The Zakharov–Kuznetsov equation in high dimensions: small initial data of critical regularity

The Zakharov-Kuznetsov equation in spatial dimension $d\geq 5$ is considered. Cauchy problem shown to be globally well-posed for small initial data critical spaces and it proved that solutions scatter free as $t \to \pm \infty$. proof based on i) novel endpoint non-isotropic Strichartz estimates which are derived from the $(d-1)$-dimensional Schr\"odinger equation, ii) transversal bilinear restriction estimates, iii) an interpolation argument function spaces. Under additional radiality assumption, a similar result obtained $d=4$.