Theoretical aspects of PDEs with applications to engineering structures, models of phase transitions and transport problems
We consider mathematical models based on partial differential equations whose interest is motivated by applications in the engineering field. The challenge is to identify simplified models which, on the one hand, allow for rigorous analytical treatment and, on the other hand, capture the essential aspects of the structures they model. An example are the fourth order PDE problems that model plates and beams and which can be employed to study the oscillatory dynamics of suspension bridges or footbridges. Another example is the use of asymptotic methods in the study of the spectral properties of partial differential operators which describe the behavior of waveguides in hydrodynamics, or the dispersion effects of acoustic and electromagnetic waves against obstacles with a periodic structure. The analysis requires a wide range of skills ranging from qualitative and quantitative analysis of PDEs and ODEs, spectral optimization, stability analysis of dynamical systems, application of homogenization techniques.
Other topics of interest are quasi-parabolic and hyperbolic type PDEs arising in models of phase transition. In particular we study the Penrose-Fife and Stefan models, also coupled with laws of phase relaxation and with the Cattaneo heat flux law.
Furthermore, we are interested in the theoretical study (existence, uniqueness, stability...) and the development of particle numerical methods for solving conservation laws modeling crowd movements or congested traffic with non-local interaction.