Biannual Report - Fachbereich Mathematik - Technische Universität ...

Objective of this project is the understanding and treatment of three phase contact line
problems. Finding an approach to such problems leads to questions under which circumstances
one can solve initial boundary value problems on domains with non-smooth
boundary. In a first step well-posedness for Stokes and Navier-Stokes equations with fullslip
boundary conditions on a wedge domain is considered, cf. [1].
Support: DFG
Contact: S. Maier, J. Saal
References
[1] S. Maier and J. Saal. Stokes and Navier-Stokes equations with perfect slip on wedge type
domains. submitted.
Project: L p -theory for the Tornado-Hurricance Equations
The Tornado-Hurricane equations represent a system of equations modeling the evolution
of cyclones. Based on a first approach given in [1] in L 2 , the objective is to develop an
L p -theory. In this setting preciser results on well-posedness as well as new results on
regularity and stability seem to be available.
Support: DFG
Contact: S. Maier, J. Saal
References
[1] J. Saal. Well-posedness of the Tornado-Hurricane equations. Discr. Cont. Dyn. Sys. - Series A,
26(2):649–664, 2010.
1.2.3 Applied Geometry
The research group "Geometry and Approximation" investigates geometric objects, typically
surfaces, as well as approximations thereof.
Classical Differential Geometry deals with curves and surfaces. Surfaces arising in the
sciences are frequently minimizers to certain functionals. In the simplest case, say for a
biological cell, they might bound a given volume in such a way that the area of the surface
is minimal. Other interfaces minimize functionals involving curvatures. Critical points
satisfy Euler equations, namely non-linear partial differential equations. Our goal is to
establish new solutions and properties of solutions, in Euclidean 3-space but also in other
Riemannian spaces, by employing analysis and Riemannian Geometry.
In Geometric Modeling, mathematical tools for the explicit description of geometric objects
are developed and analyzed. Unlike in elementary geometry, the focus is not on simple
objects like circles or spheres, but on more complex structures, as they arise in various
applications. One may think of a car body, a piece of cloth, or a dinosaur in an animated
film.
The surfaces considered in Differential Geometry and Geometric Modeling typically have a
fairly complicated structure. For further processing, it is necesary to approximate them in a
function space of reduced complexity, say a spline space. For that reason, the development
of tools for efficient approximation of geometric objects is an important task, giving rise to
interesting mathematical questions in the field of multivariate approximation theory.
34 1 Research