Maximal function estimates and local well-posedness for the generalized Zakharov--Kuznetsov equation

TL;DR

This paper establishes high-dimensional Strichartz and maximal function estimates for the Zakharov--Kuznetsov equation, enabling new results on local well-posedness and pointwise convergence in multiple dimensions.

Contribution

It extends Strichartz estimates to high dimensions, proves local well-posedness in subcritical cases for dimensions four and above, and demonstrates pointwise convergence of solutions.

Findings

High-dimensional Strichartz estimates for Zakharov--Kuznetsov

Local well-posedness in subcritical cases for d ≥ 4, k ≥ 4

Pointwise convergence of solutions in any dimension

Abstract

We prove a high-dimensional version of the Strichartz estimates for the unitary group associated to the free Zakharov--Kuznetsov equation. As a by--product, we deduce maximal estimates which allow us to prove local well-posedness for the generalized Zakharov--Kuznetsov equation in the whole subcritical case whenever $d \ge 4, k \ge 4,$ complementing the recent results of Kinoshita and Herr--Kinoshita. Finally, we use some of those maximal estimates in order to prove pointwise convergence results for the flow of the generalized Zakharov--Kuznetsov equation in any dimension, in the same spirit of a recent manuscript by Compaan, Luc\`a and Staffilani.