Biannual Report - Fachbereich Mathematik - Technische Universität ...

of the boundary which in general is assumed to be of Lipschitz type only. The underlying
model is the Boussinesq system in which the classical Navier-Stokes system is coupled with
the heat equation mainly via the buoyancy term. The heat flux through the boundary is an
important physical quantity which should be controlled, either maximized or minimized
depending on the physical problem at hand.
In the first part of the project we build up the theory of weak and strong solutions to
the Boussinesq system in unbounded domains with Lipschitz boundary. The second aim
is the analysis of the change of the boundary condition for a sequence of domains Ω k
with oscillating boundaries and decreasing amplitude, but increasing frequency. The main
tool in this analysis is the theory of Young measures. A consequence of the boundary
oscillations in the case of Robin boundary conditions for the temperature is a new weight
factor in the Robin condition depending on the way of convergence of Ω k . This result for
perturbed half spaces will be generalized to bounded domains and coupled with methods
from optimal control theory.
Partner: Cluster of Excellence at TU Darmstadt: Smart Interfaces: Understanding and
Designing Fluid Boundaries
Contact: R. Farwig, C. Komo
Project: Global L p solutions for Oldroyd-B models
We investigate existence and uniqueness of global solution for Oldroyd-B models. To be
more precise, we first prove existence of stationary solutions. In a second step we show
that global solutions exist for initial data sufficiently close to stationary solutions. Finally,
we investigate stability of stationary solutions.
Partner: Y. Shibata, Waseda Univeristy, Tokyo
Contact: M. Geissert
Project: Square roots of divergence form operators in non-smooth situations
Elliptic regularity of divergence form operators in non-smooth situations, i.e. bounded
measurable coefficients, Lipschitz domains and mixed boundary conditions, is a delicate
matter. For instance it is possible for every p > 2 to construct such an operator of second
order whose domain on the corresponding W −1,p -space is not contained in a Sobolev space
of type W 1,p , i.e. the gain in regularity when solving the corresponding equation is not two.
Nevertheless, for scalar equations we could show in [2] that the square root of such an
operator behaves nicely, which means that its domain on the same W −1,p -space is L p , i.e.
the gain in regularity is one.
Based on this result, in the future we will head for a corresponding result for systems of
equations and deal with applications to parabolic linear and quasilinear equations.
Partner: J. Rehberg (Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS),
Berlin) and P. Auscher (University of Paris-Sud (Paris XI))
Contact: M. Egert, R. Haller-Dintelmann
References
[1] P. Auscher. On necessary and sufficient conditions for L p -estimates of Riesz transforms
associated to elliptic operators on n and related estimates. Mem. Amer. Math. Soc.,
186(871):xviii+75, 2007.
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