Project Details
Projekt
Makroskopische Dynamik in diskreten Gittern
Applicant
Professor Dr. Alexander Mielke
Subject Area
Mathematics
Term
from 2000 to 2008
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 5276052
The aim of this project is to study partial differential equations whose solutions display several spatial scales. In particular we are interested in systems which develop fast spatial oscillations whose macroscopic behavior has to be described. We want to treat parabolic as well as hyperbolic problems. In particular, methods are to be developed which allow to characterize the set of all Young-measure solutions to a given evolutionary problem. Moreover, the time evolution of such measures will be investigated. Roughly spoken there are three different cases.(1) In spatially one-dimensional semilinear wave equations spatial oscillations can be transported on arbitrarily small scales; no small parameter is needed. The theory is well understood with classical Young-measures in the case of at most two characteristic speeds. We want to develop methods using nonlocal correlations which allow for the description of systems with more than two characteristic speeds. This is even an open problem in the linear setting.(2) In dispersive wave equations the theory of Modulation equations is well understood. Under quite General conditions the evolution of wave packets can be described by an amplitude equation like the nonlinear Schrödinger equation. We want to connect this theory to the theory of Young-measure solutions, which very likely will also give new relations to microscopic equations in semiconductor theory.(3) Parabolic problems usually damp out spatial oscillations on small scales. However, there are situations where a small parameter e makes these oscillations favorable. This parameter e gives the ratio between the small and the large spatial scale. In the limit e>0 we obtain a macroscopic evolution equation which may be either hyperbolic or parabolic. In particular we want to treat the case which is usually studied in the Ginzburg-Landau formalism for modulation equations.
DFG Programme
Priority Programmes
