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

TL;DR

This paper establishes local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation in certain function spaces, using advanced harmonic analysis techniques and new estimates.

Contribution

It introduces new well-posedness results for the equation in specific Sobolev spaces, extending previous understanding with innovative analytical methods.

Findings

Local well-posedness in spaces E^s for s > 2/α - 3/4

Global well-posedness in energy space E^{1/2} for α > 8/5

Application of short time Bourgain spaces and new Strichartz estimates

Abstract

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.