On the generalized Zakharov-Kuznetsov equation at critical regularity

Abstract

The Cauchy problem for the generalized Zakharov-Kuznetsov equation $$\partial_t u +\partial_x\Delta u=\partial_x u^{k+1}, \qquad \qquad u(0)=u_0$$ is considered in space dimensions $n=2$ and $n=3$ for integer exponents $k \ge 3$. For data $u_0 \in \dot{B}^{s_c}_{2,q}$, where $1\le q \le \infty$ and $s_c=\frac{n}{2}- \frac{2}{k}$ is the critical Sobolev regularity, it is shown, that this problem is locally well-posed and globally well-posed, if the data are sufficiently small. The proof follows ideas of Kenig, Ponce, and Vega and uses estimates for the corresponding linear equation, such as local smoothing effect, Strichartz estimates, and maximal function inequalities. These are inserted into the framework of the function spaces $U^p$ and $V^p$ introduced by Koch and Tataru.