Systems in one or two space dimensions

Reducing the amount of symmetry assumed leads to systems of partial differential equations in one space dimension. The best known case is that of the Gowdy class of solutions of the vacuum Einstein equations. Their dynamics have been investigated in detail by Ringström. It would be good to have some generalizations of this work so as to be able to see it in a wider context. One way of obtaining such a generalization is to increase the dimension of space in the original equations, while still maintaining enough symmetry to give an effective system in one space dimension. This is the theme of the PhD project of Aneta Barbos. Another is to add suitable matter sources. The case that the source is chosen to be an electromagnetic field was studied under additional symmetry assumptions in the diploma thesis of Nungesser. I am at present analysing the corresponding problem without additional symmetry. Reducing the symmetry yet again leads to systems where the effective space dimension is two. Results on this type of problem in the vacuum case with small data have been obtained by Yvonne Choquet-Bruhat and Vincent Moncrief. The aim of the PhD project of David Klawonn is to generalize some of these to the case with collisionless matter.

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Alan Rendall

Last modified: Fri May 16 15:44:42 MESZ 2003