Non-linear and singular partial differential equations and their applications

Project Information

Grant agreement ID: INTAS-94-2187

Project closed

Start date 1 October 1995

End date 30 September 1998

Funded under

• International Association for the promotion of cooperation with scientists from the independent states of the former Soviet Union (INTAS), 1993-

Total cost No data

EU contribution € 50 000,00

Coordinated by

Institut für Angewandte Analysis and Stochastik
Germany

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Deliverables

Non-linear and singular partial differential equations and their applications

In this network, 11 teams from FSU and 6 teams from INTAS member states joined their efforts in a combined study of nonlinear and singular partial differential equations. This field of research, which is of fundamental importance for the development, simulation and control of modern technologies, is very difficult and requires deep mathematical methods. Since the applications covered in the programme include quite different problems ranging from fluid dynamics, materials science, phase transition and wave phenomena to diffusive processes, many mathematical tools had to be developed and many different expertises were needed. The scientific concept of this network was to further develop the already existing cooperations between the different teams, with the aim to achieve a greater scientific progress than possible individually. The partial differential equations or systems studied in this programme have one distinguishing feature in common: they are nonlinear and, which makes their mathematical treament even more difficult, in some sense singular. The nature of these singularities can be quite different and depends on the actual physical background; typical cases are degenerating or even exploding coefficents in the underlying differential equations, shocks, blow-up effects or singularities that arise from the geometry such as from non-smooth domains like domains with corners. Other possible sources for nonlinearities and singularities are phase transitions which lead to boundary value problems on domains which are a prior unknown (so-called free boundary problems). Various aspects of all these problems have been studied in the network; in particular, questions of modeling, existence, uniqueness and regularity, as well as of the qualitative and asymptotic behaviour of the corresponding solutions, have been addressed. Several new results have been obtained which have been published in a total of 196 publications.