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 We also prove that the equation is “almost well-posedness” for initial data in the sense that the solution belongs to a certain intersection 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 mass for the whole range, and energy for By a limiting argument, this implies, in particular, global existence for small initial data in Finally, we study the question of almost everywhere (a.e.) convergence of solutions of the initial value problem to initial data.