Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation

Francis Ribaud (1) , Stéphane Vento (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 LAGA - Laboratoire Analyse, Géométrie et Applications

We show that the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation $$ u_t-D_x^\alpha u_{x} + u_{xyy} = uu_x,\quad (t,x,y)\in\R^3,\quad 1\le \alpha\le 2, $$ is locally well-posed in the spaces $E^s$, $s>\frac 2\alpha-\frac 34$, endowed with the norm $ \|f\|_{E^s} = \|\langle |\xi|^\alpha+\mu^2\rangle^s\hat{f}\|_{L^2(\R^2)}. $ As a consequence, we get the global well-posedness in the energy space $E^{1/2}$ as soon as $\alpha>\frac 85$. The proof is based on the approach of the short time Bourgain spaces developed by Ionescu, Kenig and Tataru \cite{IKT} combined with new Strichartz estimates and a modified energy.

Domaines: Equations aux dérivées partielles [math.AP]

Francis Ribaud, Stéphane Vento. Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation. Discrete and Continuous Dynamical Systems - Series A, 2017, 37 (1), pp.449 - 483. ⟨10.3934/dcds.2017019⟩. ⟨hal-01251033⟩