A Trilinear Strichartz estimate for the modified Zakharov-Kuznetsov equation with application in well-posedness

TL;DR

This paper establishes local well-posedness for the modified Zakharov-Kuznetsov equation in certain Sobolev spaces using a novel trilinear estimate in Bourgain spaces, advancing understanding of its nonlinear dynamics.

Contribution

Introduces a new trilinear estimate in Bourgain spaces to prove local well-posedness for the modified Zakharov-Kuznetsov equation in $H^s$ with $s > 1$.

Findings

Proves local well-posedness in $H^s$ for $s > 1$.

Develops a new trilinear estimate for the cubic nonlinearity.

Controls the nonlinear term via Bourgain space techniques.

Abstract

This paper is focused on the modified Zakharov-Kusnetsov equation. We prove the associated Cauchy problem is locally (in time) well-posed in $H^s(\R \times \T)$ for $s >1$. The new ingredient in this work is a trilinear estimate in the context of Bourgain spaces which controls the cubic non-linearity of the modified Zakharov-Kuznetsov equation in the contraction argument.