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