Séminaire d'analyse IHP-Fondation Sciences Mathématiques de Paris -
Conférence du lundi 28 mars 2011 -
Blow up for some energy critical geometrical models par Pierre Raphaël (Institut de Mathématiques de Toulouse (IMT), CNRS - Université Paul Sabatier UMR5219) -
I will consider energy critical geometric models which appear in various physical situations (cristal physics, ferromagnetism), and which are respectively the parabolic, wave and Schrodinger analogue of the stationary harmonic map problem: the harmonic heat flow, the wave map problem and the Schrodinger map, for maps from (Rtimes R^2) to S^2. For these three problems, I will describe a natural class of smooth solutions which lead to finite time blow up dynamics at some universal blow up speed. This regime is stable for wave maps and the heat flow within the classs of corotational symmetry, and a completely new phenomenon occurs for the Schrodinger map: the remaining rotation fredoom in equivariant symmetry stabilizes the system, and leads to a codimension one blow up phenomenon. This is joint work with Frank Merle, Igor Rodnianski and Remi Schweyer.