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

TL;DR

This paper establishes local well-posedness, near well-posedness, and conservation laws for the $L^2$-critical generalized Zakharov-Kuznetsov equation in three dimensions, leading to global existence for small initial data.

Contribution

It proves well-posedness results in various Sobolev spaces, conservation laws, and the convergence of solutions to initial data for the 3D $L^2$-critical generalized Zakharov-Kuznetsov equation.

Findings

Local well-posedness in $H^s$ for $s ext{ in } (3/4,1)$.

Almost well-posedness for $s ext{ in } [1,2)$ with unique solutions.

Conservation of $L^2$-mass and energy in specified Sobolev spaces.

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.