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Blow up and near soliton dynamics for the L 2 critical gKdV equation

Yvan MartelFrank MerlePierre Raphaël

Séminaire Laurent Schwartz — EDP et applications

These notes present the main results of [, , ] concerning the mass critical (gKdV) equation u t + ( u x x + u 5 ) x = 0 for initial data in H 1 close to the soliton. These works revisit the blow up phenomenon close to the family of solitons in several directions: definition of the stable blow up and classification of all possible behaviors in a suitable functional setting, description of the minimal mass blow up in H 1 , construction of various exotic blow up rates in H 1 , including grow up in infinite time.

Blow up for the critical gKdV equation. II: Minimal mass dynamics

Yvan MartelFrank MerlePierre Raphaël — 2015

Journal of the European Mathematical Society

We consider the mass critical (gKdV) equation u t + ( u x x + u 5 ) x = 0 for initial data in H 1 . We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [29].

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