On special regularity properties of solutions of the benjamin-ono-zakharov-kuznetsov (bo-zk) equation

Abstract

In this paper we study special properties of solutions of the initial value problem (IVP) associated to the Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK) equation. We prove that if initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. In other words, the extra regularity on the data propagates in the solutions in the direction of the dispersion. The method of proof to obtain our result uses weighted energy estimates arguments combined with the smoothing properties of the solutions. Hence we need to have local well-posedness for the associated IVP via compactness method. In particular, we establish a local well-posedness in the usual \begin{document}$ L^{2}( \mathbb R^2) $\end{document} -based Sobolev spaces \begin{document}$ H^s( \mathbb R^2) $\end{document} for \begin{document}$ s>\frac{5}{4} $\end{document} which coincides with the best available result in the literature proved employing more complicated tools.