Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D

Abstract

In this paper, we consider the Zakharov-Kuznetsov equation in 3D, with a dissipative term of order \begin{document}$ 0 in the \begin{document}$ x $\end{document} direction. We prove that the problem is locally well-posed in \begin{document}$ H^{s}( { I\!\!R}^3) $\end{document} , for \begin{document}$ s > 1-\frac{\alpha}{2} $\end{document} , and by an a priori energy estimate, we prove that the problem is globally well-posed in \begin{document}$ H^{1}( { I\!\!R}^3) $\end{document} .