Séminaire Analyse non linéaire et EDP

Since the basic work of Peierls it is common consensus thatweakly nonlinear wave equations with random initial data can be wellapproximated by a kinetic equation. I will explain the precise conjecture.I will also discuss my recent joint work with J. Lukkarinen, where wecontrol the kinetic limit in special cases for the lattice nonlinearSchroedinger equation.

We describe recent theoretical and experimental progress on making objects invisible to detection by electromagnetic waves, acoustic waves and matter waves. For the case of electromagnetic waves, Maxwell's equations have transformation laws that allow for design of electromagnetic materials that steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was being hidden. The object, which would have no shadow, is said to […]

Inverse transport consists of reconstructing the optical parameters in a transportequation from knowledge of a measurement operator. We review several uniqueness andstability results obtained in the context of various boundary measurements as they arisee.g. in optical tomography, a medical imaging modality. Accurate numerical reconstructionsobtained by carefully capturing the singularities of the measurement operator are also brieflypresented.In many practical settings, the typical boundary measurements available in inversetransport do not allow us to reconstruct the optical parameters with sufficient resolution.In the second part of the talk, I present recent results obtained for […]

We study global behavior of the nonlinear Klein-Gordon equation with a focusing cubic power in three dimensions, in the energy space under the restriction of radial symmetry and an energy upper bound slightly above that of the ground state. We give a complete classification of the solutions into 9 non-empty sets according to whether they blow-up, scatter to 0, or scatter to the ground states, in the forward and backward time directions, and the splitting is given in terms of the stable and the unstable manifoldsof the ground states. This […]