A note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations

TL;DR

This paper establishes local well-posedness results for the generalized 2D Zakharov-Kuznetsov equations in Sobolev spaces, using an iterative method, for various nonlinearities characterized by the parameter k.

Contribution

It provides new well-posedness thresholds in Sobolev spaces for the 2D generalized Zakharov-Kuznetsov equations, extending previous results to higher nonlinearities.

Findings

Proves local well-posedness for k=2 in H^s with s>1/4.

Establishes well-posedness for k=3 in H^s with s>5/12.

Shows for k≥4, well-posedness in H^s with s>1-2/k.

Abstract

In this note we study the generalized 2D Zakharov-Kuznetsov equations $\partial_tu+\Delta\partial_xu+u^k\partial_xu=0$ for $k\ge 2$. By an iterative method we prove the local well-posedness of these equations in the Sobolev spaces $H^s(\mathbb{R}^2)$ for $s>1/4$ if $k=2$, $s>5/12$ if $k=3$ and $s>1-2/k$ if $k\ge 4$.