Existence and stability of the log log blow up dynamics for

Niveau: Supérieur, Licence, Bac+2
Existence and stability of the log-log blow-up dynamics for the L2-critical nonlinear Schrödinger equation in a domain Fabrice Planchon Laboratoire Analyse, Géométrie & Applications UMR 7539 du CNRS, Institut Galilée Université Paris 13, 99 avenue J.B. Clément 93430 Villetaneuse France Pierre Raphaël Laboratoire de mathématiques UMR 8628 du CNRS Université Paris-Sud 91405 Orsay Cedex France July 3, 2007 Abstract Let iut = ?∆u?|u| 4 N u be the L2-critical nonlinear Schrödinger equation, in a domain ? ? RN with initial data in H10 (?) (Dirichlet boundary condition) and N ≤ 4. We prove existence and stability of finite time blow-up dynamics with the log-log blow-up speed |?u(t)|L2 ? √ log|log(T?t)| T?t . Moreover, for a suitable class of finite time blow-up solutions, we derive global rigidity properties which turn out to be modeled after the RN ones. 1 Introduction 1.1 Setup and notations We consider the L2-critical focusing nonlinear Schrödinger equation in a domain ? with Dirichlet boundary condition: (1) (NLS) ? ? ? iut = ?∆u? |u| 4 N u, (t, x) ? [0, T )? ?, u|∂? = 0, u

• finite time


• up solution


• minimal mass


• mass finite


• critical


• time blow


• u0 ?


• l2-critical nonlinear