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 $E^s$, $s\textgreater{ }\frac 2\alpha-\frac 34$, 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 $\alpha\textgreater{ }\frac 85$. 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.
Categories: math.AP