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Séminaire Math-Phys : Existence of blow-up solutions in the KdV-type equations

Mercredi 13 juin 16:15-17:15 - Svetlana Roudenko - The George Washington University

Séminaire Math-Phys : Existence of blow-up solutions in the KdV-type equations

Résumé : While the KdV equation and its generalizations with higher power nonlinearities (gKdV) have been long studied, a question about existence of blow-up solutions for higher power nonlinearities has posed lots of challenges and far from being answered. One of the main obstacles is that unlike other dispersive models such as the nonlinear Schrodinger or wave equations, the gKdV equation does not have a suitable virial quantity which is the key to prove the finite time blow-up. Only at the dawn of this century the groundbreaking works of Martel and Merle rigorously proved the existence of finite-time blow-up solutions for the quintic (critical) gKdV equation.
We consider a higher dimensional extension of the gKdV equation, called generalized Zakharov-Kuznetsov (gZK) equation (the gKdV is limited as a spatially one-dimensional model), and ask if blow-up solutions exist in the corresponding critical gZK equation. We positively answer this question for the two dimensional version of (cubic) Zakharov-Kuznetsov equation. The main ingredient of the proof is the Liouville-type theorem, which uses time-decay estimates, a la virial type quantity and spectral properties associated to it. This is a joint work with Luiz Farah, Justin Holmer and Kai Yang.

Lieu : Salle A318

Pour en savoir plus sur cet événement, consultez l'article Séminaires Math-Physique