Abstract

The project falls into the scope of the applications of mathematics to engineering problem solving. More precisely, it concentrates on models leading to the study of nonlinear partial differential equations according to the three following viewpoints :

• mathematical analysis
• numerical simulation and analysis
• modelization and applications.

Inside this field, our interest may be at various stages of the continuous spectrum going from industrial problems to their numerical treatment : mathematical modelling, theoretical study of the models, description of methods and tools of resolution, analysis of numerical algorithms and numerical simulation. Our main topics of interest can be classified as follows :

• Free boundary problems, shape optimization, control of shapes and connected problems : one of the main application is the control of surfaces in the electromagnetic treatment of liquid metals. Our research concerns the numerical computation of the shapes, their existence and stability, inverse problems like "shapability" ; our mathematical approach carries over to many other situations where free boundaries are involved like phase transitions. New applications to molecular chemistry are now also considered.
• Stabilization of flexible structures : underlying applications concern the stabilization of vibrating systems like satellite antennas or flexibles part of robots. Models are mainly wave- beam- or plate-like equations and stabilization is to be obtained by various nonlinear feedbacks.
• Nonlinear evolution problems and applications : emphasis is put on asymptotic behavior (and global existence in time) of some nonlinear evolutions : reaction-diffusion, semi-linear phenomena, models in oceanography and meteorology, predicibility.

A specific effort has been recently made to develop new competences in parallel computing on the above topics in collaboration with the Charles Hermite Center.