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 $C([0,T]:H^s({\mathbb{R}}^3))\cap X_T^s$ 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.

26 pages