PhD research topic - Piero D'Ancona
Partial differential equations, Harmonic analysis
Nonlinear waves, Dispersive equations
Understanding the structure of solutions to nonlinear evolution PDEs for large times seemed an impossible goal until 20 years ago. In recent years some exciting progress has been made, and a general picture is emerging. This progress required a combination of methods from classical PDE theory, geometry, harmonic analysis and nonlinear analysis. I am interested in analazying the asymptotic properties of solutions to nonlinear wave, Schroedinger, Dirac and Maxwell equations, in the free space or in exterior domains, and in presence of potentials. In the next years I would like in particular to focus on some very interesting specific problems. The first one is the wave maps system, which is a simplified model of general relativity where all the main difficulties are still present. I plan to study large wave maps on a curved background which may have a large curvature at infinity. The second one is the nonlinear Maxwell system, which is especially important for applications. Despite its physical relevance, modern techniques have not yet been applied to its study, and this line of research seems particularly promising.
A nice general introduction to the subject:
Linares, Felipe and Ponce, Gustavo: Introduction to nonlinear dispersive equations. Springer, New York, 2009
My papers on these subjects are on arXiv, links can be found here.