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

We show that the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation$u\_t-D\_x^αu\_{x} + u\_{xyy} = uu\_x,\quad (t,x,y)\in\R^3,\quad 1\le α\le 2,$is locally well-posed in the spaces $E^s$, $s\textgreater{}\frac 2α-\frac 34$, endowed with the norm$\|f\|\_{E^s} = \|\langle |ξ|^α+μ^2\rangle^s\hat{f}\|\_{L^2(\R^2)}.$As a consequence, we get the global well-posedness in the energy space $E^{1/2}$ as soon as $α\textgreater{}\frac 85$. 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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