedding theorems), weak solutions.
[ This material can be found in many textbooks, see e.g. 'Linear Functional Analysis' by H. W. Alt and 'Partial Differential Equations' by L. C. Evans. ]