The Cauchy problem for the $L^2-$critical generalized Zakharov-Kuznetsov equation in dimension 3

Abstract

We prove local well-posedness for the $L^2$ critical generalized Zakharov-Kuznetsov equation in $H^s, \, s \in (3/4,1).$ We also prove that the equation is "almost well-posedness" for initial data $u_0 \in H^s, \, s \in [1,2),$ in the sense that the solution belongs to a certain intersection $C([0,T] : H^s(\mathbb{R}^3)) \cap X^s_T$ and is unique within that class, where we can ensure continuity of the data-to-solution map in an only slightly larger space. We also prove that solutions satisfy the expected conservation of $L^2-$mass for the whole $s \in (3/4,2)$ range, and energy for $s \in (1,2).$ By a limiting argument, this implies, in particular, global existence for small initial data in $H^1.$ Finally, we study the question of almost everywhere (a.e.) convergence of solutions of the initial value problem to initial data.