A Trilinear Strichartz Estimate for the Modified Zakharov-Kuznetsov Equation with Application to Well-Posedness
Published in arXiv, 2024
Abstract
This paper is focused on the modified Zakharov-Kusnetsov equation
\[\partial_t u(t,x,y) + \partial_x^3 u(t,x,y) + \partial_x u^3(t,x,y)=0.\]We prove that the associated Cauchy problem is locally (in time) well-posed in \(H^s(\mathbb{R} \times \mathbb{T})\) for \(s > 1\). The new ingredient in this work is a trilinear estimate in the context of Bourgain spaces, controlling the cubic nonlinearity of the modified Zakharov-Kuznetsov equation that naturally arises in the contraction argument.
Keywords: Nonlinear Strichartz estimates, modified Zakharov-Kuznetsov equation, harmonic analysis.