Construction of multi-bubble solutions for the critical gKdV equation

Combet, Vianney; Martel, Yvan

Type de document :

Article dans une revue scientifique: Article original

DOI :

10.1137/17M1140595

Titre :

Construction of multi-bubble solutions for the critical gKdV equation

Auteur(s) :

Combet, Vianney [Auteur]

Laboratoire Paul Painlevé - UMR 8524 [LPP]
Martel, Yvan [Auteur]
Centre de Mathématiques Laurent Schwartz [CMLS]

Titre de la revue :

SIAM Journal on Mathematical Analysis

Pagination :

3715-3790

Éditeur :

Society for Industrial and Applied Mathematics

Date de publication :

2018

ISSN :

0036-1410

Discipline(s) HAL :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

Résumé en anglais : [en]

We prove the existence of solutions of the mass critical generalized Korteweg-de Vries equation $\partial_t u + \partial_x(\partial_{xx} u + u^5) = 0$ containing an arbitrary number $K\geq 2$ of blow up bubbles, for any ...
Lire la suite >We prove the existence of solutions of the mass critical generalized Korteweg-de Vries equation $\partial_t u + \partial_x(\partial_{xx} u + u^5) = 0$ containing an arbitrary number $K\geq 2$ of blow up bubbles, for any choice of sign and scaling parameters: for any $\ell_1>\ell_2>\cdots>\ell_K>0$ and $\epsilon_1,\ldots,\epsilon_K\in\{\pm1\}$, there exists an $H^1$ solution $u$ of the equation such that \[ u(t) - \sum_{k=1}^K \frac {\epsilon_k}{\lambda_k^\frac12(t)} Q\left( \frac {\cdot - x_k(t)}{\lambda_k(t)} \right) \longrightarrow 0 \quad\mbox{ in }\ H^1 \mbox{ as }\ t\downarrow 0, \] with $\lambda_k(t)\sim \ell_k t$ and $x_k(t)\sim -\ell_k^{-2}t^{-1}$ as $t\downarrow 0$. The construction uses and extends techniques developed mainly by Martel, Merle and Rapha\"el. Due to strong interactions between the bubbles, it also relies decisively on the sharp properties of the minimal mass blow up solution (single bubble case) proved by the authors in arXiv:1602.03519.Lire moins >

Langue :

Anglais

Comité de lecture :

Oui

Audience :

Internationale

Vulgarisation :

Non

Projet ANR :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Commentaire :

70 pages